Exact Group Sequential Methods for Estimating a Binomial Proportion

Abstract

We first review existing sequential methods for estimating a binomial proportion. Afterward, we propose a new family of group sequential sampling schemes for estimating a binomial proportion with prescribed margin of error and confidence level. In particular, we establish the uniform controllability of coverage probability and the asymptotic optimality for such a family of sampling schemes. Our theoretical results establish the possibility that the parameters of this family of sampling schemes can be determined so that the prescribed level of confidence is guaranteed with little waste of samples. Analytic bounds for the cumulative distribution functions and expectations of sample numbers are derived. Moreover, we discuss the inherent connection of various sampling schemes. Numerical issues are addressed for improving the accuracy and efficiency of computation. Computational experiments are conducted for comparing sampling schemes. Illustrative examples are given for applications in clinical trials.

1. Introduction

Estimating a binomial proportion is a problem of ubiquitous significance in many areas of engineering and sciences. For economical reasons and other concerns, it is important to use as fewer as possible samples to guarantee the required reliability of estimation. To achieve this goal, sequential sampling schemes can be very useful. In a sequential sampling scheme, the total number of observations is not fixed in advance. The sampling process is continued stage by stage until a prespecified stopping rule is satisfied. The stopping rule is evaluated with accumulated observations. In many applications, for administrative feasibility, the sampling experiment is performed in a group fashion. Similar to group sequential tests [1, Section 8], [2], an estimation method based on taking samples by groups and evaluating them sequentially is referred to as a group sequential estimation method. It should be noted that group sequential estimation methods are general enough to include fixed-sample-size and fully sequential procedures as special cases. Particularly, a fixed-sample-size method can be viewed as a group sequential procedure of only one stage. If the increment between the sample sizes of consecutive stages is equal to 1, then the group sequential method is actually a fully sequential method.

It is a common contention that statistical inference, as a unique science to quantify the uncertainties of inferential statements, should avoid errors in the quantification of uncertainties, while minimizing the sampling cost. That is, a statistical inferential method is expected to be exact and efficient. The conventional notion of exactness is that no approximation is involved, except the round-off error due to finite word length of computers. Existing sequential methods for estimating a binomial proportion are dominantly of asymptotic nature (see, e.g., [3–7] and the references therein). Undoubtedly, asymptotic techniques provide approximate solutions and important insights for the relevant problems. However, any asymptotic method inevitably introduces unknown error in the resultant approximate solution due to the necessary use of a finite number of samples. In the direction of nonasymptotic sequential estimation, the primary goal is to ensure that the true coverage probability is above the prespecified confidence level for any value of the associated parameter, while the required sample size is as low as possible. In this direction, Mendo and Hernando [8] developed an inverse binomial sampling scheme for estimating a binomial proportion with relative precision. Tanaka [9] developed a rigorous method for constructing fixed-width sequential confidence intervals for a binomial proportion. Although no approximation is involved, Tanaka’s method is very conservative due to the bounding techniques employed in the derivation of sequential confidence intervals. Franzén [10] studied the construction of fixed-width sequential confidence intervals for a binomial proportion. However, no effective method for defining stopping rules is proposed in [10]. In his later paper [11], Franzén proposed to construct fixed-width confidence intervals based on sequential probability ratio tests (SPRTs) invented by Wald [12]. His method can generate fixed-sample-size confidence intervals based on SPRTs. Unfortunately, he made a fundamental flaw by mistaking that if the width of the fixed-sample-size confidence interval decreases to be smaller than the prespecified length as the number of samples is increasing, then the fixed-sample-size confidence interval at the termination of sampling process is the desired fixed-width sequential confidence interval guaranteeing the prescribed confidence level. More recently, Frey published a paper [13] in The American Statistician (TAS) on the classical problem of sequentially estimating a binomial proportion with prescribed margin of error and confidence level. Before Frey submitted his original manuscript to TAS in July 2009, a general framework of multistage parameter estimation had been established by Chen [14–18], which provides exact methods for estimating parameters of common distributions with various error criterion. This framework is also proposed in [19]. The approach of Frey [13] is similar to that of Chen [14–18] for the specific problem of estimating a binomial proportion with prescribed margin of error and confidence level.

In this paper, our primary interests are in the exact sequential methods for the estimation of a binomial proportion with prescribed margin of error and confidence level. We first introduce the exact approach established in [14–18]. In particular, we introduce the inclusion principle proposed in [18] and its applications to the construction of concrete stopping rules. We investigate the connection among various stopping rules. Afterward, we propose a new family of stopping rules which are extremely simple and accommodate some existing stopping rules as special cases. We provide rigorous justification for the feasibility and asymptotic optimality of such stopping rules. We prove that the prescribed confidence level can be guaranteed uniformly for all values of a binomial proportion by choosing appropriate parametric values for the stopping rule. We show that as the margin of error tends to be zero, the sample size tends to the attainable minimum as if the binomial proportion were exactly known. We derive analytic bounds for distributions and expectations of sample numbers. In addition, we address some critical computational issues and propose methods to improve the accuracy and efficiency of numerical calculation. We conduct extensive numerical experiment to study the performance of various stopping rules. We determine parametric values for the proposed stopping rules to achieve unprecedentedly efficiency while guaranteeing prescribed confidence levels. We attempt to make our proposed method as user-friendly as possible so that it can be immediately applicable even for layer persons.

The remainder of the paper is organized as follows. In Section 2, we introduce the exact approach proposed in [14–18]. In Section 3, we discuss the general principle of constructing stopping rules. In Section 4, we propose a new family of sampling schemes and investigate their feasibility, optimality, and analytic bounds of the distribution and expectation of sample numbers. In Section 5, we compare various computational methods. In particular, we illustrate why the natural method of evaluating coverage probability based on gridding parameter space is neither rigorous nor efficient. In Section 6, we present numerical results for various sampling schemes. In Section 7, we illustrate the applications of our group sequential method in clinical trials. Section 8 is the conclusion. The proofs of theorems are given in appendices. Throughout this paper, we shall use the following notations. The empty set is denoted by ∅. The set of positive integers is denoted by ℕ. The ceiling function is denoted by ⌈⋅⌉. The notation Pr{𝐸∣𝜃} denotes the probability of the event 𝐸 associated with parameter 𝜃. The expectation of a random variable is denoted by 𝔼[⋅]. The standard normal distribution is denoted by Φ(⋅). For 𝛼∈(0,1), the notation 𝒵_𝛼 denotes the critical value such that Φ(𝒵_𝛼)=1−𝛼. For 𝑛∈ℕ, in the case that 𝑋₁,…,𝑋_𝑛 are i.i.d. samples of 𝑋, we denote the sample mean (∑^𝑛_𝑖=1𝑋_𝑖)/𝑛 by 𝑋_𝑛, which is also called the relative frequency when 𝑋 is a Bernoulli random variable. The other notations will be made clear as we proceed.

2. How Can It Be Exact?

In many areas of scientific investigation, the outcome of an experiment is of dichotomy nature and can be modeled as a Bernoulli random variable 𝑋, defined in probability space (Ω,Pr,ℱ), such that Pr{𝑋=1}=1−Pr{𝑋=0}=𝑝∈(0,1), (1) where 𝑝 is referred to as a binomial proportion. In general, there is no analytic method for evaluating the binomial proportion 𝑝. A frequently used approach is to estimate 𝑝 based on i.i.d. samples 𝑋₁,𝑋₂,… of 𝑋. To reduce the sampling cost, it is appropriate to estimate 𝑝 by a multistage sampling procedure. More formally, let 𝜀∈(0,1) and 1−𝛿, with 𝛿∈(0,1), be the prespecified margin of error and confidence level, respectively. The objective is to construct a sequential estimator 𝐩 for 𝑝 based on a multistage sampling scheme such that Pr𝐩−𝑝<𝜀∣𝑝≥1−𝛿, (2) for any 𝑝∈(0,1). Throughout this paper, the probability Pr{|𝐩−𝑝|<𝜀∣𝑝} is referred to as the coverage probability. Accordingly, the probability Pr{|𝐩−𝑝|≥𝜀∣𝑝} is referred to as the complementary coverage probability. Clearly, a complete construction of a multistage estimation scheme needs to determine the number of stages, the sample sizes for all stages, the stopping rule, and the estimator for 𝑝. Throughout this paper, we let 𝑠 denote the number of stages and let 𝑛_ℓ denote the number of samples at the ℓth stages. That is, the sampling process consists of 𝑠 stages with sample sizes 𝑛₁<𝑛₂<⋯<𝑛_𝑠. For ℓ=1,2,…,𝑠, define 𝐾_ℓ=∑^𝑛ℓ_𝑖=1𝑋_𝑖 and 𝐩_ℓ=𝐾_ℓ/𝑛_ℓ. The stopping rule is to be defined in terms of 𝐩_ℓ,ℓ=1,…,𝑠. Of course, the index of stage at the termination of the sampling process, denoted by 𝐥, is a random number. Accordingly, the number of samples at the termination of the experiment, denoted by 𝐧, is a random number which equals 𝑛_𝐥. Since for each ℓ, 𝐩_ℓ is a maximum-likelihood and minimum-variance unbiased estimator of 𝑝, the sequential estimator for 𝑝 is taken as 𝐩=𝐩_𝐥=∑^𝑛𝐥_𝑖=1𝑋_𝑖𝑛_𝐥=∑^𝐧_𝑖=1𝑋_𝑖𝐧. (3) In the above discussion, we have outlined the general characteristics of a multistage sampling scheme for estimating a binomial proportion. It remains to determine the number of stages, the sample sizes for all stages, and the stopping rule so that the resultant estimator 𝐩 satisfies (2) for any 𝑝∈(0,1).

Actually, the problem of sequential estimation of a binomial proportion has been treated by Chen [14–18] in a general framework of multistage parameter estimation. The techniques of [14–18] are sufficient to offer exact solutions for a wide range of sequential estimation problems, including the estimation of a binomial proportion as a special case. The central idea of the approach in [14–18] is the control of coverage probability by a single parameter 𝜁, referred to as the coverage tuning parameter, and the adaptive rigorous checking of coverage guarantee by virtue of bounds of coverage probabilities. It is recognized in [14–18] that, due to the discontinuity of the coverage probability on parameter space, the conventional method of evaluating the coverage probability for a finite number of parameter values is neither rigorous not computationally efficient for checking the coverage probability guarantee.

As mentioned in the introduction, Frey published an article [13] in TAS on the sequential estimation of a binomial proportion with prescribed margin of error and confidence level. For clarity of presentation, the comparison of the works of Chen and Frey is given in Section 5.4. In the remainder of this section, we shall only introduce the idea and techniques of [14–18], which had been precedentially developed by Chen before Frey submitted his original manuscript to TAS in July 2009. We will introduce the approach of [14–18] with a focus on the special problem of estimating a binomial proportion with prescribed margin of error and confidence level.

2.1. Four Components Suffice

The exact methods of [14–18] for multistage parameter estimation have four main components as follows.(i)Stopping rules parameterized by the coverage tuning parameter 𝜁>0 such that the associated coverage probabilities can be made arbitrarily close to 1 by choosing 𝜁>0 to be a sufficiently small number.(ii)Recursively computable lower and upper bounds for the complementary coverage probability for a given 𝜁 and an interval of parameter values.(iii)Adapted branch and bound algorithm.(iv)Bisection coverage tuning.

Without looking at the technical details, one can see that these four components are sufficient for constructing a sequential estimator so that the prescribed confidence level is guaranteed. The reason is as follows. As lower and upper bounds for the complementary coverage probability are available, the global optimization technique, branch and bound (B&B) algorithm [20], can be used to compute exactly the maximum of complementary coverage probability on the whole parameter space. Thus, it is possible to check rigorously whether the coverage probability associated with a given 𝜁 is no less than the prespecified confidence level. Since the coverage probability can be controlled by 𝜁, it is possible to determine 𝜁 as large as possible to guarantee the desired confidence level by a bisection search. This process is referred to as bisection coverage tuning in [14–18]. Since a critical subroutine needed for bisection coverage tuning is to check whether the coverage probability is no less than the prespecified confidence level, it is not necessary to compute exactly the maximum of the complementary coverage probability. Therefore, Chen revised the standard B&B algorithm to reduce the computational complexity and called the improved algorithm as the adapted B&B Algorithm. The idea is to adaptively partition the parameter space as many subintervals. If for all subintervals, the upper bounds of the complementary coverage probability are no greater than 𝛿, then declare that the coverage probability is guaranteed. If there exists a subinterval for which the lower bound of the complementary coverage probability is greater than 𝛿, then declare that the coverage probability is not guaranteed. Continue partitioning the parameter space if no decision can be made. The four components are illustrated in the sequel under the headings of stopping rules, interval bounding, adapted branch and bound, and bisection coverage tuning.

2.2. Stopping Rules

The first component for the exact sequential estimation of a binomial proportion is the stopping rule for constructing a sequential estimator such that the coverage probability can be controlled by the coverage tuning parameter 𝜁. For convenience of describing some concrete stopping rules, define ℳ()=𝜃𝑧,𝜃𝑧ln𝑧+(1−𝑧)ln1−𝜃(),(),()(),(),[],(),𝑆=1−𝑧for𝑧∈0,1𝜃∈0,1ln1−𝜃for𝑧=0,𝜃∈0,1ln𝜃for𝑧=1,𝜃∈0,1−∞for𝑧∈0,1𝜃∉0,1𝑘,𝑙,𝑛,𝑝^𝑙_𝑖∑_=𝑘𝑛𝑖𝑝^𝑖1−𝑝^𝑛−𝑖(),(),for𝑝∈0,10for𝑝∉0,1 (4) where 𝑘 and 𝑙 are integers such that 0≤𝑘≤𝑙≤𝑛. Assume that 0<𝜁𝛿<1. For the purpose of controlling the coverage probability Pr{|𝐩−p|<𝜀∣𝑝} by the coverage tuning parameter, Chen has proposed four stopping rules as follows.

Stopping Rule A. Continue sampling until ℳ((1/2)−|(1/2)−𝐩_ℓ|,(1/2)−|(1/2)−𝐩_ℓ|+𝜀)≤(ln(𝜁𝛿))/𝑛_ℓ for some ℓ∈{1,…,𝑠}.

Stopping Rule B. Continue sampling until (|𝐩_ℓ−(1/2)|−(2/3)𝜀)²≥(1/4)+(𝜀²𝑛_ℓ/2ln(𝜁𝛿)) for some ℓ∈{1,…,𝑠}.

Stopping Rule C. Continue sampling until 𝑆(𝐾_ℓ,𝑛_ℓ,𝑛_ℓ,𝐩_ℓ−𝜀)≤𝜁𝛿 and 𝑆(0,𝐾_ℓ,𝑛_ℓ,𝐩_ℓ+𝜀)≤𝜁𝛿 for some ℓ∈{1,…,𝑠}.

Stopping Rule D. Continue sampling until 𝑛_ℓ≥𝐩_ℓ(1−𝐩_ℓ)(2/𝜀²)ln(1/𝜁𝛿) for some ℓ∈{1,…,𝑠}.

Stopping Rule A was first proposed in [14, Theorem7] and restated in [15, Theorem16]. Stopping Rule B was first proposed in [16, Theorem1] and represented as the third stopping rule in [21, Section4.1.1]. Stopping Rule C originated from [17, Theorem1] and was restated as the first stopping rule in [21, Section4.1.1]. Stopping Rule D was described in the remarks following Theorem7 of [22]. All these stopping rules can be derived from the general principles proposed in [18, Section3] and [19, Section2.4].

Given that a stopping rule can be expressed in terms of 𝐩_ℓ and 𝑛_ℓ for ℓ=1,…,𝑠, it is possible to find a bivariate function 𝒟(⋅,⋅) on {(𝑧,𝑛)∶𝑧∈[0,1],𝑛∈ℕ}, taking values from {0,1}, such that the stopping rule can be stated as the following: continue sampling until 𝒟(𝐩_ℓ,𝑛_ℓ)=1 for some ℓ∈{1,…,𝑠}. It can be checked that such representation applies to Stopping Rules A, B, C, and D. For example, Stopping Rule B can be expressed in this way by virtue of function 𝒟(⋅,⋅) such that 𝒟()=1𝑧,𝑛1if𝑧−2−23𝜀²≥14+𝜀²𝑛(),2ln𝜁𝛿0otherwise. (5) The motivation of introducing function 𝒟(⋅,⋅) is to parameterize the stopping rule in terms of design parameters. Function 𝒟(⋅,⋅) determines the form of the stopping rule and, consequently, the sample sizes for all stages can be chosen as functions of design parameters. Specifically, let 𝑁_min𝑘=min𝑛∈ℕ∶𝒟𝑛,𝑁,𝑛=1forsomenonnegativeinteger𝑘notexceeding𝑛_max𝑘=min𝑛∈ℕ∶𝒟𝑛.,𝑛=1forallnonnegativeinteger𝑘notexceeding𝑛 (6)(7) To avoid unnecessary checking of the stopping criterion and thus reduce administrative cost, there should be a possibility that the sampling process is terminated at the first stage. Hence, the minimum sample size 𝑛₁ should be chosen to ensure that {𝐧=𝑛₁}≠∅. This implies that the sample size 𝑛₁ for the first stage can be taken as 𝑁_min. On the other hand, since the sampling process must be terminated at or before the 𝑠th stage, the maximum sample size 𝑛_𝑠 should be chosen to guarantee that {𝐧>𝑛_𝑠}=∅. This implies that the sample size 𝑛_𝑠 for the last stage can be taken as 𝑁_max. If the number of stages 𝑠 is given, then the sample sizes for stages in between 1 and 𝑠 can be chosen as 𝑠−2 integers between 𝑁_min and 𝑁_max. Particularly, if the group sizes are expected to be approximately equal, then the sample sizes can be taken as 𝑛_ℓ=𝑁_min+ℓ−1𝑁𝑠−1^max−𝑁_min,ℓ=1,…,𝑠. (8) Since the stopping rule is associated with the coverage tuning parameter 𝜁, it follows that the number of stages 𝑠 and the sample sizes 𝑛₁,𝑛₂,…,𝑛_𝑠 can be expressed as functions of 𝜁. In this sense, it can be said that the stopping rule is parameterized by the coverage tuning parameter 𝜁. The above method of parameterizing stopping rules has been used in [14–17] and proposed in [21, Section2.1, page 9].

2.3. Interval Bounding

The second component for the exact sequential estimation of a binomial proportion is the method of bounding the complementary coverage probability Pr{|𝐩−𝑝|≥𝜀∣𝑝} for 𝑝 in an interval [𝑎,𝑏] contained by interval (0,1). Applying Theorem8 of [15] to the special case of a Bernoulli distribution immediately yields {}{}{}{},Pr𝐩≤𝑎−𝜀∣𝑏+Pr𝐩≥𝑏+𝜀∣𝑎≤Pr𝐩−𝑝≥𝜀∣𝑝≤Pr𝐩≤𝑏−𝜀∣𝑎+Pr𝐩≥𝑎+𝜀∣𝑏 (9) for all 𝑝∈[𝑎,𝑏]⊆(0,1). The bounds of (9) can be shown as follows. Note that Pr{𝐩≤𝑎−𝜀∣𝑝}+Pr{𝐩≥𝑏+𝜀∣𝑝}≤Pr{|𝐩−𝑝|≥𝜀∣𝑝}=Pr{𝐩≤𝑝−𝜀∣𝑝}+Pr{𝐩≥𝑝+𝜀∣𝑝}≤Pr{𝐩≤𝑏−𝜀∣𝑝}+Pr{𝐩≥𝑎+𝜀∣𝑝} for 𝑝∈[𝑎,𝑏]⊆(0,1). As a consequence of the monotonicity of Pr{𝐩≥𝜗∣𝑝} and Pr{𝐩≤𝜗∣𝑝} with respect to 𝑝, where 𝜗 is a real number independent of 𝑝, the lower and upper bounds of Pr{|𝐩−𝑝|≥𝜀∣𝑝} for 𝑝∈[𝑎,𝑏]⊆(0,1) can be given as Pr{𝐩≤𝑎−𝜀∣𝑏}+Pr{𝐩≥𝑏+𝜀∣𝑎} and Pr{𝐩≤𝑏−𝜀∣𝑎}+Pr{𝐩≥𝑎+𝜀∣𝑏}, respectively.

In page 15, equation (1) of [15], Chen proposed to apply the recursive method of Schultz et al. [23, Section2] to compute the lower and upper bounds of Pr{|𝐩−𝑝|≥𝜀∣𝑝} given by (9). It should be pointed out that such lower and upper bounds of Pr{|𝐩−𝑝|≥𝜀∣𝑝} can also be computed by the recursive path-counting method of Franzén [10, page 49].

2.4. Adapted Branch and Bound

The third component for the exact sequential estimation of a binomial proportion is the adapted B&B algorithm, which was proposed in [15, Section2.8], for quick determination of whether the coverage probability is no less than 1−𝛿 for any value of the associated parameter. Such a task of checking the coverage probability is also referred to as checking the coverage probability guarantee. Given that lower and upper bounds of the complementary coverage probability on an interval of parameter values can be obtained by the interval bounding techniques, this task can be accomplished by applying the B&B algorithm [20] to compute exactly the maximum of the complementary coverage probability on the parameter space. However, in our applications, it suffices to determine whether the maximum of the complementary coverage probability Pr{|𝐩−𝑝|≥𝜀∣𝑝} with respect to 𝑝∈(0,1) is greater than the confidence parameter 𝛿. For fast checking whether the maximal complementary coverage probability exceeds 𝛿, Chen proposed to reduce the computational complexity by revising the standard B&B algorithm as the Adapted B&B Algorithm in [15, Section2.8]. To describe this algorithm, let ℐ_init denote the parameter space (0,1). For an interval ℐ⊆ℐ_init, let maxΨ(ℐ) denote the maximum of the complementary coverage probability Pr{|𝐩−𝑝|≥𝜀∣𝑝} with respect to 𝑝∈ℐ. Let Ψ_lb(ℐ) and Ψ_ub(ℐ) be, respectively, the lower and upper bounds of Ψ(ℐ), which can be obtained by the interval bounding techniques introduced in Section 2.3. Let 𝜂>0 be a prespecified tolerance, which is much smaller than 𝛿. The adapted B&B algorithm of [15] is represented with a slight modification as in Algorithm 1.

∇ Let 𝑘←0,  𝑙0←Ψlb(ℐinit) and 𝑢0←Ψub(ℐinit). ∇ Let 𝒮0←{ℐinit} if 𝑢0>𝛿. Otherwise, let 𝒮0 be empty. ∇ While 𝒮𝑘 is nonempty, 𝑙𝑘<𝛿 and 𝑢𝑘 is greater than max{𝑙𝑘+𝜂,𝛿}, do the following:    ⋄ Split each interval in 𝒮𝑘 as two new intervals of equal length.        Let 𝑆𝑘 denote the set of all new intervals obtained from this splitting procedure.    ⋄ Eliminate any interval ℐ from 𝑆𝑘 such that Ψub(ℐ)≤𝛿.    ⋄ Let 𝒮𝑘+1 be the set 𝑆𝑘 processed by the above elimination procedure.    ⋄ Let 𝑙𝑘+1←maxℐ∈𝒮𝑘+1Ψlb(ℐ) and 𝑢𝑘+1←maxℐ∈𝒮𝑘+1Ψub(ℐ). Let 𝑘←𝑘+1. ∇ If 𝒮𝑘 is empty and 𝑙𝑘<𝛿, then declare max Ψ (ℐinit)≤𝛿.  Otherwise, declare max Ψ(ℐinit)>𝛿.

Algorithm 1  

It should be noted that for a sampling scheme of symmetrical stopping boundary, the initial interval ℐ_init may be taken as (0,1/2) for the sake of efficiency. In Section 5.1, we will illustrate why the adapted B&B algorithm is superior than the direct evaluation based on gridding parameter space. As will be seen in Section 5.2, the objective of the adapted B&B algorithm can also be accomplished by the Adaptive Maximum Checking Algorithm due to Chen [21, Section3.3] and rediscovered by Frey [13, Appendix]. An explanation is given in Section 5.3 for the advantage of working with the complementary coverage probability.

2.5. Bisection Coverage Tuning

The fourth component for the exact sequential estimation of a binomial proportion is Bisection Coverage Tuning. Based on the adaptive rigorous checking of coverage probability, Chen proposed in [14, Section2.7] and [15, Section2.6] to apply a bisection search method to determine maximal 𝜁 such that the coverage probability is no less than 1−𝛿 for any value of the associated parameter. Moreover, Chen has developed asymptotic results in [15, page 21, Theorem18] for determining the initial interval of 𝜁 needed for the bisection search. Specifically, if the complementary coverage probability Pr{|𝐩−𝑝|≥𝜀∣𝑝} associated with 𝜁=𝜁₀ tends to 𝛿 as 𝜀→0, then the initial interval of 𝜁 can be taken as [𝜁₀2^𝑖,𝜁₀2^𝑖+1], where 𝑖 is the largest integer such that the complementary coverage probability associated with 𝜁=𝜁₀2^𝑖 is no greater than 𝛿 for all 𝑝∈(0,1). By virtue of a bisection search, it is possible to obtain 𝜁^∗∈[𝜁₀2^𝑖,𝜁₀2^𝑖+1] such that the complementary coverage probability associated with 𝜁=𝜁^∗ is guaranteed to be no greater than 𝛿 for all 𝑝∈(0,1).

3. Principle of Constructing Stopping Rules

In this section, we shall illustrate the inherent connection between various stopping rules. It will be demonstrated that a lot of stopping rules can be derived by virtue of the inclusion principle proposed by Chen [18, Section3].

3.1. Inclusion Principle

The problem of estimating a binomial proportion can be considered as a special case of parameter estimation for a random variable 𝑋 parameterized by 𝜃∈Θ, where the objective is to construct a sequential estimator 𝜽 for 𝜃 such that Pr{|𝜽−𝜃|<𝜀∣𝜃}≥1−𝛿 for any 𝜃∈Θ. Assume that the sampling process consists of 𝑠 stages with sample sizes 𝑛₁<𝑛₂<⋯<𝑛_𝑠. For ℓ=1,…,𝑠, define an estimator 𝜽_ℓ for 𝜃 in terms of samples 𝑋₁,…,𝑋_𝑛ℓ of 𝑋. Let [𝐿_ℓ,𝑈_ℓ],ℓ=1,2,…,𝑠 be a sequence of confidence intervals such that for any ℓ, [𝐿_ℓ,𝑈_ℓ] is defined in terms of 𝑋₁,…,𝑋_𝑛ℓ and that the coverage probability Pr{𝐿_ℓ≤𝜃≤𝑈_ℓ∣𝜃} can be made arbitrarily close to 1 by choosing 𝜁>0 to be a sufficiently small number. In Theorem2 of [18], Chen proposed the following general stopping rule: Continuesamplinguntil𝑈_ℓ−𝜀≤𝜽_ℓ≤𝐿_ℓ{}.+𝜀forsomeℓ∈1,…,𝑠 (10) At the termination of the sampling process, a sequential estimator for 𝜃 is taken as 𝜽=𝜽_𝐥, where 𝐥 is the index of stage at the termination of sampling process.

Clearly, the general stopping rule (10) can be restated as follows.

Continue sampling until the confidence interval [𝐿_ℓ,𝑈_ℓ] is included by interval [𝜽_ℓ−𝜀,𝜽_ℓ+𝜀] for some ℓ∈{1,…,𝑠}.

The sequence of confidence intervals are parameterized by 𝜁 for purpose of controlling the coverage probability Pr{|𝜽−𝜃|<𝜀∣𝜃}. Due to the inclusion relationship [𝐿_ℓ,𝑈_ℓ]⊆[𝜽_ℓ−𝜀,𝜽_ℓ+𝜀], such a general methodology of using a sequence of confidence intervals to construct a stopping rule for controlling the coverage probability is referred to as the inclusion principle. It is asserted by Theorem2 of [18] that Pr𝜽−𝜃<𝜀∣𝜃≥1−𝑠𝜁𝛿,∀𝜃∈Θ, (11) provided that Pr{𝐿_ℓ<𝜃<𝑈_ℓ∣𝜃}≥1−𝜁𝛿 for ℓ=1,…,𝑠 and 𝜃∈Θ. This demonstrates that if the number of stages 𝑠 is bounded respective to 𝜁, then the coverage probability Pr{|𝜽−𝜃|<𝜀∣𝜃} associated with the stopping rule derived from the inclusion principle can be controlled by 𝜁. Actually, before explicitly proposing the inclusion principle in [18], Chen had extensively applied the inclusion principle in [14–17] to construct stopping rules for estimating parameters of various distributions such as binomial, Poisson, geometric, hypergeometric, and normal distributions. A more general version of the inclusion principle is proposed in [19, Section2.4]. For simplicity of the stopping rule, Chen had made effort to eliminate the computation of confidence limits.

In the context of estimating a binomial proportion 𝑝, the inclusion principle immediately leads to the following general stopping rule: Continuesamplinguntil𝐩_ℓ−𝜀≤𝐿_ℓ≤𝑈_ℓ≤𝐩_ℓ{}.+𝜀forsomeℓ∈1,…,𝑠 (12) Consequently, the sequential estimator for 𝑝 is taken as 𝐩 according to (3). It should be pointed out that the stopping rule (12) had been rediscovered by Frey in Section 2, the 1st paragraph of [13]. The four stopping rules considered in his paper follow immediately from applying various confidence intervals to the general stopping rule (12).

In the sequel, we will illustrate how to apply (12) to the derivation of Stopping Rules A, B, C, and D introduced in Section 2.2 and other specific stopping rules.

3.2. Stopping Rule from Wald Intervals

By virtue of Wald’s method of interval estimation for a binomial proportion 𝑝, a sequence of confidence intervals [𝐿_ℓ,𝑈_ℓ],ℓ=1,…,𝑠 for 𝑝 can be constructed such that 𝐿_ℓ=𝐩_ℓ−𝒵_𝜁𝛿𝐩_ℓ1−𝐩_ℓ𝑛_ℓ,𝑈_ℓ=𝐩_ℓ+𝒵_𝜁𝛿𝐩_ℓ1−𝐩_ℓ𝑛_ℓ,ℓ=1,…,𝑠, (13) and that Pr{𝐿_ℓ≤𝑝≤𝑈_ℓ∣𝑝}≈1−2𝜁𝛿 for ℓ=1,…,𝑠 and 𝑝∈(0,1). Note that, for ℓ=1,…,𝑠, the event {𝐩_ℓ−𝜀≤𝐿_ℓ≤𝑈_ℓ≤𝐩_ℓ+𝜀} is the same as the event {(𝐩_ℓ−1/2)²≥(1/4)−𝑛_ℓ(𝜀/𝒵_𝜁𝛿)²}. So, applying this sequence of confidence intervals to (12) results in the stopping rule “continue sampling until (𝐩_ℓ−1/2)²≥(1/4)−𝑛_ℓ(𝜀/𝒵_𝜁𝛿)² for some ℓ∈{1,…,𝑠}”. Since for any 𝜁∈(0,1/𝛿), there exists a unique number 𝜁^′∈(0,1/𝛿) such that 𝒵_𝜁𝛿=2ln(1/𝜁^′𝛿), this stopping rule is equivalent to “Continue sampling until (𝐩_ℓ−1/2)²≥(1/4)+(𝜀²𝑛_ℓ/2ln(𝜁𝛿)) for some ℓ∈{1,…,𝑠}.” This stopping rule is actually the same as Stopping Rule D, since {(𝐩_ℓ−1/2)²≥(1/4)+(𝜀²𝑛_ℓ/2ln(𝜁𝛿))}={𝑛_ℓ≥𝐩_ℓ(1−𝐩_ℓ)(2/𝜀²)ln(1/𝜁𝛿)} for ℓ∈{1,…,𝑠}.

3.3. Stopping Rule from Revised Wald Intervals

Define 𝐩_ℓ=(𝑛_ℓ𝐩_ℓ+𝑎)/(𝑛_ℓ+2𝑎) for ℓ=1,…,𝑠, where 𝑎 is a positive number. Inspired by Wald’s method of interval estimation for 𝑝, a sequence of confidence intervals [𝐿_ℓ,𝑈_ℓ],ℓ=1,…,𝑠 can be constructed such that 𝐿_ℓ=𝐩_ℓ−𝒵_𝜁𝛿𝐩_ℓ1−𝐩_ℓ𝑛_ℓ,𝑈_ℓ=𝐩_ℓ+𝒵_𝜁𝛿𝐩_ℓ1−𝐩_ℓ𝑛_ℓ (14) and that Pr{𝐿_ℓ≤𝑝≤𝑈_ℓ∣𝑝}≈1−2𝜁𝛿 for ℓ=1,…,𝑠 and 𝑝∈(0,1). This sequence of confidence intervals was applied by Frey [13] to the general stopping rule (12). As a matter of fact, such idea of revising Wald interval [𝑋_𝑛−𝒵_𝜁𝛿(𝑋_𝑛(1−𝑋_𝑛))/𝑛,𝑋_𝑛+𝒵_𝜁𝛿(𝑋_𝑛(1−𝑋_𝑛))/𝑛] by replacing the relative frequency 𝑋_𝑛=(∑^𝑛_𝑖=1𝑋_𝑖)/𝑛 involved in the confidence limits with 𝑝_𝑎=(𝑛𝑋_𝑛+𝑎)/(𝑛+2𝑎) had been proposed by Chen [24, Section4].

As can be seen from Section2, page 243, of Frey [13], applying (12) with the sequence of revised Wald intervals yields the stopping rule “Continue sampling until (𝐩_ℓ−1/2)²≥(1/4)+(𝜀²𝑛_ℓ/2ln(𝜁𝛿)) for some ℓ∈{1,…,𝑠}.” Clearly, replacing 𝐩_ℓ in Stopping Rule D with 𝐩_ℓ=(𝑎+𝑛_ℓ𝐩_ℓ)/(𝑛_ℓ+2𝑎) also leads to this stopping rule.

3.4. Stopping Rule from Wilson’s Confidence Intervals

Making use of the interval estimation method of Wilson [25], one can obtain a sequence of confidence intervals [𝐿_ℓ,𝑈_ℓ],ℓ=1,…,𝑠 for 𝑝 such that𝐿_ℓ𝐩=max0,^ℓ+𝒵²_𝜁𝛿/2𝑛_ℓ1+𝒵²_𝜁𝛿/𝑛_ℓ−𝒵_𝜁𝛿𝐩_ℓ1−𝐩_ℓ/𝑛_ℓ+𝒵_𝜁𝛿/2𝑛_ℓ²1+𝒵²_𝜁𝛿/𝑛_ℓ,𝑈_ℓ𝐩=min1,^ℓ+𝒵²_𝜁𝛿/2𝑛_ℓ1+𝒵²_𝜁𝛿/𝑛_ℓ+𝒵_𝜁𝛿𝐩_ℓ1−𝐩_ℓ/𝑛_ℓ+𝒵_𝜁𝛿/2𝑛_ℓ²1+𝒵²_𝜁𝛿/𝑛_ℓ (15) and that Pr{𝐿_ℓ≤𝑝≤𝑈_ℓ∣𝑝}≈1−2𝜁𝛿 for ℓ=1,…,𝑠 and 𝑝∈(0,1). It should be pointed out that the sequence of Wilson’s confidence intervals has been applied by Frey [13, Section2, page 243] to the general stopping rule (12) for estimating a binomial proportion.

Since a stopping rule directly involves the sequence of Wilson’s confidence intervals is cumbersome, it is desirable to eliminate the computation of Wilson’s confidence intervals in the stopping rule. For this purpose, we need to use the following result.

Theorem 1. Assume that 0<𝜁𝛿<1 and 0<𝜀<1/2. Then, Wilson’s confidence intervals satisfy {𝐩_ℓ−𝜀≤𝐿_ℓ≤𝑈_ℓ≤𝐩_ℓ+𝜀}={(|𝐩_ℓ−1/2|−𝜀)²≥(1/4)−𝑛_ℓ(𝜀/𝒵_𝜁𝛿)²} for ℓ=1,…,𝑠.

See Appendix A for a proof. As a consequence of Theorem 1 and the fact that for any 𝜁∈(0,1/𝛿), there exists a unique number 𝜁^′∈(0,1/𝛿) such that 𝒵_𝜁𝛿=2ln(1/𝜁^′𝛿), applying the sequence of Wilson’s confidence intervals to (12) leads to the following stopping rule.

Continue sampling until 𝐩_ℓ−12−𝜀²≥14+𝜀²𝑛_ℓ(),2ln𝜁𝛿 (16) for  someℓ∈{1,…,𝑠}.

3.5. Stopping Rule from Clopper-Pearson Confidence Intervals

Applying the interval estimation method of Clopper-Pearson [26], a sequence of confidence intervals [𝐿_ℓ,𝑈_ℓ],ℓ=1,…,𝑠 for 𝑝 can be obtained such that Pr{𝐿_ℓ≤𝑝≤𝑈_ℓ∣𝑝}≥1−2𝜁𝛿 for ℓ=1,…,𝑠 and 𝑝∈(0,1), where the upper confidence limit 𝑈_ℓ satisfies the equation 𝑆(0,𝐾_ℓ,𝑛_ℓ,𝑈_ℓ)=𝜁𝛿 if 𝐾_ℓ<𝑛_ℓ; and the lower confidence limit 𝐿_ℓ satisfies the equation 𝑆(𝐾_ℓ,𝑛_ℓ,𝑛_ℓ,𝐿_ℓ)=𝜁𝛿 if 𝐾_ℓ>0. The well-known equation (10.8) in [27, page 173] implies that 𝑆(0,𝑘,𝑛,𝑝), with 0≤𝑘<𝑛, is decreasing with respect to 𝑝∈(0,1) and that 𝑆(𝑘,𝑛,𝑛,𝑝), with 0<𝑘≤𝑛, is increasing with respect to 𝑝∈(0,1). It follows that 𝐩_ℓ−𝜀≤𝐿_ℓ=0<𝐩_ℓ−𝜀≤𝐿_ℓ∪𝐩_ℓ=𝐩≤𝜀_ℓ>𝜀,𝑆𝐾_ℓ,𝑛_ℓ,𝑛_ℓ,𝐩_ℓ∪𝐩−𝜀≤𝜁𝛿_ℓ=𝐩≤𝜀_ℓ>𝜀,𝑆𝐾_ℓ,𝑛_ℓ,𝑛_ℓ,𝐩_ℓ∪𝐩−𝜀≤𝜁𝛿_ℓ≤𝜀,𝑆𝐾_ℓ,𝑛_ℓ,𝑛_ℓ,𝐩_ℓ=𝑆𝐾−𝜀≤𝜁𝛿_ℓ,𝑛_ℓ,𝑛_ℓ,𝐩_ℓ,𝐩−𝜀≤𝜁𝛿_ℓ+𝜀≥𝑈_ℓ=1>𝐩_ℓ+𝜀≥𝑈_ℓ∪𝐩_ℓ=𝐩≥1−𝜀_ℓ<1−𝜀,𝑆0,𝐾_ℓ,𝑛_ℓ,𝐩_ℓ∪𝐩+𝜀≤𝜁𝛿_ℓ=𝐩≥1−𝜀_ℓ<1−𝜀,𝑆0,𝐾_ℓ,𝑛_ℓ,𝐩_ℓ∪𝐩+𝜀≤𝜁𝛿_ℓ≥1−𝜀,𝑆0,𝐾_ℓ,𝑛_ℓ,𝐩_ℓ=𝑆+𝜀≤𝜁𝛿0,𝐾_ℓ,𝑛_ℓ,𝐩_ℓ+𝜀≤𝜁𝛿, (17) for ℓ=1,…,𝑠. Consequently, 𝐩_ℓ−𝜀≤𝐿_ℓ≤𝑈_ℓ≤𝐩_ℓ=𝑆𝐾+𝜀_ℓ,𝑛_ℓ,𝑛_ℓ,𝐩_ℓ−𝜀≤𝜁𝛿,𝑆0,𝐾_ℓ,𝑛_ℓ,𝐩_ℓ+𝜀≤𝜁𝛿, (18) for ℓ=1,…,𝑠. This demonstrates that applying the sequence of Clopper-Pearson confidence intervals to the general stopping rule (12) gives Stopping Rule C.

It should be pointed out that Stopping Rule C was rediscovered by Frey as the third stopping rule in Section2, page 243 of his paper [13].

3.6. Stopping Rule from Fishman’s Confidence Intervals

By the interval estimation method of Fishman [28], a sequence of confidence intervals [𝐿_ℓ,𝑈_ℓ],ℓ=1,…,𝑠 for 𝑝 can be obtained such that 𝐿_ℓ=0if𝐩_ℓ𝜃=0,_ℓ∈0,𝐩_ℓ∶ℳ𝐩_ℓ,𝜃_ℓ=ln(𝜁𝛿)𝑛_ℓif𝐩_ℓ𝑈>0,_ℓ=1if𝐩_ℓ𝜃=1,_ℓ∈𝐩_ℓ,1∶ℳ𝐩_ℓ,𝜃_ℓ=ln(𝜁𝛿)𝑛_ℓif𝐩_ℓ<1. (19) Under the assumption that 0<𝜁𝛿<1 and 0<𝜀<1/2, by similar techniques as the proof of Theorem7 of [22], it can be shown that {𝐩_ℓ−𝜀≤𝐿_ℓ≤𝑈_ℓ≤𝐩_ℓ+𝜀}={ℳ((1/2)−|(1/2)−𝐩_ℓ|,(1/2)−|(1/2)−𝐩_ℓ|+𝜀)≤(ln(𝜁𝛿))/𝑛_ℓ} for ℓ=1,…,𝑠. Therefore, applying the sequence of confidence intervals of Fishman to the general stopping rule (12) gives Stopping Rule A.

It should be noted that Fishman’s confidence intervals are actually derived from the Chernoff bounds of the tailed probabilities of the sample mean of Bernoulli random variable. Hence, Stopping Rule A is also referred to as the stopping rule from Chernoff bounds in this paper.

3.7. Stopping Rule from Confidence Intervals of Chen et al.  

Using the interval estimation method of Chen et al. [29], a sequence of confidence intervals [𝐿_ℓ,𝑈_ℓ],ℓ=1,…,𝑠 for 𝑝 can be obtained such that 𝐿_ℓ=max0,𝐩_ℓ+34×1−2𝐩_ℓ−1+9𝑛_ℓ/2ln(1/𝜁𝛿)𝐩_ℓ1−𝐩_ℓ1+9𝑛_ℓ(),𝑈/8ln1/𝜁𝛿_ℓ=min1,𝐩_ℓ+34×1−2𝐩_ℓ+1+9𝑛_ℓ/2ln(1/𝜁𝛿)𝐩_ℓ1−𝐩_ℓ1+9𝑛_ℓ/8ln(1/𝜁𝛿) (20) and that Pr{𝐿_ℓ≤𝑝≤𝑈_ℓ∣𝑝}≥1−2𝜁𝛿 for ℓ=1,…,𝑠 and 𝑝∈(0,1). Under the assumption that 0<𝜁𝛿<1 and 0<𝜀<1/2, by similar techniques as the proof of Theorem1 of [30], it can be shown that {𝐩_ℓ−𝜀≤𝐿_ℓ≤𝑈_ℓ≤𝐩_ℓ+𝜀}={(|𝐩_ℓ−1/2|−(2/3)𝜀)²≥(1/4)+(𝜀²𝑛_ℓ/2ln(𝜁𝛿))} for ℓ=1,…,𝑠. This implies that applying the sequence of confidence intervals of Chen et al. to the general stopping rule (12) leads to Stopping Rule B.

Actually, the confidence intervals of Chen et al. [29] are derived from Massart’s inequality [31] on the tailed probabilities of the sample mean of Bernoulli random variable. For this reason, Stopping Rule B is also referred to as the stopping rule from Massart’s inequality in [21, Section4.1.1].

4. Double-Parabolic Sequential Estimation

From Sections 2.2, 3.2, and 3.7, it can be seen that, by introducing a new parameter 𝜌∈[0,1] and letting 𝜌 take values 2/3 and 0, respectively, Stopping Rules B and D can be accommodated as special cases of the following general stopping rule.

Continue the sampling process until 𝐩_ℓ−12−𝜌𝜀²≥14+𝜀²𝑛_ℓ2ln(𝜁𝛿) (21) for some ℓ∈{1,2,…,𝑠}, where 𝜁∈(0,1/𝛿).

Moreover, as can be seen from (16), the stopping rule derived from applying Wilson’s confidence intervals to (12) can also be viewed as a special case of such general stopping rule with 𝜌=1.

From the stopping condition (21), it can be seen that the stopping boundary is associated with the double-parabolic function 𝑓(𝑥)=(2/𝜀²)ln(𝜁𝛿)[(1/4)−(|𝑥−1/2|−𝜌𝜀)²] such that 𝑥 and 𝑓(𝑥) correspond to the sample mean and sample size, respectively. For 𝜀=0.1, 𝛿=0.05, and 𝜁=1, stopping boundaries with various 𝜌 are shown by Figure 1.

Figure 1 

Double-parabolic sampling.

For fixed 𝜀 and 𝛿, the parameters 𝜌 and 𝜁 affect the shape of the stoping boundary in a way as follows. As 𝜌 increases, the span of stopping boundary is increasing in the axis of sample mean. By decreasing 𝜁, the stopping boundary can be dragged toward the direction of increasing sample size. Hence, the parameter 𝜌 is referred to as the dilation coefficient. The parameter 𝜁 is referred to as the coverage tuning parameter. Since the stopping boundary consists of two parabolas, this approach of estimating a binomial proportion is referred to as the double-parabolic sequential estimation method.

4.1. Parametrization of the Sampling Scheme

In this section, we shall parameterize the double-parabolic sequential sampling scheme by the method described in Section 2.2. From the stopping condition (21), the stopping rule can be restated as follows. Continue sampling until 𝒟(𝐩_ℓ,𝑛_ℓ)=1 for some ℓ∈{1,…,𝑠}, where the function 𝒟(⋅,⋅) is defined by 𝒟()=1𝑧,𝑛1if𝑧−2−𝜌𝜀²≥14+𝜀²𝑛(),2ln𝜁𝛿0otherwise. (22) Clearly, the function 𝒟(⋅,⋅) associated with the double-parabolic sequential sampling scheme depends on the design parameters 𝜌,𝜁,𝜀 and 𝛿. Applying the function 𝒟(⋅,⋅) defined by (22) to (6) yields 𝑁_min𝑘=min𝑛∈ℕ∶𝑛−12−𝜌𝜀²≥14+𝜀²𝑛().2ln𝜁𝛿forsomenonnegativeinteger𝑘notexceeding𝑛 (23) Since 𝜀 is usually small in practical applications, we restrict 𝜀 to satisfy 0<𝜌𝜀≤1/4. As a consequence of 0≤𝜌𝜀≤1/4 and the fact that |𝑧−1/2|≤1/2 for any 𝑧∈[0,1], it must be true that (|𝑧−1/2|−𝜌𝜀)²≤((1/2)−𝜌𝜀)² for any 𝑧∈[0,1]. It follows from (23) that ((1/2)−𝜌𝜀)²≥(1/4)+(𝜀²𝑁_min/2ln(𝜁𝛿)), which implies that the minimum sample size can be taken as 𝑁_min=12𝜌𝜀1−𝜌ln.𝜁𝛿 (24) On the other hand, applying the function 𝒟(⋅,⋅) defined by (22) to (7) gives 𝑁_max𝑘=min𝑛∈ℕ∶𝑛−12−𝜌𝜀²≥14+𝜀²𝑛().2ln𝜁𝛿forallnonnegativeinteger𝑘notexceeding𝑛 (25) Since (|𝑧−1/2|−𝜌𝜀)²≥0 for any 𝑧∈[0,1], it follows from (25) that (1/4)+(𝜀²𝑁_max/2ln(𝜁𝛿))≤0, which implies that maximum sample size can be taken as 𝑁_max=12𝜀²1ln.𝜁𝛿 (26) Therefore, the sample sizes 𝑛₁,…,𝑛_𝑠 can be chosen as functions of 𝜌,𝜁,𝜀, and 𝛿 which satisfy the following constraint: 𝑁_min≤𝑛₁<⋯<𝑛_𝑠−1<𝑁_max≤𝑛_𝑠. (27) In particular, if the number of stages 𝑠 is given and the group sizes are expected to be approximately equal, then the sample sizes, 𝑛₁,…,𝑛_𝑠, for all stages can be obtained by substituting 𝑁_min defined by (24) and 𝑁_max defined by (26) into (8). For example, if the values of design parameters are 𝜀=0.05,𝛿=0.05,𝜌=3/4,𝜁=2.6759 and 𝑠=7, then the sample sizes of this sampling scheme are calculated as 𝑛₁=59,𝑛₂=116,𝑛₃=173,𝑛₄𝑛=231,₅=288,𝑛₆=345,𝑛₇=403. (28) The stopping rule is completely determined by substituting the values of design parameters into (21).

4.2. Uniform Controllability of Coverage Probability

Clearly, for prespecified 𝜀,𝛿, and 𝜌, the coverage probability Pr{|𝐩−𝑝|<𝜀∣𝑝} depends on the parameter 𝜁, the number of stages 𝑠, and the sample sizes 𝑛₁,…,𝑛_𝑠. As illustrated in Section 4.1, the number of stages 𝑠 and the sample sizes 𝑛₁,…,𝑛_𝑠 can be defined as functions of 𝜁∈(0,1/𝛿). That is, the stopping rule can be parameterized by 𝜁. Accordingly, for any 𝑝∈(0,1), the coverage probability Pr{|𝐩−𝑝|<𝜀∣𝑝} becomes a function of 𝜁. The following theorem shows that it suffices to choose 𝜁∈(0,1/𝛿) small enough to guarantee the prespecified confidence level.

Theorem 2. Let 𝜀,𝛿∈(0,1) and 𝜌∈(0,1] be fixed. Assume that the number of stages 𝑠 and the sample sizes 𝑛₁,…,𝑛_𝑠 are functions of 𝜁∈(0,1/𝛿) such that the constraint (27) is satisfied. Then, Pr{|𝐩−𝑝|<𝜀∣𝑝} is no less than 1−𝛿 for any 𝑝∈(0,1) provided that 10<𝜁≤𝛿()expln𝛿/2+ln1−exp−2𝜀².4𝜀𝜌1−𝜌𝜀 (29)

See Appendix B for a proof. For Theorem 2 to be valid, the choice of sample sizes is very flexible. Particularly, the sample sizes can be arithmetic or geometric progressions or any others, as long as the constraint (27) is satisfied. It can be seen that for the coverage probability to be uniformly controllable, the dilation coefficient 𝜌 must be greater than 0. Theorem 2 asserts that there exists 𝜁>0 such that the coverage probability is no less than 1−𝛿, regardless of the associated binomial proportion 𝑝. For the purpose of reducing sampling cost, we want to have a value of 𝜁 as large as possible such that the prespecified confidence level is guaranteed for any 𝑝∈(0,1). This can be accomplished by the technical components introduced in Sections 2.1, 2.3, 2.4, and 2.5. Clearly, for every value of 𝜌, we can obtain a corresponding value of 𝜁 (as large as possible) to ensure the desired confidence level. However, the performance of resultant stopping rules are different. Therefore, we can try a number of values of 𝜌 and pick the best resultant stopping rule for practical use.

4.3. Asymptotic Optimality of Sampling Schemes

Now we shall provide an important reason why we propose the sampling scheme of that structure by showing its asymptotic optimality. Since the performance of a group sampling scheme will be close to its fully sequential counterpart, we investigate the optimality of the fully sequential sampling scheme. In this scenario, the sample sizes 𝑛₁,𝑛₂,…,𝑛_𝑠 are consecutive integers such that 12𝜌𝜀1−𝜌ln𝜁𝛿=𝑛₁<𝑛₂<⋯<𝑛_𝑠−1<𝑛_𝑠=12𝜀²1ln.𝜁𝛿 (30) The fully sequential sampling scheme can be viewed as a special case of a group sampling scheme of 𝑠=𝑛_𝑠−𝑛₁+1 stages and group size 1. Clearly, if 𝛿,𝜁 and 𝜌 are fixed, the sampling scheme is dependent only on 𝜀. Hence, for any 𝑝∈(0,1), if we allow 𝜀 to vary in (0,1), then the coverage probability Pr{|𝐩−𝑝|<𝜀∣𝑝} and the average sample number 𝔼[𝐧] are functions of 𝜀. We are interested in knowing the asymptotic behavior of these functions as 𝜀→0, since 𝜀 is usually small in practical situations. The following theorem provides us the desired insights.

Theorem 3. Assume that 𝛿∈(0,1),𝜁∈(0,1/𝛿) and 𝜌∈(0,1] are fixed. Define 𝑁(𝑝,𝜀,𝛿,𝜁)=(2𝑝(1−𝑝)ln(1/𝜁𝛿))/𝜀² for 𝑝∈(0,1) and 𝜀∈(0,1). Then, Prlim_𝜀→0𝐧𝑁𝑝,𝜀,𝛿,𝜁=1∣𝑝=1,lim_𝜀→0Pr𝐩−𝑝<𝜀∣𝑝=2Φ12ln𝜁𝛿−1,lim_𝜀→0𝔼[𝐧]𝑁𝑝,𝜀,𝛿,𝜁=1, (31)(32)(33) for any 𝑝∈(0,1).

See Appendix C for a proof. From (32), it can be seen that lim_𝜀→0Pr{|𝐩−𝑝|<𝜀∣𝑝}=1−𝛿 for any 𝑝∈(0,1) if 𝜁=(1/𝛿)exp(−(1/2)𝒵²_𝛿/2). Such value can be taken as an initial value for the coverage tuning parameter 𝜁. In addition to providing guidance on the coverage tuning techniques, Theorem 3 also establishes the optimality of the sampling scheme. To see this, let 𝒩(𝑝,𝜀,𝛿) denote the minimum sample size 𝑛 required for a fixed-sample-size procedure to guarantee that Pr{|𝑋_𝑛−𝑝|<𝜀∣𝑝}≥1−𝛿 for any 𝑝∈(0,1), where 𝑋_𝑛=(∑^𝑛_𝑖=1𝑋_𝑖)/𝑛. It is well known that from the central limit theorem, lim_𝜀→0𝒩𝑝,𝜀,𝛿𝑝1−𝑝𝒵_𝛿/2/𝜀²=1. (34) Applying (33), (34), and letting 𝜁=(1/𝛿)exp(−(1/2)𝒵²_𝛿/2), we have lim_𝜀→0(𝒩(𝑝,𝜀,𝛿)/𝑁(𝑝,𝜀,𝛿,𝜁))=1 for 𝑝∈(0,1) and 𝛿∈(0,1), which implies the asymptotic optimality of the double-parabolic sampling scheme. By virtue of (33), an approximate formula for computing the average sample number is given as follows: 𝔼[𝐧]=()≈𝑁𝑝,𝜀,𝛿,𝜁2𝑝1−𝑝ln1/𝜁𝛿𝜀², (35) for 𝑝∈(0,1) and 𝜀∈(0,1). From (34), one obtains 𝒩(𝑝,𝜀,𝛿)≈𝑝(1−𝑝)(𝒵_𝛿/2/𝜀)², which is a well-known result in statistics. In situations that no information of 𝑝 is available, one usually uses 𝑁=_normal^def14𝒵_𝛿/2𝜀² (36) as the sample size for estimating the binomial proportion 𝑝 with prescribed margin of error 𝜀 and confidence level 1−𝛿. Since the sample size formula (36) can lead to under-coverage, researchers in many areas are willing to use a more conservative but rigorous sample size formula 𝑁=_ch^defln(2/𝛿)2𝜀², (37) which is derived from the Chernoff-Hoeffding bound [32, 33]. Comparing (35) and (37), one can see that under the premise of guaranteeing the prescribed confidence level 1−𝛿, the double-parabolic sampling scheme can lead to a substantial reduction of sample number when the unknown binomial proportion 𝑝 is close to 0 or 1.

4.4. Bounds on Distribution and Expectation of Sample Number

We shall derive analytic bounds for the cumulative distribution function and expectation of the sample number 𝐧 associated with the double-parabolic sampling scheme. In this direction, we have obtained the following results.

Theorem 4. Let 𝑝∈(0,1/2]. Define 𝑎_ℓ=(1/2)−𝜌𝜀−(1/4)+(𝜀²𝑛_ℓ/2ln(𝜁𝛿)) for ℓ=1,…,𝑠. Let 𝜏 denote the index of stage such that 𝑎_𝜏−1≤𝑝<𝑎_𝜏. Then, Pr{𝐧>𝑛_ℓ∣𝑝}≤exp(𝑛_ℓℳ(𝑎_ℓ,𝑝)) for 𝜏≤ℓ<𝑠. Moreover, 𝔼[𝐧]≤𝑛_𝜏+∑^𝑠−1_ℓ=𝜏(𝑛_ℓ+1−𝑛_ℓ)exp(𝑛_ℓℳ(𝑎_ℓ,𝑝)).

See Appendix D for a proof. By the symmetry of the double-parabolic sampling scheme, similar analytic bounds for the distribution and expectation of the sample number can be derived for the case that 𝑝∈[1/2,1).

5. Comparison of Computational Methods

In this section, we shall compare various computational methods. First, we will illustrate why a frequently used method of evaluating the coverage probability based on gridding the parameter space is not rigorous and is less efficient as compared to the adapted B&B algorithm. Second, we will introduce the Adaptive Maximum Checking Algorithm of [21] which has better computational efficiency as compared to the adapted B&B algorithm. Third, we will explain that it is more advantageous in terms of numerical accuracy to work with the complementary coverage probability as compared to direct evaluation of the coverage probability. Finally, we will compare the computational methods of Chen [14–18] and Frey [13] for the design of sequential procedures for estimating a binomial proportion.

5.1. Verifying Coverage Guarantee without Gridding Parameter Space

For purpose of constructing a sampling scheme so that the prescribed confidence level 1−𝛿 is guaranteed, an essential task is to determine whether the coverage probability Pr{|𝐩−𝑝|<𝜀∣𝑝} associated with a given stopping rule is no less than 1−𝛿. In other words, it is necessary to compare the infimum of coverage probability with 1−𝛿. To accomplish such a task of checking coverage guarantee, a natural method is to evaluate the infimum of coverage probability as follows:(i)choose 𝑚 grid points 𝑝₁,…,𝑝_𝑚 from parameter space (0,1);(ii)compute 𝑐_𝑗=Pr{|𝐩−𝑝|<𝜀∣𝑝_𝑗} for 𝑗=1,…,𝑚;(iii)Take min{𝑐₁,…,𝑐_𝑚} as inf_𝑝∈(0,1)Pr{|𝐩−𝑝|<𝜀∣𝑝}.

This method can be easily mistaken as an exact approach and has been frequently used for evaluating coverage probabilities in many problem areas.

It is not hard to show that if the sample size 𝐧 of a sequential procedure has a support 𝒮, then the coverage probability Pr{|𝐩−𝑝|<𝜀∣𝑝} is discontinuous at 𝑝∈𝒫∩(0,1), where 𝒫={(𝑘/𝑛)±𝜀∶𝑘 is a nonnegative integer no greater than 𝑛∈𝒮}. The set 𝒫 typically has a large number of parameter values. Due to the discontinuity of the coverage probability as a function of 𝑝, the coverage probabilities can differ significantly for two parameter values which are extremely close. This implies that an intolerable error can be introduced by taking the minimum of coverage probabilities of a finite number of parameter values as the infimum of coverage probability on the whole parameter space. So, if one simply uses the minimum of the coverage probabilities of a finite number of parameter values as the infimum of coverage probability to check the coverage guarantee, the sequential estimator 𝐩 of the resultant stopping rule will fail to guarantee the prescribed confidence level.

In addition to the lack of rigorousness, another drawback of checking coverage guarantee based on the method of gridding parameter space is its low efficiency. A critical issue is on the choice of the number, 𝑚, of grid points. If the number 𝑚 is too small, the induced error can be substantial. On the other hand, choosing a large number for 𝑚 results in high computational complexity.

In contrast to the method based on gridding parameter space, the adapted B&B algorithm is a rigorous approach for checking coverage guarantee as a consequence of the mechanism for comparing the bounds of coverage probability with the prescribed confidence level. The algorithm is also efficient due to the mechanism of pruning branches.

5.2. Adaptive Maximum Checking Algorithm

As illustrated in Section 2, the techniques developed in [14–18] are sufficient to provide exact solutions for a wide range of sequential estimation problems. However, one of the four components, the adapted B&B algorithm, requires computing both the lower and upper bounds of the complementary coverage probability. To further reduce the computational complexity, it is desirable to have a checking algorithm which needs only one of the lower and upper bounds. For this purpose, Chen had developed the Adaptive Maximum Checking Algorithm (AMCA) in [21, Section3.3] and [19, Section2.7]. In the following introduction of the AMCA, we shall follow the description of [21]. The AMCA can be applied to a wide class of computational problems dependent on the following critical subroutine.

Determine whether a function 𝐶(𝜃) is smaller than a prescribed number 𝛿 for every value of 𝜃 contained in interval [𝜃,𝜃].

Particularly, for checking the coverage guarantee in the context of estimating a binomial proportion, the parameter 𝜃 is the binomial proportion 𝑝 and the function 𝐶(𝜃) is actually the complementary coverage probability. In many situations, it is impossible or very difficult to evaluate 𝐶(𝜃) for every value of 𝜃 in interval [𝜃,𝜃], since the interval may contain infinitely many or an extremely large number of values. Similar to the adapted B&B algorithm, the purpose of AMCA is to reduce the computational complexity associated with the problem of determining whether the maximum of 𝐶(𝜃) over [𝜃,𝜃] is less than 𝛿. The only assumption required for AMCA is that, for any interval [𝑎,𝑏]⊆[𝜃,𝜃], it is possible to compute an upper bound 𝐶(𝑎,𝑏) such that 𝐶(𝜃)≤𝐶(𝑎,𝑏) for any 𝜃∈[𝑎,𝑏] and that the upper bound converges to 𝐶(𝜃) as the interval width 𝑏−𝑎 tends to 0. The backward AMCA proceeds as in Algorithm 2.

∇ Choose initial step size 𝑑>𝜂. ∇ Let 𝐹←0, 𝑇←0 and 𝑏←𝜃. ∇ While 𝐹=𝑇=0, do the following:      ⋄ Let st←0 and ℓ←2;      ⋄ While st = 0, do the following:      ⋆ Let ℓ←ℓ−1 and 𝑑←𝑑2ℓ.      ⋆ If 𝑏−𝑑>𝜃, then let 𝑎←𝑏−𝑑 and 𝑇←0.       Otherwise, let 𝑎←𝜃 and 𝑇←1.      ⋆ If 𝐶(𝑎,𝑏)<𝛿, then let st ← 1 and 𝑏←𝑎.      ⋆ If 𝑑<𝜂, then let st ← 1 and F ← 1. ∇ Return 𝐹.

Algorithm 2  

The output of the backward AMCA is a binary variable 𝐹 such that “𝐹=0” means “𝐶(𝜃)<𝛿” and “𝐹=1” means “𝐶(𝜃)≥𝛿.” An intermediate variable 𝑇 is introduced in the description of AMCA such that “𝑇=1” means that the left endpoint of the interval is reached. The backward AMCA starts from the right endpoint of the interval (i.e., 𝑏=𝜃) and attempts to find an interval [𝑎,𝑏] such that 𝐶(𝑎,𝑏)<𝛿. If such an interval is available, then, attempt to go backward to find the next consecutive interval with twice width. If doubling the interval width fails to guarantee 𝐶(𝑎,𝑏)<𝛿, then try to repeatedly cut the interval width in half to ensure that 𝐶(𝑎,𝑏)<𝛿. If the interval width becomes smaller than a prescribed tolerance 𝜂, then AMCA declares that “𝐹=1.” For our relevant statistical problems, if 𝐶(𝜃)≥𝛿 for some 𝜃∈[𝜃,𝜃], it is sure that “𝐹=1” will be declared. On the other hand, it is possible that “𝐹=1” is declared even though 𝐶(𝜃)<𝛿 for any 𝜃∈[𝜃,𝜃]. However, such situation can be made extremely rare and immaterial if we choose 𝜂 to be a very small number. Moreover, this will only introduce negligible conservativeness in the evaluation of 𝐶(𝜃) if 𝜂 is chosen to be sufficiently small (e.g., 𝜂=10⁻¹⁵). Clearly, the backward AMCA can be easily modified as forward AMCA. Moreover, the AMCA can also be easily modified as Adaptive Minimum Checking Algorithm (forward and backward). For checking the maximum of complementary coverage probability Pr{|𝐩−𝑝|≥𝜀∣𝑝}, one can use the AMCA with 𝐶(𝑝)=Pr{|𝐩−𝑝|≥𝜀∣𝑝} over interval [0,1/2]. We would like to point out that, in contrast to the adapted B&B algorithm, it seems difficult to generalize the AMCA to problems involving multidimensional parameter spaces.

5.3. Working with Complementary Coverage Probability

We would like to point out that, instead of evaluating the coverage probability as in [13], it is better to evaluate the complementary coverage probability for purpose of reducing numerical error. The advantage of working on the complementary coverage probability can be explained as follows. Note that, in many cases, the coverage probability is very close to 1 and the complementary coverage probability is very close to 0. Since the absolute precision for computing a number close to 1 is much lower than the absolute precision for computing a number close to 0, the method of directly evaluating the coverage probability will lead to intolerable numerical error for problems involving small 𝛿. As an example, consider a situation that the complementary coverage probability is in the order of 10⁻⁵. Direct computation of the coverage probability can easily lead to an absolute error of the order of 10⁻⁵. However, the absolute error of computing the complementary coverage probability can be readily controlled at the order of 10⁻⁹.

5.4. Comparison of Approaches of Chen and Frey

As mentioned in the introduction, Frey published a paper [13] in The American Statistician (TAS) on the sequential estimation of a binomial proportion with prescribed margin of error and confidence level. The approaches of Chen and Frey are based on the same strategy as follows. First, construct a family of stopping rules parameterized by 𝛾 (and possibly other design parameters) so that the associated coverage probability Pr{|𝐩−𝑝|<𝜀∣𝑝} can be controlled by parameter 𝛾 in the sense that the coverage probability can be made arbitrarily close to 1 by increasing 𝛾. Second, apply a bisection search method to determine the parameter 𝛾 so that the coverage probability is no less than the prescribed confidence level 1−𝛿 for any 𝑝∈(0,1).

For the purpose of controlling the coverage probability, Frey [13] applied the inclusion principle previously proposed in [18, Section3] and used in [14–17]. As illustrated in Section 3, the central idea of inclusion principle is to use a sequence of confidence intervals to construct stopping rules so that the sampling process is continued until a confidence interval is included by an interval defined in terms of the estimator and margin of error. Due to the inclusion relationship, the associated coverage probability can be controlled by the confidence coefficients of the sequence of confidence intervals. The critical value 𝛾 used by Frey plays the same role for controlling coverage probabilities as that of the coverage tuning parameter 𝜁 used by Chen. Frey [13] stated stopping rules in terms of confidence limits. This way of expressing stopping rules is straightforward and insightful, since one can readily see the principle behind the construction. For convenience of practical use, Chen proposed to eliminate the necessity of computing confidence limits.

Similar to the AMCA proposed in [21, Section3.3], the algorithm of Frey [13, Appendix] for checking coverage guarantee adaptively scans the parameter space based on interval bounding. The adaptive method used by Frey for updating step size is essentially the same as that of the AMCA. Ignoring the number 0.01 in Frey’s expression “𝜖_𝑖=min{0.01,2(𝑝_𝑖−1−𝑝_𝑖−2)},” which has very little impact on the computational efficiency, Frey’s step size 𝜖_𝑖 can be identified as the adaptive step size 𝑑 in the AMCA. The operation associated with “𝜖_𝑖=min{0.01,2(𝑝_𝑖−1−𝑝_i−2)}” has a similar function as that of the command “Let st←0 and ℓ←2” in the outer loop of the AMCA. The operation associated with Frey’s expression “𝑝_𝑖−1+𝜖_𝑖/2^𝑗, 𝑗≥0” is equivalent to that of the command “Let ℓ←ℓ−1 and 𝑑←𝑑2^ℓ” in the inner loop of the AMCA. Frey proposed to declare a failure of coverage guarantee if “the distance from 𝑝_𝑖−1 to the candidate value for 𝑝_𝑖 falls below 10⁻¹⁴.” The number “10⁻¹⁴” actually plays the same role as “𝜂” in the AMCA, where “𝜂=10⁻¹⁵” is recommended by [21].

6. Numerical Results

In this section, we shall illustrate the proposed double-parabolic sampling scheme through examples. As demonstrated in Sections 2.2 and 4, the double-parabolic sampling scheme can be parameterized by the dilation coefficient 𝜌 and the coverage tuning parameter 𝜁. Hence, the performance of the resultant stopping rule can be optimized with respect to 𝜌∈(0,1] and 𝜁 by choosing various values of 𝜌 from interval (0,1] and determining the corresponding values of 𝜁 by the computational techniques introduced in Section 2 to guarantee the desired confidence interval.

6.1. Asymptotic Analysis May Be Inadequate

For fully sequential cases, we have evaluated the double-parabolic sampling scheme with 𝜀=0.1, 𝛿=0.05, 𝜌=0.1, and 𝜁=(1/𝛿)exp(−(1/2)𝒵²_𝛿/2)≈2.93. The stopping boundary is displayed in the left side of Figure 2. The function of coverage probability with respect to the binomial proportion is shown in the right side of Figure 2, which indicates that the coverage probabilities are generally substantially lower than the prescribed confidence level 1−𝛿=0.05. By considering 𝜀=0.1 as a small number and applying the asymptotic theory, the coverage probability associated with the sampling scheme is expected to be close to 0.95. This numerical example demonstrates that although the asymptotic method is insightful and involves virtually no computation, it may not be adequate.

Figure 2 

Double-parabolic sampling with 𝜀=0.1, 𝛿=0.05, 𝜌=1/10, and 𝜁=2.93.

In general, the main drawback of an asymptotic method is that there is no guarantee of coverage probability. Although an asymptotical method asserts that if the margin of error 𝜀 tends to 0, the coverage probability will tend to the prespecified confidence level 1−𝛿, it is difficult to determine how small the margin of error 𝜀 is sufficient for the asymptotic method to be applicable. Note that 𝜀→0 implies the average sample size tends to ∞. However, in reality, the sample sizes must be finite. Consequently, an asymptotic method inevitably introduces unknown statistical error. Since an asymptotic method does not necessarily guarantee the prescribed confidence level, it is not fair to compare its associated sample size with that of an exact method, which guarantees the prespecified confidence level.

This example also indicates that, due to the discrete nature of the problem, the coverage probability is a discontinuous and erratic function of 𝑝, which implies that Monte Carlo simulation is not suitable for evaluating the coverage performance.

6.2. Parametric Values of Fully Sequential Schemes

For fully sequential cases, to allow direct application of our double-parabolic sequential method, we have obtained values of coverage tuning parameter 𝜁, which guarantee the prescribed confidence levels, for double-parabolic sampling schemes with 𝜌=3/4 and various combinations of (𝜀,𝛿) as shown in Table 1. We used the computational techniques introduced in Section 2 to obtain this table.

Table 1 

Coverage tuning parameter.

To illustrate the use of Table 1, suppose that one wants a fully sequential sampling procedure to ensure that Pr{|𝐩−𝑝|<0.1∣𝑝}>0.95 for any 𝑝∈(0,1). This means that one can choose 𝜀=0.1, 𝛿=0.05 and the range of sample size is given by (30). From Table 1, it can be seen that the value of 𝜁 corresponding to 𝜀=0.1 and 𝛿=0.05 is 2.4174. Consequently, the stopping rule is completely determined by substituting the values of design parameters 𝜀=0.1, 𝛿=0.05, 𝜌=3/4, and 𝜁=2.4174 into its definition. The stopping boundary of this sampling scheme is displayed in the left side of Figure 3. The function of coverage probability with respect to the binomial proportion is shown in the right side of Figure 3.

Figure 3 

Double-parabolic sampling with 𝜀=0.1, 𝛿=0.05, 𝜌=3/4, and 𝜁=2.4174.

6.3. Parametric Values of Group Sequential Schemes

In many situations, especially in clinical trials, it is desirable to use group sequential sampling schemes. In Tables 2 and 3, assuming that sample sizes satisfy (8) for the purpose of having approximately equal group sizes, we have obtained parameters for concrete schemes by the computational techniques introduced in Section 2.

Table 2 

Coverage tuning parameter.

Table 3 

Coverage tuning parameter.

For dilation coefficient 𝜌=3/4 and confidence parameter 𝛿=0.05, we have obtained values of coverage tuning parameter 𝜁, which guarantee the prescribed confidence level 0.95, for double-parabolic sampling schemes, with the number of stages 𝑠 ranging from 3 to 10, as shown in Table 2.

For dilation coefficient 𝜌=3/4 and confidence parameter 𝛿=0.01, we have obtained values of coverage tuning parameter 𝜁, which guarantee the prescribed confidence level 0.99, for double-parabolic sampling schemes, with the number of stages 𝑠 ranging from 3 to 10, as shown in Table 3.

To illustrate the use of these tables, suppose that one wants a ten-stage sampling procedure of approximately equal group sizes to ensure that Pr{|𝐩−𝑝|<0.01∣𝑝}>0.99 for any 𝑝∈(0,1). This means that one can choose 𝜀=𝛿=0.01, 𝑠=10 and sample sizes satisfying (8). To obtain appropriate parameter values for the sampling procedure, one can look at Table 3 to find the coverage tuning parameter 𝜁 corresponding to 𝜀=0.01 and 𝑠=10. From Table 3, it can be seen that 𝜁 can be taken as 3.5753. Consequently, the stopping rule is completely determined by substituting the values of design parameters 𝜀=0.01, 𝛿=0.01, 𝜌=3/4, 𝜁=3.5753, and 𝑠=10 into its definition and (8). The stopping boundary of this sampling scheme and the function of coverage probability with respect to the binomial proportion are displayed, respectively, in the left and right sides of Figure 4.

Figure 4 

Double-parabolic sampling with 𝜀=𝛿=0.01, 𝑠=10, 𝜌=3/4, and 𝜁=3.5753.

6.4. Comparison of Sampling Schemes

We have conducted numerical experiments to investigate the impact of dilation coefficient 𝜌 on the performance of our double-parabolic sampling schemes. Our computational experiences indicate that the dilation coefficient 𝜌=3/4 is frequently a good choice in terms of average sample number and coverage probability. For example, consider the case that the margin of error is given as 𝜀=0.1 and the prescribed confidence level is 1−𝛿 with 𝛿=0.05. For the double-parabolic sampling scheme with the dilation coefficient 𝜌 chosen as 2/3,3/4, and 1, we have determined that, to ensure the prescribed confidence level 1−𝛿=0.95, it suffices to set the coverage tuning parameter 𝜁 as 2.1,2.4 and 2.4, respectively. The average sample numbers of these sampling schemes and the coverage probabilities as functions of the binomial proportion are shown, respectively, in the left and right sides of Figure 5. From Figure 5, it can be seen that a double-parabolic sampling scheme with dilation coefficient 𝜌=3/4 has better performance in terms of average sample number and coverage probability as compared to that of the double-parabolic sampling scheme with smaller or larger values of dilation coefficient.

Figure 5 

Double-parabolic sampling with various dilation coefficients.

We have investigated the impact of confidence intervals on the performance of fully sequential sampling schemes constructed from the inclusion principle. We have observed that the stopping rule derived from Clopper-Pearson intervals generally outperforms the stopping rules derived from other types of confidence intervals. However, via appropriate choice of the dilation coefficient, the double-parabolic sampling scheme can perform uniformly better than the stopping rule derived from Clopper-Pearson intervals. To illustrate, consider the case that 𝜀=0.1 and 𝛿=0.05. For stopping rules derived from Clopper-Pearson intervals, Fishman’s intervals, Wilson’s intervals, and revised Wald intervals with 𝑎=4, we have determined that to guarantee the prescribed confidence level 1−𝛿=0.95, it suffices to set the coverage tuning parameter 𝜁 as 0.5,1,2.4, and 0.37, respectively. For the stopping rule derived from Wald intervals, we have determined 𝜁=0.77 to ensure the confidence level, under the condition that the minimum sample size is taken as ⌈(1/𝜀)ln(1/𝜁𝛿)⌉. Recall that for the double-parabolic sampling scheme with 𝜌=3/4, we have obtained 𝜁=2.4 for purpose of guaranteeing the confidence level. The average sample numbers of these sampling schemes are shown in Figure 6. From these plots, it can be seen that as compared to the stopping rule derived from Clopper-Pearson intervals, the stopping rule derived from the revised Wald intervals performs better in the region of 𝑝 close to 0 or 1, but performs worse in the region of 𝑝 in the middle of (0,1). The performance of stopping rules from Fishman’s intervals (i.e., from Chernoff bound) and Wald intervals are obviously inferior as compared to that of the stopping rule derived from Clopper-Pearson intervals. It can be observed that the double-parabolic sampling scheme uniformly outperforms the stopping rule derived from Clopper-Pearson intervals.

Figure 6 

Comparison of average sample numbers.

6.5. Estimation with High Confidence Level

In some situations, we need to estimate a binomial proportion with a high confidence level. For example, one might want to construct a sampling scheme such that, for 𝜀=0.05 and 𝛿=10⁻¹⁰, the resultant sequential estimator 𝐩 satisfies Pr{|𝐩−𝑝|<𝜀∣𝑝}>1−𝛿 for any 𝑝∈(0,1). By working with the complementary coverage probability, we determined that it suffices to let the dilation coefficient 𝜌=3/4 and the coverage tuning parameter 𝜁=7.65. The stopping boundary and the function of coverage probability with respect to the binomial proportion are displayed, respectively, in the left and right sides of Figure 7. As addressed in Section 5.3, it should be noted that it is impossible to obtain such a sampling scheme without working with the complementary coverage probability.

Figure 7 

Double-parabolic sampling with 𝜀=0.05, 𝛿=10−10, 𝜌=3/4, and 𝜁=7.65.

7. Illustrative Examples for Clinical Trials

In this section, we shall illustrate the applications of our double-parabolic group sequential estimation method in clinical trials.

An example of our double-parabolic sampling scheme can be illustrated as follows. Assume that 𝜀=𝛿=0.05 is given and that the sampling procedure is expected to have 7 stages with sample sizes satisfying (8). Choosing 𝜌=3/4, we have determined that it suffices to take 𝜁=2.6759 to guarantee that the coverage probability is no less than 1−𝛿=0.95 for all 𝑝∈(0,1). Accordingly, the sample sizes of this sampling scheme are calculated as 59,116,173,231,288,345, and 403. This sampling scheme, with a sample path, is shown in the left side of Figure 8. In this case, the stopping rule can be equivalently described by virtue of Figure 8 as the following: continue sampling until (𝐩_ℓ,𝑛_ℓ) hit a green line at some stage. The coverage probability is shown in the right side of Figure 8.

Figure 8 

Double-parabolic sampling with 𝜀=𝛿=0.05, 𝑠=7,  𝜌=3/4, and 𝜁=2.6759.

To apply this estimation method in a clinical trial for estimating the proportion 𝑝 of a binomial response with margin of error 0.05 and confidence level 95%, we can have seven groups of patients with group sizes 59,57,57,58,57,57, and 58. In the first stage, we conduct experiment with the 59 patients of the first group. We observe the relative frequency of response and record it as 𝐩₁. Suppose that there are 12 patients having positive responses, then the relative frequency at the first stage is 𝐩₁=12/59=0.2034. With the values of (𝐩₁,𝑛₁)=(0.2034,59), we check if the stopping rule is satisfied. This is equivalent to see if the point (𝐩₁,𝑛₁) hits a green line at the first stage. For such value of (𝐩₁,𝑛₁), it can be seen that the stopping condition is not fulfilled. So, we need to conduct the second stage of experiment with the 57 patients of the second group. We observe the response of these 57 patients. Suppose we observe that 5 patients among this group have positive responses. Then, we add 5 with 12, the number of positive responses before the second stage, to obtain 17 positive responses among 𝑛₂=59+57=116 patients. So, at the second stage, we get the relative frequency 𝐩₂=17/116=0.1466. Since the stopping rule is not satisfied with the values of (𝐩₂,𝑛₂)=(0.1466,116), we need to conduct the third stage of experiment with the 57 patients of the third group. Suppose we observe that 14 patients among this group have positive responses. Then, we add 14 with 17, the number of positive responses before the third stage, to get 31 positive responses among 𝑛₃=59+57+57=173 patients. So, at the third stage, we get the relative frequency 𝐩₃=31/173=0.1792. Since the stopping rule is not satisfied with the values of (𝐩₃,𝑛₃)=(0.1792,173), we need to conduct the fourth stage of experiment with the 58 patients of the fourth group. Suppose we observe that 15 patients among this group have positive responses. Then, we add 15 with 31, the number of positive responses before the fourth stage, to get 46 positive responses among 𝑛₄=59+57+57+58=231 patients. So, at the fourth stage, we get the relative frequency 𝐩₄=46/231=0.1991. Since the stopping rule is not satisfied with the values of (𝐩₄,𝑛₄)=(0.1991,231), we need to conduct the fifth stage of experiment with the 57 patients of the fifth group. Suppose we observe that 6 patients among this group have positive responses. Then, we add 6 with 46, the number of positive responses before the fifth stage, to get 52 positive responses among 𝑛₅=59+57+57+58+57=288 patients. So, at the fifth stage, we get the relative frequency 𝐩₅=52/288=0.1806. It can be seen that the stopping rule is satisfied with the values of (𝐩₅,𝑛₅)=(0.1806,288). Therefore, we can terminate the sampling experiment and take 𝐩=52/288=0.1806 as an estimate of the proportion of the whole population having positive responses. With a 95% confidence level, one can believe that the difference between the true value of 𝑝 and its estimate 𝐩=0.1806 is less than 0.05.

In this experiment, we only use 288 samples to obtain the estimate for 𝑝. Except the round-off error, there is no other source of error for reporting statistical accuracy, since no asymptotic approximation is involved. As compared to fixed-sample-size procedure, we achieved a substantial save of samples. To see this, one can check that using the rigorous formula (37) gives a sample size 738, which is overly conservative. From the classical approximate formula (35), the sample size is determined as 385, which has been known to be insufficient to guarantee the prescribed confidence level 95%. The exact method of [34] shows that at least 391 samples are needed. As compared to the best-fixed-sample size obtained by the method of [34], the reduction of sample sizes resulted from our double-parabolic sampling scheme is 391−288=103. It can be seen that the fixed-sample-size procedure wastes 103/288=35.76% samples as compared to our group sequential method, which is also an exact method. This percentage may not be serious if it were a save of a number of simulation runs. However, as the number count is for patients, the reduction of samples is important for ethical and economical reasons. Using our group sequential method, the worst-case sample size is equal to 403, which is only 12 more than the minimum sample size of fixed-sample procedure. However, a lot of samples can be saved in the average case.

As 𝜀 or 𝛿 become smaller, the reduction of samples is more significant. For example, let 𝜀=0.02 and 𝛿=0.05, we have a double-parabolic sample scheme with 10 stages. The sampling scheme, with a sample path, is shown in the left side of Figure 9. The coverage probability is shown in the right side of Figure 9.

Figure 9 

Double-parabolic sampling with 𝜀=0.02, 𝛿=0.05, 𝑠=10, 𝜌=3/4, and 𝜁=2.6725.

8. Conclusion

In this paper, we have reviewed recent development of group sequential estimation methods for a binomial proportion. We have illustrated the inclusion principle and its applications to various stopping rules. We have introduced computational techniques in the literature, which suffice for determining parameters of stopping rules to guarantee desired confidence levels. Moreover, we have proposed a new family of sampling schemes with stopping boundary of double-parabolic shape, which are parameterized by the coverage tuning parameter and the dilation coefficient. These parameters can be determined by the exact computational techniques to reduce the sampling cost, while ensuring prescribed confidence levels. The new family of sampling schemes are extremely simple in structure and asymptotically optimal as the margin of error tends to 0. We have established analytic bounds for the distribution and expectation of the sample number at the termination of the sampling process. We have obtained parameter values via the exact computational techniques for the proposed sampling schemes such that the confidence levels are guaranteed and that the sampling schemes are generally more efficient as compared to existing ones.

Appendices

A. Proof of Theorem 1

Consider function 𝑔(𝑥,𝑧)=(𝑥−𝑧)²/𝑥(1−𝑥) for 𝑥∈(0,1) and 𝑧∈[0,1]. It can be checked that 𝜕𝑔(𝑥,𝑧)/𝜕𝑥=(𝑥−𝑧)[𝑧(1−𝑥)+𝑥(1−𝑧)][𝑥(1−𝑥)]⁻², which shows that for any fixed 𝑧∈[0,1], −𝑔(𝑥,𝑧) is a unimodal function of 𝑥∈(0,1), with a maximum attained at 𝑥=𝑧. By such a property of 𝑔(𝑥,𝑧) and the definition of Wilson’s confidence intervals, we have 𝐩_ℓ−𝜀≤𝐿_ℓ=0<𝐩_ℓ−𝜀≤𝐿_ℓ∪𝐩_ℓ=𝐩≤𝜀0<_ℓ−𝜀≤𝐿_ℓ≤𝐩_ℓ,𝑔𝐿_ℓ,𝐩_ℓ=𝒵²_𝜁𝛿𝑛_ℓ∪𝐩_ℓ=𝐩≤𝜀_ℓ𝜀>𝜀,²𝐩_ℓ−𝜀1−𝐩_ℓ≥𝒵−𝜀𝑛^2𝜁𝛿_ℓ∪𝐩_ℓ,𝐩≤𝜀_ℓ+𝜀≥𝑈_ℓ=1>𝐩_ℓ+𝜀≥𝑈_ℓ∪𝐩_ℓ=𝐩+𝜀≥11>_ℓ+𝜀≥𝑈_ℓ≥𝐩_ℓ,𝑔𝑈_ℓ,𝐩_ℓ=𝒵²_𝜁𝛿𝑛_ℓ∪𝐩_ℓ+𝜀≥1=𝐩_ℓ𝜀<1−𝜀,²𝐩_ℓ+𝜀1−𝐩_ℓ≥𝒵+𝜀𝑛^2𝜁𝛿_ℓ∪𝐩_ℓ≥1−𝜀, (A.1) for ℓ=1,…,𝑠, where we have used the fact that {𝐩_ℓ>𝜀}⊆{𝐿_ℓ>0},{𝐩_ℓ<1−𝜀}⊆{𝑈_ℓ<1} and 0≤𝐿_ℓ≤𝐩_ℓ≤𝑈_ℓ≤1. Recall that 0<𝜀<1/2. It follows that 𝐩_ℓ−𝜀≤𝐿_ℓ≤𝑈_ℓ≤𝐩_ℓ=𝐩+𝜀𝜀<_ℓ𝜀<1−𝜀,²𝐩_ℓ−𝜀1−𝐩_ℓ≥𝒵−𝜀𝑛^2𝜁𝛿_ℓ,𝜀²𝐩_ℓ+𝜀1−𝐩_ℓ≥𝒵+𝜀𝑛^2𝜁𝛿_ℓ∪𝐩_ℓ𝜀≤𝜀,²𝐩_ℓ+𝜀1−𝐩_ℓ≥𝒵+𝜀𝑛^2𝜁𝛿_ℓ∪𝐩_ℓ𝜀≥1−𝜀,²𝐩_ℓ−𝜀1−𝐩_ℓ≥𝒵−𝜀𝑛^2𝜁𝛿_ℓ=𝜀<𝐩_ℓ<1−𝜀,𝐩_ℓ−12−𝜀²≥14−𝑛_ℓ𝜀𝒵_𝜁𝛿²∪𝐩_ℓ≤𝜀,𝐩_ℓ−12−𝜀²≥14−𝑛_ℓ𝜀𝒵_𝜁𝛿²∪𝐩_ℓ≥1−𝜀,𝐩_ℓ−12−𝜀²≥14−𝑛_ℓ𝜀𝒵_𝜁𝛿²=𝐩_ℓ−12−𝜀²≥14−𝑛_ℓ𝜀𝒵_𝜁𝛿², (A.2) for ℓ=1,…,𝑠. This completes the proof of the theorem.

B. Proof of Theorem 2

By the assumption that 𝑛_𝑠≥(1/2𝜀²)ln(1/𝜁𝛿), we have (1/4)+(𝜀²𝑛_𝑠/2ln(𝜁𝛿))≤0 and, consequently, Pr{(|𝐩_𝑠−1/2|−𝜌𝜀)²≥(1/4)+(𝜀²𝑛_𝑠/2ln(𝜁𝛿))}=1. It follows from the definition of the sampling scheme that the sampling process must stop at or before the 𝑠th stage. In other words, Pr{𝐥≤𝑠}=1. This allows one to write Pr𝐩−𝑝≥𝜀∣𝑝=^𝑠_ℓ=1=Pr𝐩−𝑝≥𝜀,𝐥=ℓ∣𝑝_𝑠ℓ=1Pr𝐩_ℓ≤−𝑝≥𝜀,𝐥=ℓ∣𝑝_𝑠ℓ=1Pr𝐩_ℓ−𝑝≥𝜀∣𝑝, (B.1) for 𝑝∈(0,1). By virtue of the well-known Chernoff-Hoeffding bound [32, 33], we have Pr𝐩_ℓ−𝑝≥𝜀∣𝑝≤2exp−2𝑛_ℓ𝜀², (B.2) for ℓ=1,…,𝑠. Making use of (B.1), (B.2), and the fact that 𝑛₁≥2𝜌((1/𝜀)−𝜌)ln(1/𝜁𝛿) as can be seen from (30), we have Pr𝐩−𝑝≥𝜀∣𝑝≤2^𝑠_ℓ=1exp−2𝑛_ℓ𝜀²≤2^∞_𝑚=𝑛1exp−2𝑚𝜀²=2exp−2𝑛₁𝜀²1−exp−2𝜀²≤2exp−2𝜀²×2𝜌(1/𝜀)−𝜌ln(1/𝜁𝛿)1−exp−2𝜀²=2exp4𝜀𝜌1−𝜌𝜀ln(𝜁𝛿)1−exp−2𝜀², (B.3) for any 𝑝∈(0,1). Therefore, to guarantee that Pr{|𝐩−𝑝|<𝜀∣𝑝}≥1−𝛿 for any 𝑝∈(0,1), it is sufficient to choose 𝜁 such that 2exp(4𝜀𝜌(1−𝜌𝜀)ln(𝜁𝛿))≤𝛿[1−exp(−2𝜀²)]. This inequality can be written as 4𝜀𝜌(1−𝜌𝜀)ln(𝜁𝛿)≤ln(𝛿/2)+ln[1−exp(−2𝜀²)] or, equivalently, 𝜁≤(1/𝛿)exp((ln(𝛿/2)+ln[1−exp(−2𝜀²)])/4𝜀𝜌(1−𝜌𝜀)). The proof of the theorem is thus completed.

C. Proof of Theorem 3

First, we need to show that Pr{lim_𝜀→0(𝐧/𝑁(𝑝,𝜀,𝛿,𝜁))=1∣𝑝}=1 for any 𝑝∈(0,1). Clearly, the sample number 𝐧 is a random number dependent on 𝜀. Note that for any 𝜔∈Ω, the sequences {𝑋_𝐧(𝜔)(𝜔)}_𝜀∈(0,1) and {𝑋_{𝐧(𝜔)−1}(𝜔)}_𝜀∈(0,1) are subsets of {𝑋_𝑚(𝜔)}^∞_𝑚=1. By the strong law of large numbers, for almost every 𝜔∈Ω, the sequence {𝑋_𝑚(𝜔)}^∞_𝑚=1 converges to 𝑝. Since every subsequence of a convergent sequence must converge, it follows that the sequences {𝑋_𝐧(𝜔)(𝜔)}_𝜀∈(0,1) and {𝑋_{𝐧(𝜔)−1}(𝜔)}_𝜀∈(0,1) converge to 𝑝 as 𝜀→0 provided that 𝐧(𝜔)→∞ as 𝜀→0. Since it is certain that 𝐧≥2𝜌((1/𝜀)−𝜌)ln(1/𝜁𝛿)→∞ as 𝜀→0, we have that {lim_𝜀→0((𝐧−1)/𝐧)=1} is a sure event. It follows that 𝐵={lim_𝜀→0𝑋_𝐧−1=𝑝,lim_𝜀→0𝑋_𝐧=𝑝,lim_𝜀→0((𝐧−1)/𝐧)=1} is an almost sure event. By the definition of the sampling scheme, we have 𝐴=𝑋_𝐧−1−12−𝜌𝜀²<14+𝜀²(𝐧−1)(),2ln𝜁𝛿𝑋_𝐧−12−𝜌𝜀²≥14+𝜀²𝐧2ln(𝜁𝛿) (C.1) as a sure event. Hence, 𝐴∩𝐵 is an almost sure event. Define 𝐶={lim_𝜀→0(𝐧/𝑁(𝑝,𝜀,𝛿,𝜁))=1}. We need to show that 𝐶 is an almost sure event. For this purpose, we let 𝜔∈𝐴∩𝐵 and expect to show that 𝜔∈𝐶. As a consequence of 𝜔∈𝐴∩𝐵, 𝐧(𝜔)𝑁<𝐧(𝜔)𝑝,𝜀,𝛿,𝜁𝐧(𝜔)×)−−1(1/4𝑋_{𝐧(𝜔)−1}(𝜔)−(1/2)−𝜌𝜀²𝑝,1−𝑝lim_𝜀→0𝑋_{𝐧(𝜔)−1}(𝜔)=𝑝,lim_𝜀→0𝐧(𝜔)−1𝐧(𝜔)=1. (C.2) By the continuity of the function |𝑥−1/2|−𝜌𝜀 with respect to 𝑥 and 𝜀, we have limsup_𝜀→0𝐧(𝜔)𝑁𝑝,𝜀,𝛿,𝜁≤lim_𝜀→0𝐧(𝜔)𝐧(𝜔)×)−−1(1/4lim_𝜀→0𝑋_{𝐧(𝜔)−1}(𝜔)−(1/2)−lim_𝜀→0𝜌𝜀²𝑝1−𝑝=1. (C.3) On the other hand, as a consequence of 𝜔∈𝐴∩𝐵, 𝐧(𝜔)𝑁≥)−𝑝,𝜀,𝛿,𝜁(1/4𝑋_𝐧(𝜔)(𝜔)−(1/2)−𝜌𝜀²𝑝,1−𝑝lim_𝜀→0𝑋_𝐧(𝜔)(𝜔)=𝑝. (C.4) Making use of the continuity of the function |𝑥−1/2|−𝜌𝜀 with respect to 𝑥 and 𝜀, we have liminf_𝜀→0𝐧(𝜔)𝑁≥)−𝑝,𝜀,𝛿,𝜁(1/4lim_𝜀→0𝑋_𝐧(𝜔)(𝜔)−(1/2)−lim_𝜀→0𝜌𝜀²𝑝1−𝑝=1. (C.5) Combining (C.3) and (C.5) yields lim_𝜀→0(𝐧(𝜔)/𝑁(𝑝,𝜀,𝛿,𝜁))=1 and thus 𝐴∩𝐵⊆𝐶. This implies that 𝐶 is an almost sure event and thus Pr{lim_𝜀→0(𝐧/𝑁(𝑝,𝜀,𝛿,𝜁))=1∣𝑝}=1 for 𝑝∈(0,1).

Next, we need to show that lim_𝜀→0Pr{|𝐩−𝑝|<𝜀∣𝑝}=2Φ(2ln(1/𝜁𝛿))−1 for any 𝑝∈(0,1). For simplicity of notations, let 𝜎=𝑝(1−𝑝) and 𝑎=2ln(1/𝜁𝛿). Note that Pr{|𝐩−𝑝|<𝜀∣𝑝}=Pr{|𝑋_𝐧−𝑝|<𝜀∣𝑝}=Pr{√𝐧|𝑋_𝐧−𝑝|/𝜎<𝜀√𝐧/𝜎}. Clearly, for any 𝜂∈(0,𝑎), √Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎√≤Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎,𝜀√𝐧𝜎∈𝜀√𝑎−𝜂,𝑎+𝜂+Pr𝐧𝜎∉√𝑎−𝜂,𝑎+𝜂≤Pr𝐧𝑋_𝐧−𝑝𝜎𝜀√<𝑎+𝜂,𝐧𝜎∈𝜀√𝑎−𝜂,𝑎+𝜂+Pr𝐧𝜎∉√𝑎−𝜂,𝑎+𝜂≤Pr𝐧𝑋_𝐧−𝑝𝜎𝜀√<𝑎+𝜂+Pr𝐧𝜎∉,√𝑎−𝜂,𝑎+𝜂Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎√≥Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎,𝜀√𝐧𝜎∈√𝑎−𝜂,𝑎+𝜂≥Pr𝐧𝑋_𝐧−𝑝𝜎𝜀√<𝑎−𝜂,𝐧𝜎∈√𝑎−𝜂,𝑎+𝜂≥Pr𝐧𝑋_𝐧−𝑝𝜎𝜀√<𝑎−𝜂−Pr𝐧𝜎∉𝑎−𝜂,𝑎+𝜂. (C.6)(C.7) Recall that we have established that 𝐧/𝑁(𝑝,𝜀,𝛿,𝜁)→1 almost surely as 𝜀→0. This implies that (𝜀√𝐧/𝜎)→𝑎 and 𝐧/𝑁(𝑝,𝜀,𝛿,𝜁)→1 in probability as 𝜀 tends to zero. It follows from Anscombe’s random central limit theorem [35] that as 𝜀 tends to zero, √𝐧(𝑋_𝐧−𝑝)/𝜎 converges in distribution to a Gaussian random variable with zero mean and unit variance. Hence, from (C.6), limsup_𝜀→0√Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎≤lim_𝜀→0√Pr𝐧𝑋_𝐧−𝑝𝜎<𝑎+𝜂+lim_𝜀→0𝜀√Pr𝐧𝜎∉𝑎−𝜂,𝑎+𝜂=2Φ𝑎+𝜂−1 (C.8) and from (C.7), liminf_𝜀→0√Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎≥lim_𝜀→0√Pr𝐧𝑋_𝐧−𝑝𝜎<𝑎−𝜂−lim_𝜀→0𝜀√Pr𝐧𝜎∉𝑎−𝜂,𝑎+𝜂=2Φ𝑎−𝜂−1. (C.9) Since this argument holds for arbitrarily small 𝜂∈(0,𝑎), it must be true that liminf_𝜀→0√Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎=limsup_𝜀→0√Pr𝐧𝑋_𝐧−𝑝𝜎<𝜀√𝐧𝜎=2Φ(𝑎)−1. (C.10) So, lim_𝜀→0Pr{|𝐩−𝑝|<𝜀∣𝑝}=lim_𝜀→0Pr{√𝐧|𝑋_𝐧−𝑝|/𝜎<𝜀√𝐧/𝜎}=2Φ(𝑎)−1=2Φ(2ln(1/𝜁𝛿))−1 for any 𝑝∈(0,1).

Now, we focus our attention to show that lim_𝜀→0(𝔼[𝐧]/𝑁(𝑝,𝜀,𝛿,𝜁))=1 for any 𝑝∈(0,1). For this purpose, it suffices to show that 1−𝜂≤liminf_𝜀→0𝔼[𝐧]𝑁𝑝,𝜀,𝛿,𝜁≤limsup_𝜀→0𝔼[𝐧]𝑁(),𝑝,𝜀,𝛿,𝜁≤1+𝜂,∀𝑝∈0,1 (C.11) for any 𝜂∈(0,1). For simplicity of notations, we abbreviate 𝑁(𝑝,𝜀,𝛿,𝜁) as 𝑁 in the sequel. Since we have established Pr{lim_𝜀→0(𝐧/𝑁(𝑝,𝜀,𝛿,𝜁))=1}=1, we can conclude that lim_𝜀→0Pr1−𝜂𝑁≤𝐧≤1+𝜂𝑁=1. (C.12) Noting that 𝔼[𝐧]=^∞_𝑚=0{}≥𝑚Pr𝐧=𝑚_{𝑁1−𝜂𝑁≤𝑚≤1+𝜂}{}≥𝑚Pr𝐧=𝑚1−𝜂𝑁_{𝑁1−𝜂𝑁≤𝑚≤1+𝜂}Pr{𝐧=𝑚}, (C.13) we have 𝔼[𝐧]≥1−𝜂𝑁Pr1−𝜂𝑁≤𝐧≤1+𝜂𝑁. (C.14) Combining (C.12) and (C.14) yields liminf_𝜀→0𝔼[𝐧]𝑁≥𝑝,𝜀,𝛿,𝜁1−𝜂lim_𝜀→0Pr1−𝜂𝑁≤𝐧≤1+𝜂𝑁=1−𝜂. (C.15) On the other hand, using 𝔼[𝐧]=∑^∞_𝑚=0Pr{𝐧>𝑚}, we can write 𝔼[𝐧]=_{𝑁0≤𝑚<1+𝜂}Pr{𝐧>𝑚}+_{𝑁𝑚≥1+𝜂}{}≤𝑁+Pr𝐧>𝑚1+𝜂_{𝑁𝑚≥1+𝜂}Pr{𝐧>𝑚}. (C.16) Since limsup_𝜀→0(⌈(1+𝜂)𝑁⌉/𝑁(𝑝,𝜀,𝛿,𝜁))=1+𝜂, for the purpose of establishing limsup_𝜀→0(𝔼[𝐧]/𝑁(𝑝,𝜀,𝛿,𝜁))≤1+𝜂, it remains to show that limsup_𝜀→0∑_{𝑚≥(1+𝜂)𝑁}Pr{𝐧>𝑚}𝑁𝑝,𝜀,𝛿,𝜁=0. (C.17) Consider functions 𝑓(𝑥)=(1/4)−(|𝑥−1/2|−𝜌𝜀)² and 𝑔(𝑥)=𝑥(1−𝑥) for 𝑥∈[0,1]. Note that 𝑓(𝑥)−𝑔(𝑥)=(𝑥−1/2)²−|𝑥−1/2|−𝜌𝜀²=𝜌𝜀|2𝑥−1|−𝜌𝜀≤𝜌𝜀1+𝜌𝜀, (C.18) for all 𝑥∈[0,1]. For 𝑝∈(0,1), there exists a positive number 𝛾<min{𝑝,1−𝑝} such that |𝑔(𝑥)−𝑔(𝑝)|<(𝜂/2)𝑝(1−𝑝) for any 𝑥∈(𝑝−𝛾,𝑝+𝛾), since 𝑔(𝑥) is a continuous function of 𝑥. From now on, let 𝜀>0 be sufficiently small such that 𝜌𝜀(1+𝜌𝜀)<(𝜂/2)𝑝(1−𝑝). Then, 𝑓(𝑥)(𝑥)𝑝+𝜂≤𝑔+𝜌𝜀1+𝜌𝜀<𝑔2𝑝<𝑝,1−𝑝+𝜌𝜀1+𝜌𝜀1+𝜂1−𝑝 (C.19) for all 𝑥∈(𝑝−𝛾,𝑝+𝛾). This implies that 𝑋_𝑚∈⊆𝑝≥1𝑝−𝛾,𝑝+𝛾1+𝜂1−𝑝4−𝑋_𝑚−12−𝜌𝜀² (C.20) for all 𝑚>0. Taking complementary events on both sides of (C.20) leads to 𝑝<11+𝜂1−𝑝4−𝑋_𝑚−12−𝜌𝜀²⊆𝑋_𝑚∉𝑝−𝛾,𝑝+𝛾, (C.21) for all 𝑚>0. Since (1+𝜂)𝑝(1−𝑝)=((1+𝜂)𝑁𝜀²/2ln(1/𝜁𝛿))≤(𝑚𝜀²/2ln(1/𝜁𝛿)) for all 𝑚≥(1+𝜂)𝑁, it follows that 𝑚𝜀²()<12ln1/𝜁𝛿4−𝑋_𝑚−12−𝜌𝜀²⊆𝑋_𝑚∉𝑝−𝛾,𝑝+𝛾, (C.22) for all 𝑚≥(1+𝜂)𝑁. Therefore, we have shown that if 𝜀 is sufficiently small, then there exists a number 𝛾>0 such that {𝐧>𝑚}⊆𝑋_𝑚−12−𝜌𝜀²<14+𝑚𝜀²()⊆2ln𝜁𝛿𝑋_𝑚∉𝑝−𝛾,𝑝+𝛾, (C.23) for all 𝑚≥(1+𝜂)𝑁. Using this inclusion relationship and the Chernoff-Hoeffding bound [32, 33], we have Pr{𝐧>𝑚}≤Pr𝑋_𝑚∉𝑝−𝛾,𝑝+𝛾≤2exp−2𝑚𝛾², (C.24) for all 𝑚≥(1+𝜂)𝑁 provided that 𝜀>0 is sufficiently small. Letting 𝑘=⌈(1+𝜂)𝑁⌉ and using (C.24), we have Pr{_{𝑚≥(1+𝜂)𝑁}𝐧>𝑚}=_𝑚≥𝑘Pr{𝐧>𝑚}≤_𝑚≥𝑘2exp−2𝑚𝛾²=2exp−2𝑘𝛾²1−exp−2𝛾², (C.25) provided that 𝜀 is sufficiently small. Consequently, limsup_𝜀→0∑_{𝑚≥(1+𝜂)𝑁}Pr{𝐧>𝑚}𝑁𝑝,𝜀,𝛿,𝜁≤limsup_𝜀→02𝑁exp−2𝑘𝛾²1−exp−2𝛾²=0, (C.26) since 𝑘→∞ and 𝑁→∞ as 𝜀→0. So, we have established (C.11). Since the argument holds for arbitrarily small 𝜂>0, it must be true that lim_𝜀→0(𝔼[𝐧]/𝑁(𝑝,𝜀,𝛿,𝜁))=1 for any 𝑝∈(0,1). This completes the proof of the theorem.

D. Proof of Theorem 4

Recall that 𝐥 denotes the index of stage at the termination of the sampling process. Observing that 𝑛_𝑠−𝑛₁Pr{𝐥=1}=𝑛_𝑠Pr{𝐥≤𝑠}−𝑛₁{}=Pr𝐥≤1_𝑠ℓ=2𝑛_ℓPr{𝐥≤ℓ}−𝑛_ℓ−1{}=Pr𝐥<ℓ_𝑠ℓ=2𝑛_ℓ({}{+Pr𝐥≤ℓ−Pr𝐥<ℓ})_𝑠ℓ=2𝑛_ℓ−𝑛_ℓ−1{}=Pr𝐥<ℓ_𝑠ℓ=2𝑛_ℓ{}+Pr𝐥=ℓ_{𝑠−1ℓ=1}𝑛_ℓ+1−𝑛_ℓPr{𝐥≤ℓ}, (D.1) we have 𝑛_𝑠−∑^𝑠_ℓ=1𝑛_ℓPr{𝐥=ℓ}=∑^𝑠−1_ℓ=1(𝑛_ℓ+1−𝑛_ℓ)Pr{𝐥≤ℓ}. Making use of this result and the fact 𝑛_𝑠=𝑛₁+∑^𝑠−1_ℓ=1(𝑛_ℓ+1−𝑛_ℓ), we have 𝔼[𝐧]=^𝑠_ℓ=1𝑛_ℓPr{𝐥=ℓ}=𝑛_𝑠−𝑛_𝑠−^𝑠_ℓ=1𝑛_ℓPr{𝐥=ℓ}=𝑛₁+^𝑠−1_ℓ=1𝑛_ℓ+1−𝑛_ℓ−^𝑠−1_ℓ=1𝑛_ℓ+1−𝑛_ℓPr{𝐥≤ℓ}=𝑛₁+^𝜏−1_ℓ=1𝑛_ℓ+1−𝑛_ℓ{}+Pr𝐥>ℓ_{𝑠−1ℓ=𝜏}𝑛_ℓ+1−𝑛_ℓPr{𝐥>ℓ}. (D.2) By the definition of the stopping rule, we have {𝐥>ℓ}⊆𝐩_ℓ−12−𝜌𝜀²<14+𝜀²𝑛_ℓ()=2ln𝜁𝛿𝜌𝜀−14+𝜀²𝑛_ℓ()<𝐩2ln𝜁𝛿^ℓ−12<𝜌𝜀+14+𝜀²𝑛_ℓ()=2ln𝜁𝛿𝜌𝜀−14+𝜀²𝑛_ℓ()<12ln𝜁𝛿2−𝐩_ℓ<𝜌𝜀+14+𝜀²𝑛_ℓ(),𝐩2ln𝜁𝛿^ℓ≤12∪𝜌𝜀−14+𝜀²𝑛_ℓ()<𝐩2ln𝜁𝛿^ℓ−12<𝜌𝜀+14+𝜀²𝑛_ℓ(),𝐩2ln𝜁𝛿^ℓ>12⊆𝑎_ℓ<𝐩_ℓ<𝑏_ℓ∪1−𝑏_ℓ<𝐩_ℓ<1−𝑎_ℓ, (D.3) for 1≤ℓ<𝑠, where 𝑏_ℓ=(1/2)−𝜌𝜀+(1/4)+(𝜀²𝑛_ℓ/2ln(𝜁𝛿)) for ℓ=1,…,𝑠−1. By the assumption that 𝜀 and 𝜌 are nonnegative, we have 1−𝑏_ℓ−𝑎_ℓ=2𝜌𝜀≥0 for ℓ=1,…,𝑠−1. It follows from (D.3) that {𝐥>ℓ}⊆{𝐩_ℓ>𝑎_ℓ} for ℓ=1,…,𝑠−1. By the definition of 𝜏, we have 𝑝<𝑎_ℓ for 𝜏≤ℓ<𝑠. Making use of this fact, the inclusion relationship {𝐥>ℓ}⊆{𝐩_ℓ>𝑎_ℓ},  ℓ=1,…,𝑠−1, and Chernoff-Hoeffding bound [32, 33], we have Pr𝐧>𝑛_ℓ𝐩∣𝑝=Pr𝐥>ℓ∣𝑝≤Pr_ℓ>𝑎_ℓ∣𝑝≤exp𝑛_ℓℳ𝑎_ℓ,𝑝 (D.4) for 𝜏≤ℓ<𝑠. It follows from (D.2) and (D.4) that 𝔼[𝐧]≤𝑛₁+^𝜏−1_ℓ=1𝑛_ℓ+1−𝑛_ℓ+^𝑠−1_ℓ=𝜏𝑛_ℓ+1−𝑛_ℓPr{𝐥>ℓ}=𝑛_𝜏+^𝑠−1_ℓ=𝜏𝑛_ℓ+1−𝑛_ℓPr{𝐥>ℓ}≤𝑛_𝜏+^𝑠−1_ℓ=𝜏𝑛_ℓ+1−𝑛_ℓexp𝑛_ℓℳ𝑎_ℓ,𝑝. (D.5) This completes the proof of the theorem.

Acknowledgment

This paper is supported in part by NIH/NCI Grants no. 1 P01 CA116676, P30 CA138292-01 and 5 P50 CA128613.