Lower Confidence Bounds for the Probabilities of Correct Selection

Abstract

We extend the results of Gupta and Liang (1998), derived for location parameters, to obtain lower confidence bounds for the probability of correctly selecting the 𝑡 best populations (PCS_𝑡) simultaneously for all 𝑡=1,…,𝑘−1 for the general scale parameter models, where 𝑘 is the number of populations involved in the selection problem. The application of the results to the exponential and normal probability models is discussed. The implementation of the simultaneous lower confidence bounds for PCS_𝑡 is illustrated through real-life datasets.

1. Introduction

The population Π_𝑖 is characterized by an unknown scale parameter 𝜃_𝑖(>0), 𝑖=1,…,𝑘. Let 𝑇_𝑖 be an appropriate statistic for 𝜃_𝑖, based on a random sample of size 𝑛 from population Π_𝑖, having the probability density function (pdf) 𝑓_𝜃𝑖(𝑥)=(1/𝜃_𝑖)𝑓(𝑥/𝜃_𝑖) with the corresponding cumulative distribution function (cdf) 𝐹_𝜃𝑖(𝑥)=𝐹(𝑥/𝜃_𝑖), 𝑥>0, 𝜃_𝑖>0, 𝑖=1,…,𝑘.𝐹(⋅) is an arbitrary continuous cdf with pdf 𝑓(⋅). Let the ordered values of 𝑇_𝑖’s and 𝜃_𝑖’s be denoted by 𝑇_[1],…,𝑇_[𝑘] and 𝜃_[1],…,𝜃_[𝑘], respectively. Let 𝑇_(𝑖) be the statistic having a scale parameter 𝜃_[𝑖]. Let Π_(𝑖) denote the population associated with 𝜃_[𝑖], the 𝑖th smallest of 𝜃_𝑖’s. Any other population or sample quantity associated with Π_(𝑖) will be denoted by the subscript (𝑖) attached to it. Throughout, we assume that there is no prior knowledge about which of Π₁,…,Π_𝑘 is Π_(𝑖), 𝑖=1,…,𝑘 and that 𝜃₁,…,𝜃_𝑘 are unknown. Call the populations Π_(𝑘),Π_(𝑘−1),…,Π_{(𝑘−𝑡+1)} as the 𝑡 best populations.

In practice, the interest is to select the populations Π_(𝑘),Π_(𝑘−1),…,Π_{(𝑘−𝑡+1)}, that is, the populations associated with the largest unknown parameters 𝜃_[𝑘],𝜃_[𝑘−1],…,𝜃_{[𝑘−𝑡+1]}. For this, the natural selection rule “select the populations corresponding to 𝑡 largest 𝑇_𝑖’s, that is, 𝑇_[𝑘],𝑇_[𝑘−1],…,𝑇_{[𝑘−𝑡+1]} as the 𝑡 best populations” is used. However, it is possible that selected populations according to the natural selection rule may not be the best. Therefore, a question which naturally arises is: what kind of confidence statement can be made about these selection results? Motivated by this, we make an effort to answer this question.

Let CS_𝑡 (a correct selection of the 𝑡 best populations) denote the event that 𝑡 best populations are actually selected. Then, the probability of correct selection of the 𝑡 best populations (PCS_𝑡) is: PCS_𝑡(𝜃)=𝑃max_{1≤𝑖≤𝑘−𝑡}𝑇_(𝑖)<min_{𝑘−𝑡+1≤𝑗≤𝑘}𝑇_(𝑗)=_𝑘−𝑡_𝑖=1𝐹𝑦𝜃_[𝑖]𝑑1−^𝑘_{−𝑗=𝑘−𝑡+1}𝐹𝑦=𝜃_[𝑗](1.1a)_𝑘𝑗=_{𝑘−𝑡+1}𝐹𝑦𝜃_[𝑗]𝑑^𝑘−𝑡_𝑖=𝑖𝐹𝑦𝜃_[𝑖],(1.1b)where 𝐹(⋅)=1−𝐹(⋅) and 𝜃=(𝜃₁,…,𝜃_𝑘).

For the 𝑘 populations differing in their location parameters 𝜇₁,…,𝜇_𝑘, Gupta and Liang [1] provided a novel idea to construct simultaneous lower confidence bounds for the PCS_𝑡 for all 𝑡=1,…,𝑘−1. Their result was applied to the selection of the 𝑡 best means of normal populations. For other references under location set up, one may refer to the papers cited therein.

For other relevant references, one may refer to Gupta et al. [2], Gupta and Panchpakesan [3], Mukhopadhyay and Solanky [4], and the review papers by Gupta and Panchapakesan [5, 6], Khamnei and Kumar [7], and the references cited therein.

In this article, we use the methodology and results of Gupta and Liang [1] to derive simultaneous lower confidence bounds for the PCS_t for all 𝑡=1,…,𝑘−1 under the general scale parameter models. Section 2 deals with obtaining such intervals. The application of the results to the exponential and normal probability models is discussed in Section 3. In the case of an exponential distribution, Type-II censored data is also considered. In Section 4, we have given some numerical examples, based on real life data sets, to illustrate the procedure of finding out simultaneous lower confidence bounds for the probability of correctly selecting the 𝑡 best populations (PCS_𝑡).

2. Simultaneous Lower Confidence Bounds for PCS_𝑡

Most of the results in this Section are as a simple consequence of the results obtained by Gupta and Liang [1].

From (1.1a), the PCS_𝑡(𝜃) can be expressed as PCS_𝑡(𝜃)=^𝑘_𝑗=_{𝑘−𝑡+1}𝑃_𝑡𝑗(𝜃),(2.1) where for each 𝑗=𝑘−𝑡+1,…,𝑘,𝑃_𝑡𝑗(𝜃)=^𝑘−𝑡_𝑖=1𝐹𝑦Δ_𝑡𝑗𝑖(1)^𝑗−1_𝑚=_{𝑘−𝑡+1}𝐹𝑦Δ_𝑡𝑗𝑚(2)^𝑘_𝑙_=𝑗+1𝐹𝑦Δ_𝑡𝑗𝑙(3)𝑑𝐹(𝑦),(2.2) where Δ_𝑡𝑗𝑖(1)=𝜃_[𝑗]/𝜃_[𝑖]≥1 for 1≤𝑖≤𝑘−𝑡<𝑗; Δ_𝑡𝑗𝑚(2)=𝜃_[𝑗]/𝜃_[𝑚]≥1 for 𝑘−𝑡+1≤𝑚<𝑗 and Δ_𝑡𝑗𝑙(3)=𝜃_[𝑗]/𝜃_[𝑙]≤1 for 𝑘−𝑡+1≤𝑗<𝑙≤𝑘. Here, ∏^𝑡𝑠≡1 if 𝑡<𝑠. Note that for each 𝑗(𝑘−𝑡+1≤𝑗≤𝑘), 𝑃_𝑡𝑗(𝜃) is increasing in Δ_𝑡𝑗𝑖(1), and decreasing in Δ_𝑡𝑗𝑚(2) and Δ_𝑡𝑗𝑙(3), respectively. Thus, if we develop simultaneous lower confidence bounds for Δ_𝑡𝑗𝑖(1), 1≤𝑖≤𝑘−𝑡 and upper confidence bounds for Δ_𝑡𝑗𝑚(2) and Δ_𝑡𝑗𝑙(3), 𝑘−𝑡+1≤𝑚≤𝑗≤𝑙≤𝑘, 𝑚≠𝑗, 𝑙≠𝑗 for all 𝑡=1,…,𝑘−1, then, simultaneous lower confidence bounds for PCS_𝑡(𝜃) for all 𝑡=1,…,𝑘−1 can be established.

Also, from (1.1b), the PCS_𝑡(𝜃) can be expressed as PCS_𝑡(𝜃)=^𝑘−𝑡_𝑖=1𝑄_𝑡𝑖(𝜃),(2.3) where for each 𝑖=1,…,𝑘−𝑡,𝑄_𝑡𝑖(𝜃)=^𝑖−1_𝑚=1𝐹𝑧𝛿_𝑡𝑖𝑚(1)^𝑘−𝑡_𝑙_=𝑖+1𝐹𝑧𝛿_𝑡𝑖𝑙(2)^𝑘_𝑗=_{𝑘−𝑡+1}𝐹𝑧𝛿_𝑡𝑖𝑗(3)𝑑𝐹(𝑧)(2.4) and 𝛿_𝑡𝑖𝑚(1)=𝜃_[𝑖]/𝜃_[𝑚]≥1 for 1≤𝑚<𝑖≤𝑘−𝑡; 𝛿_𝑡𝑖𝑙(2)=𝜃_[𝑖]/𝜃_[𝑙]≤1 for 1≤𝑖<𝑙≤𝑘−𝑡; and 𝛿_𝑡𝑖𝑗(3)=𝜃_[𝑖]/𝜃_[𝑗]≤1 for 𝑖≤𝑘−𝑡<𝑗≤𝑘. Note that for each 𝑖=1,…,𝑘−𝑡,𝑄_𝑡𝑖(𝜃) is increasing in 𝛿_𝑡𝑖𝑚(1), 𝛿_𝑡𝑖𝑙(2), and decreasing in 𝛿_𝑡𝑖𝑗(3), respectively. Thus, if simultaneous lower confidence bounds for 𝛿_𝑡𝑖𝑚(1) and 𝛿_𝑡𝑖𝑙(2), 1≤𝑚≤𝑖≤𝑙≤𝑘−𝑡, 𝑚≠𝑖, 𝑙≠𝑖 and upper confidence bounds for 𝛿_𝑡𝑖𝑙(3), 𝑖≤𝑘−𝑡<𝑗≤𝑘 can be obtained, and, thereafter, by using (2.3) and (2.4), we can obtain simultaneous lower confidence bounds for the PCS_𝑡(𝜃) for all 𝑡=1,…,𝑘−1.

We use the results of Gupta and Liang [1] to construct simultaneous lower confidence bounds for all Δ_𝑡𝑗𝑖(1), 𝛿_𝑡𝑖𝑚(1), 𝛿_𝑡𝑖𝑙(2), and upper confidence bounds for all Δ_𝑡𝑗𝑚(2), Δ_𝑡𝑗𝑙(3), and 𝛿_𝑡𝑖𝑙(3) for all 𝑡=1,…,𝑘−1.

For each 𝑃^∗(0<𝑃^∗<1), let 𝑐(𝑘,𝑛,𝑃^∗) be the value such that 𝑃_𝜃−max^{1≤𝑖≤𝑘}𝑇_𝑖/𝜃_𝑖min_{1≤𝑗≤𝑘}𝑇_𝑗/𝜃_𝑗≤𝑐𝑘,𝑛,𝑃^∗=𝑃^∗.(2.5) Note that since 𝑇_𝑖 has a distribution function 𝐹(𝑦/𝜃_𝑖), 𝑖=1,…,𝑘, the value of 𝑐=𝑐(𝑘,𝑛,𝑃^∗) is independent of the parameter 𝜃. Let𝐸=max^{1≤𝑖≤𝑘}𝑇_𝑖/𝜃_𝑖𝐸min_{1≤𝑗≤𝑘}𝑇_𝑗/𝜃_𝑗≤𝑐₁,=𝑇^[𝑖]𝑐𝑇_[𝑗]⁺≤𝜃^[𝑖]𝜃_[𝑗]≤𝑐𝑇^[𝑖]𝐸𝑇_[𝑗],∀1≤𝑗<𝑖≤𝑘₂,=𝑇^[𝑖]𝑐𝑇_[𝑗]≤𝜃𝜃^[𝑖]_[𝑗]≤𝑐𝑇^[𝑖]𝑇_[𝑗]⁻,∀1≤𝑖<𝑗≤𝑘,(2.6) where 𝑦⁺=max(1,𝑦) and 𝑦⁻=min(1,𝑦).

Lemma 2.1. (a)𝐸⊂𝐸₁∩𝐸₂ and, therefore,
(b)𝑃_𝜃{𝐸₁∩𝐸₂}≥𝑃_𝜃{𝐸}=𝑃^∗ for all 𝜃.

Proof. Part (a) follows on the lines of Lemma  2.1 of Gupta and Liang [1] by noting that 𝜃_[𝑖]/𝜃_[𝑗]≥1 as 𝑗<𝑖 and 𝜃_[𝑖]/𝜃_[𝑗]≤1 for 𝑖<𝑗, we have 𝐸⊂𝐸₁ and𝐸⊂𝐸₂. Therefore, 𝐸⊂𝐸₁∩𝐸₂.
Part (b) follows immediately from part (a) and (2.5).
For each 𝑡=1,…,𝑘−1 and 𝑗=𝑘−𝑡+1,…,𝑘, let Δ_𝑡𝑗𝑖(1)=𝑇^[𝑗]𝑐𝑇_[𝑖]⁺Δfor1≤𝑖≤𝑘−𝑡;_𝑡𝑗𝑚(2)=𝑐𝑇^[𝑗]Δ𝑇_[𝑚]for𝑘−𝑡+1≤𝑚<𝑗;_𝑡𝑗𝑙(3)=𝑐𝑇^[𝑗]𝑇_[𝑙]⁻for𝑗<𝑙≤𝑘.(2.7)
Also, for each 𝑡=1,…,𝑘−1 and 𝑖=1,…,𝑘−𝑡, let 𝛿̂_𝑡𝑖𝑚(1)=𝑇^[𝑖]𝑐𝑇_[𝑚]⁺𝛿̂for1≤𝑚≤𝑖−1;_𝑡𝑖𝑙(2)=𝑇^[𝑖]𝛿̂𝑐𝑇_[𝑙]for𝑖+1≤𝑙≤𝑘−𝑡;_𝑡𝑖𝑗(3)=𝑐𝑇^[𝑖]𝑇_[𝑗]⁻for𝑘−𝑡+1≤𝑗≤𝑘.(2.8)

The following Lemma is a direct result of Lemma 2.1.

Lemma 2.2. With probability at least 𝑃^∗, the following (A1) and (A2) hold simultaneously.
(A1) For each 𝑡=1,…,𝑘−1 and each 𝑗=𝑘−𝑡+1,…,𝑘, ΔΔ_𝑡𝑗𝑖(1)≥Δ_𝑡𝑗𝑖(1),∀𝑖=1,…,𝑘−𝑡;_𝑡𝑗𝑚(Δ2)≤Δ_𝑡𝑗𝑚(2),∀𝑘−𝑡+1≤𝑚<𝑗;_𝑡𝑗𝑙(3)≤Δ_𝑡𝑗𝑙(3),∀𝑗<𝑙≤𝑘.(2.9)
(A2) For each 𝑡=1,…,𝑘−1 and each 𝑖=1,…,𝑘−𝑡, 𝛿𝛿_𝑡𝑖𝑚(1)≥𝛿̂_𝑡𝑖𝑚(1),∀1≤𝑚≤𝑖−1;_𝑡𝑖𝑙𝛿(2)≥𝛿̂_𝑡𝑖𝑙(2),∀𝑖+1≤𝑙≤𝑘−𝑡;_𝑡𝑖𝑗(3)≤𝛿̂_𝑡𝑖𝑗(3),∀𝑘−𝑡+1≤𝑗≤𝑘.(2.10) Now, for each 𝑡=1,…,𝑘−1 and each 𝑗=𝑘−𝑡+1,…,𝑘, define 𝑃_𝑡𝑗=^𝑘−𝑡_𝑖=1𝐹𝑦Δ_𝑡𝑗𝑖(1)^𝑗−1_𝑚=_{𝑘−𝑡+1}𝐹𝑦Δ_𝑡𝑗𝑚(2)^𝑘_𝑙_=𝑗+1𝐹𝑦Δ_𝑡𝑗𝑙(3)𝑑𝐹(𝑦),(2.11) and for each 𝑡=1,…,𝑘−1, define 𝑃_𝑡=^𝑘_𝑗=_{𝑘−𝑡+1}𝑃_𝑡𝑗.(2.12) Also, for each 𝑡=1,…,𝑘−1 and each 𝑖=1,…,𝑘−𝑡, define 𝑄_𝑡𝑖=^𝑖−1_𝑚=1𝐹𝑧𝛿̂_𝑡𝑖𝑚(1)^𝑘−𝑡_𝑙_=𝑖+1𝐹𝑧𝛿̂_𝑡𝑖𝑙(2)^𝑘_𝑗=_{𝑘−𝑡+1}𝑄𝐹𝑧𝛿̂_𝑡𝑖𝑗(3)𝑑𝐹(𝑧),(2.13)_𝑡=^𝑘−𝑡_𝑖=1𝑄_𝑡𝑖.(2.14) Define 𝑃_𝑡𝐿=max𝑃_𝑡,𝑄_𝑡.(2.15) The authors propose 𝑃_𝑡𝐿=max(𝑃_𝑡,𝑄_𝑡) as an estimator of a lower confidence bound of the PCS_𝑡(𝜃) for each 𝑡=1,…,𝑘−1. The authors have the following theorem.

Theorem 2.3. 𝑃_𝜃−{PCS_𝑡(𝜃)≥𝑃_𝑡𝐿 for all 𝑡=1,…,𝑘−1}≥𝑃^∗ for all 𝜃.

Proof. Note that 𝑃_𝑡𝑗(𝜃) is increasing in Δ_𝑡𝑗𝑖(1) and decreasing in Δ_𝑡𝑗𝑚(2) andΔ_𝑡𝑗𝑙(3). Also, 𝑄_𝑡𝑖(𝜃) is increasing in 𝛿_𝑡𝑖𝑚(1), 𝛿_𝑡𝑖𝑙(2) and decreasing in 𝛿_𝑡𝑖𝑗(3). Then, by using (2.2), (2.4), (2.11), (2.13), and Lemma 2.2, we have 𝑃_𝜃𝑃^𝑡𝑗(𝜃)≥𝑃_𝑡𝑗,∀𝑗=𝑘−𝑡+1,...,𝑘,and𝑄_𝑡𝑖(𝜃)≥𝑄_𝑡𝑖,∀𝑖=1,…,𝑘−𝑡,𝑡=1,…,𝑘−1≥𝑃^∗.(2.16) Then, by (2.1), (2.3), (2.12), (2.14), and (2.16), we have 𝑃^∗=≤𝑃𝑃_𝜃PCS_𝑡(𝜃)≥𝑃_𝑡,PCS_𝑡(𝜃)≥𝑄_𝑡,∀𝑡=1,…,𝑘−1PCS_𝑡(𝜃)≥𝑃_𝑡𝐿∀𝑡=1,…,𝑘−1.(2.17) This proves the theorem.

3. Applications to Exponential and Normal Distributions

3.1. Exponential Distribution

(i)  Complete Data
Let 𝑋_𝑖𝑗, 𝑗=1,…,𝑛 denote a random sample of size 𝑛 from the two-parameter exponential population Π_𝑖 having pdf 𝑓(𝑥)=(1/𝜃_𝑖)exp{−(𝑥−𝜇_𝑖)/𝜃_𝑖}, 𝑖=1,…,𝑘. Let 𝑀_𝑖=min_{1≤𝑗≤𝑛}𝑋_𝑖𝑗 and 𝑌_𝑖=∑^𝑛𝑗=1(𝑋_𝑖𝑗−𝑀_𝑖). Here, (𝑀_𝑖,𝑌_𝑖) is a sufficient statistic for (𝜇_𝑖,𝜃_𝑖), 𝑖=1,…,𝑘.𝑌_𝑖/𝜃_𝑖 has a standardized gamma distribution with shape parameter 𝜃=𝑛−1, 𝑖=1,…,𝑘. Then, based on statistics 𝑌₁,…,𝑌_𝑘 by applying the natural selection rule for each 𝑡=1,…,𝑘−1, the associated PCS_t is PCS_𝑡(𝜃)=^𝑘_𝑗=_{𝑘−𝑡+1}=𝑃_𝑡𝑗(𝜃)_𝑘−𝑡_𝑖=1𝑄_𝑡𝑖(𝜃),(3.1) where 𝑃_𝑡𝑗(𝜃)=^𝑘−𝑡_𝑖=1𝐹𝑦Δ_𝑡𝑗𝑖(1)^𝑗−1_𝑚=_{𝑘−𝑡+1}𝐹𝑦Δ_𝑡𝑗𝑚(2)^𝑘_𝑙_=𝑗+1𝑄𝐹𝑦Δ_𝑡𝑗𝑙(3)𝑑𝐹(𝑦),_𝑡𝑖(𝜃)=^𝑖−1_𝑚=1𝐹𝑧𝛿_𝑡𝑖𝑚(1)^𝑘−𝑡_𝑙_=𝑖+1𝐹𝑧𝛿_𝑡𝑖𝑙(2)^𝑘_𝑗=_{𝑘−𝑡+1}𝐹𝑧𝛿_𝑡𝑖𝑗(3)𝑑𝐹(𝑧),(3.2) and 𝐹(⋅) is the distribution function of the standardized gamma distribution with shape parameter 𝜃=𝑛−1.

For each 𝑃^∗(0<𝑃^∗<1), let 𝑐=𝑐(𝑘,𝑃^∗,𝑛) be the 𝑃^∗ quantile of the distribution of the random variable Z defined as 𝑍={max_{1≤𝑖≤𝑘}(𝑌_𝑖/𝜃_𝑖)}/{min_{1≤𝑖≤𝑘}(𝑌_𝑖/𝜃_𝑖)}, the extreme quotient of independent and identically distributed random variables 𝑌_𝑖.

Given 𝑘,𝑛,𝑃^∗ the value of 𝑐 can be obtained from the tables of Hartley’s ratio 𝑍 with 2(𝑛−1) degrees of freedom refer to Pearson and Hartley [8].

For each 𝑡=1,…,𝑘−1 and each 𝑗=𝑘−𝑡+1,…,𝑘, let𝑃_𝑡𝑗=^𝑘−𝑡_𝑖=1𝐹𝑦Δ_𝑡𝑗𝑖(1)^𝑗−1_𝑚=_{𝑘−𝑡+1}𝐹𝑦Δ_𝑡𝑗𝑚(2)^𝑘₋_𝑙=𝑗+1𝐹𝑦Δ_𝑡𝑗𝑙(3)𝑑𝐹(𝑦),(3.3) and for each 𝑡=1,…,𝑘−1 and each 𝑖=1,…,𝑘−𝑡, let 𝑄_𝑡𝑖=^𝑖−1_𝑚=1𝐹𝑧𝛿̂_𝑡𝑖𝑚(1)^𝑘−𝑡_𝑙_=𝑖+1𝐹𝑧𝛿̂_𝑡𝑖𝑙(2)^𝑘_𝑗=_{𝑘−𝑡+1}𝐹𝑧𝛿̂_𝑡𝑖𝑗(3)𝑑𝐹(𝑧),(3.4) where Δ_𝑡𝑗𝑖(1), Δ_𝑡𝑗𝑚(2), and Δ_𝑡𝑗𝑙(3) are defined as (2.7) and 𝛿̂_𝑡𝑖𝑚(1), 𝛿̂_𝑡𝑖𝑙(2), and 𝛿̂_𝑡𝑖𝑗(3) are defined in (2.8) with 𝑐 chosen from Pearson and Hartley’s tables.

For each 𝑡=1,…,𝑘−1, let 𝑃_𝑡=^𝑘_𝑗=_{𝑘−𝑡+1}𝑄_𝑡𝑃_𝑡𝑗,=^𝑘−𝑡_𝑖=1𝑄_𝑡𝑖.(3.5) Then, by Theorem 2.3, we can conclude the following.

Theorem 3.1. 𝑃_𝜃{PCS_𝑡(𝜃)≥max(𝑃_𝑡,𝑄_𝑡) for all 𝑡=1,…,𝑘−1}≥𝑃^∗ for all 𝜃.

(ii) Type-II Censored Data
From each population Π_𝑖, 𝑖=1,…,𝑘, we take a sample of 𝑛 items. Let 𝑋_𝑖[1],…,𝑋_𝑖[𝑛] denote the order statistic representing the failure times of 𝑛 items from population Π_𝑖, 𝑖=1,…,𝑘. Let 𝑟 be a fixed integer such that 1≤𝑟≤𝑛. Under Type-II censoring, the first 𝑟 failures from each population Π_𝑖 are to be observed. The observations from population Π_𝑖 cease after observing 𝑋_𝑖[𝑟]. The (𝑛−𝑟) items whose failure times are not observable beyond 𝑋_𝑖[𝑟] become the censored observations. Type-II censoring was investigated by Epstein and Sobel [9]. The sufficient statistic for 𝜃_𝑖, when location parameters are known, is 𝑈_𝑖=^𝑟_𝑗=1𝑋_𝑖[𝑗]+(𝑛−𝑟)𝑋_𝑖[𝑟],𝑖=1,…,𝑘.(3.6)𝑈_𝑖 is called the total time on test (TTOT) statistic. It is easy to verify that 𝑈_𝑖/𝜃_𝑖 has standardized gamma distribution with shape parameter 𝑟,𝑖=1,…,𝑘. Again, the results of complete data can be applied simply by taking 𝜗=𝑟.

3.2. Normal Distribution

Let Π_𝑖 denote the normal population with mean 𝜇_𝑖 and variance 𝜃_𝑖 (both unknown), 𝑖=1,…,𝑘. The sufficient statistic for 𝜃_𝑖 based on a random sample 𝑋_𝑖1,…,𝑋_𝑖𝑛 of size 𝑛 from Π_𝑖 is 𝑌_𝑖∗=(1/(𝑛−1))∑^𝑛𝑗=1(𝑋_𝑖𝑗−𝑋_𝑖)², where 𝑋_𝑖=(1/𝑛)∑^𝑛𝑗=1𝑋_𝑖𝑗, 𝑖=1,…,𝑘. It can be verified that {(𝑛−1)𝑌_𝑖∗}/(2𝜃_𝑖) is a standardized gamma variate with shape parameter (𝑛−1)/2, 𝑖=1,…,𝑘. Once again, the above results of exponential distribution can be used with 𝜗=(𝑛−1)/2.

To illustrate the implementation of the simultaneous lower confidence bounds for the probability of correctly selecting the 𝑡 best populations (PCS_𝑡), we consider the following examples.

4. Examples

Example 4.1. Hill et al. [10] considered data on survival days of patients with inoperable lung cancer, who were subjected to a test chemotherapeutic agent. The patients are divided into the following four categories depending on the histological type of their tumor: squamous, small, adeno, and large denoted by 𝜋_1,, 𝜋₂, 𝜋₃, and 𝜋₄, respectively, in this article. The data are a part of a larger data set collected by the Veterans Administrative Lung Cancer Study Group in the USA.
We consider a random sample of eleven survival times from each group, and they are given in Table 1.
Using the standard results of reliability (refer to Lawless [11]), one can check the validity of the two-parameter exponential model for Table 1. In this example, the populations with larger survival times (i.e., larger Y_i’s) are desirable.
For Table 1 data set: 𝑌₁=3841,𝑌₂=383,𝑌₃=361,𝑌₄=1374.(4.1) Hence, according to natural selection rule, the populations 𝜋₁,𝜋₂, and 𝜋₄ are selected as the 𝑡 (𝑡=1,2,3) best populations, that is, for 𝑡=1, population 𝜋₁ which has largest survival time is the best; for 𝑡=2, populations 𝜋₁ and 𝜋₄ which have the two largest survival times are the best; and for 𝑡=3, populations 𝜋_1,, 𝜋₂, and 𝜋₄ which have the three largest survival times are the best. However, it i,s possible that selected populations according to the natural selection rule may not be the best. Therefore, we wish to find out a confidence statement that can be made about the probability of correctly selecting the 𝑡 best populations (PCS_𝑡) simultaneously for all 𝑡=1,2,3.
Here, 𝑘=4, 𝑛=11, and, by taking 𝑃^∗=0.95, we get, from the tables of Pearson and Hartley [8], 𝑐=𝑐(𝑘,𝑛,𝑃^∗)=3.29.
Then, 𝑃_𝑡 and 𝑄_𝑡 computed for the above data set using (3.5) are given in Table 2. From Table 2, we have, with at least a 95% confidence coefficient, that simultaneously PCS₁(𝜃)≥0.551725, PCS₂(𝜃)≥0.33380, and PCS₃(𝜃)≥0.174162.

Table 1 

Table 2 

Example 4.2. Nelson [12] considered the data which represent times to breakdown in minutes of an insulating fluid subjected to high voltage stress. The times in their observed order were divided into three groups. After analyzing the data, it was shown to follow an exponential distribution. We consider the following data based on a random sample of size 11 each from the three groups and the observations are in Table 4.
For the above data set: 𝑌₁=20.82,𝑌₂=21.17,𝑌₃=20.67.(4.2) Hence, according to natural selection rule, the populations 𝜋₁, 𝜋₂ are selected as the 𝑡 (𝑡=1,2) best populations, that is, for 𝑡=1, population 𝜋₁ which has largest survival time is the best; and for 𝑡=2, populations 𝜋₁ and 𝜋₂ which have the two largest survival times are the best. However, it is possible that selected populations according to the natural selection rule may not be the best. Therefore, we wish to find out a confidence statement that can be made about the probability of correctly selecting the 𝑡 best populations (PCS_𝑡) simultaneously for all 𝑡=1,2.
Here, 𝑘=3, 𝑛=11, and, by taking 𝑃^∗=0.95, we get, from the tables of Pearson and Hartley [8], 𝑐=𝑐(𝑘,𝑛,𝑃^∗)=2.95.
Then, 𝑃_𝑡 and 𝑄_𝑡 computed for the above data set using (3.5) are given in Table 3.
From Table 3, we have, with at least a 95% confidence coefficient, that simultaneously PCS₁(𝜃)≥0.424471 and PCS₂(𝜃)≥0.248274.

Table 3 

Table 4 

Example 4.3. Proschan [13] considered the data on intervals between failures (in hours) of the air-conditioning system of a fleet of 13 Boeing 720 jet air planes. After analyzing the data, he found that the failure distributions of the air-conditioning system for each of the planes was well approximated as exponential. We consider the following data based on four random samples of size seven each, and the observations in the samples are mentioned in Table 5.
For the above data set: 𝑌₁=1046,𝑌₂=96,𝑌₃=226,𝑌₄=139.(4.3) Hence, according to natural selection rule, the populations 𝜋₁,𝜋₃, and 𝜋₄ are selected as the 𝑡 (𝑡=1,2,3) best populations.
Here, 𝑘=4, 𝑛=7 and, by taking 𝑃^∗=0.99, we get, from the tables of Pearson and Hartley [8], 𝑐=𝑐(𝑘,𝑛,𝑃^∗)=6.90.
Proceeding on the lines similar to Examples 4.1 and 4.2, we have, with at least a 99% confidence coefficient, that simultaneously PCS₁(𝜃)≥0.360517, PCS₂(𝜃)≥0.217558, and PCS₃(𝜃)≥0.154598.

Table 5 

Acknowledgments

The authors thank the editor, the associate editor, and an anonymous referee for their helpful comments which led to the improvement of this paper.