The Beta-Half-Cauchy Distribution

Abstract

On the basis of the half-Cauchy distribution, we propose the called beta-half-Cauchy distribution for modeling lifetime data. Various explicit expressions for its moments, generating and quantile functions, mean deviations, and density function of the order statistics and their moments are provided. The parameters of the new model are estimated by maximum likelihood, and the observed information matrix is derived. An application to lifetime real data shows that it can yield a better fit than three- and two-parameter Birnbaum-Saunders, gamma, and Weibull models.

1. Introduction

The statistics literature is filled with hundreds of continuous univariate distributions (see, e.g., [1, 2]). Numerous classical distributions have been extensively used over the past decades for modeling data in several areas such as engineering, actuarial, environmental and medical sciences, biological studies, demography, economics, finance, and insurance. However, in many applied areas like lifetime analysis, finance, and insurance, there is a clear need for extended forms of these distributions, that is, new distributions which are more flexible to model real data in these areas, since the data can present a high degree of skewness and kurtosis. So, we can give additional control over both skewness and kurtosis by adding new parameters, and hence, the extended distributions become more flexible to model real data. Recent developments focus on new techniques for building meaningful distributions, including the generator approach pioneered by Eugene et al. [3]. In particular, these authors introduced the beta normal (BN) distribution, denoted by BN(𝜇,𝜎,𝑎,𝑏), where 𝜇∈ℝ,𝜎>0 and 𝑎 and 𝑏 are positive shape parameters. These parameters control skewness through the relative tail weights. The BN distribution is symmetric if 𝑎=𝑏, and it has negative skewness when 𝑎<𝑏 and positive skewness when 𝑎>𝑏. For 𝑎=𝑏>1, it has positive excess kurtosis, and for 𝑎=𝑏<1, it has negative excess kurtosis et al. [3]. An application of this distribution to dose-response modeling is presented in Razzaghi [4].

In this paper, we use the generator approach suggested by Eugene et al. [3] to define a new model called the beta-half-Cauchy (BHC) distribution, which extends the half-Cauchy (HC) model. In addition, we investigate some mathematical properties of the new model, discuss maximum likelihood estimation of its parameters, and derive the observed information matrix. The proposed model is much more flexible than the HC distribution and can be used effectively for modeling lifetime data.

The HC distribution is derived from the Cauchy distribution by mirroring the curve on the origin so that only positive values can be observed. Its cumulative distribution function (cdf) is𝐺_𝜙(𝑡)=2𝜋arctan𝜙𝑡,𝑡>0,(1.1) where 𝜙>0 is a scale parameter. The probability density function (pdf) corresponding to (1.1) is 𝑔_𝜙(𝑡)=2𝜋𝜙1+𝜙𝑡²⁻¹,𝑡>0.(1.2) For 𝑘<1, the 𝑘th moment comes from (1.2) as 𝜇^𝑘=𝜙^𝑘sec(𝑘𝜋/2). As a heavy-tailed distribution, the HC distribution has been used as an alternative to model dispersal distances [5], since the former predicts more frequent long-distance dispersal events than the latter. Additionally, Paradis et al. [6] used the HC distribution to model ringing data on two species of tits (Parus caeruleus and Parus major) in Britain and Ireland.

The paper is outlined as follows. In Section 2, we introduce the BHC distribution and plot the density and hazard rate functions. Explicit expressions for the density and cumulative functions, moments, moment generating function (mgf), a power series expansion for the quantile function, mean deviations, order statistics, and Rényi entropy are derived in Section 3. In Section 4, we discuss maximum likelihood estimation and inference. An application in Section 5 shows the usefulness of the new distribution for lifetime data modeling. Finally, concluding remarks are addressed in Section 6.

2. The BHC Distribution

Consider starting from an arbitrary baseline cumulative function 𝐺(𝑡), Eugene et al. [3] demonstrated that any parametric family of distributions can be incorporated into larger families through an application of the probability integral transform. They defined the beta generalized (beta-G) cumulative distribution by 𝐹(𝑡)=𝐼_𝐺(𝑡)(𝑎,𝑏)=1𝐵(𝑎,𝑏)^0𝐺(𝑡)𝜔^𝑎−1(1−𝜔)^𝑏−1𝑑𝜔,(2.1) where 𝑎>0 and 𝑏>0 are additional shape parameters whose role is to introduce skewness and to vary tail weight, 𝐵(𝑎,𝑏)=Γ(𝑎)Γ(𝑏)/Γ(𝑎+𝑏) is the beta function, Γ(𝑎)=∫^∞0𝑡^𝑎−1𝑒^−𝑡d𝑡 is the gamma function, 𝐼_𝑦(𝑎,𝑏)=𝐵_𝑦(𝑎,𝑏)/𝐵(𝑎,𝑏) is the incomplete beta function ratio, and 𝐵_𝑦(𝑎,𝑏)=∫^𝑦0𝜔^𝑎−1(1−𝜔)^𝑏−1𝑑𝜔 is the incomplete beta function. This mechanism for generating distributions from (2.1) is particularly attractive when 𝐺(𝑡) has a closed-form expression. One major benefit of the beta-G distribution is its ability of fitting skewed data that cannot be properly fitted by existing distributions.

The density function corresponding to (2.1) is 𝑓(𝑡)=𝑔(𝑡)𝐵(𝑎,𝑏)𝐺(𝑡)^𝑎−1{1−𝐺(𝑡)}^𝑏−1,(2.2) where 𝑔(𝑡)=𝑑𝐺(𝑡)/𝑑𝑡 is the baseline density function. The density function 𝑓(𝑡) will be most tractable when both functions 𝐺(𝑡) and 𝑔(𝑡) have simple analytic expressions. Except for some special choices of these functions, 𝑓(𝑡) could be too complicated to deal with in full generality.

By using the probability integral transform (2.1), some beta-G distributions have been proposed in the last few years. In particular, Eugene et al. [3], Nadarajah and Gupta [7], Nadarajah and Kotz [8], Nadarajah and Kotz [9], Lee et al. [10], and Akinsete et al. [11] defined the BN, beta Fréchet, beta Gumbel, beta exponential, beta Weibull, and beta Pareto distributions by taking 𝐺(𝑡) to be the cdf of the normal, Fréchet, Gumbel, exponential, Weibull, and Pareto distributions, respectively. More recently, Barreto-Souza et al. [12], Pescim et al. [13], Silva et al. [14], Paranaíba et al. [15], and Cordeiro and Lemonte [16, 17] defined the beta generalized exponential, beta generalized half-normal, beta modified Weibull, beta Burr XII, beta Birnbaum-Saunders, and beta Laplace distributions, respectively.

In the same way, we can extend the HC distribution, because it has closed-form cumulative function. By inserting (1.1) and (1.2) in (2.2), the BHC density function (for 𝑡>0) with three positive parameters 𝜙, 𝑎, and 𝑏, say BHC(𝜙,𝑎,𝑏), follows as𝑓(𝑡)=2^𝑎𝜙𝜋^𝑎𝐵(𝑎,𝑏)1+𝜙𝑡²⁻¹arctan𝜙𝑡^𝑎−11−𝜋2arctan𝜙𝑡^𝑏−1.(2.3) Evidently, the density function (2.3) does not involve any complicated function. Also, there is no functional relationship between the parameters, and they vary freely in the parameter space. The density function (2.3) extends a few known distributions. The HC distribution arises as the basic exemplar when 𝑎=𝑏=1. The new model called the exponentiated half-Cauchy (EHC) distribution is obtained when 𝑏=1. For 𝑎 and 𝑏 positive integers, the BHC density function reduces to the density function of the 𝑎th order statistic from the HC distribution in a sample of size 𝑎+𝑏−1. However, (2.3) can also alternatively be extended, when 𝑎 and 𝑏 are real nonintegers, to define fractional HC order statistic distributions.

The cdf and hazard rate function corresponding to (2.3) are 𝐹(𝑡)=𝐼_{(2/𝜋)arctan(𝑡/𝜙)}(𝑎,𝑏),(2.4)ℎ(𝑡)=2^𝑎𝜙𝜋^𝑎𝐵(𝑎,𝑏)[arctan(𝑡/𝜙)]^𝑎−1{1−(2/𝜋)[arctan(𝑡/𝜙)]}^𝑏−11+(𝑡/𝜙)²1−𝐼_{(2/𝜋)arctan(𝑡/𝜙)}(𝑎,𝑏),(2.5) respectively.

The BHC distribution can present several forms depending on the parameter values. In Figure 1, we illustrate some possible shapes of the density function (2.3) for selected parameter values. From Figure 1, we can see how changes in the parameters 𝑎 and 𝑏 modify the form of the density function. It is evident that the BHC distribution is much more flexible than the HC distribution. Plots of the hazard rate function (2.5) for some parameter values are shown in Figure 2. The new model is easily simulated as follows: if 𝑉 is a beta random variable with parameters 𝑎 and 𝑏, then 𝑇=𝜙tan(𝜋𝑉/2) has the BHC(𝜙,𝑎,𝑏) distribution. This scheme is useful because of the existence of fast generators for beta random variables in statistical software.

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Figure 1 

Plots of the density function (2.3) for some parameter values; 𝜙=1.

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Figure 2 

Plots of the hazard rate function (2.5) for some parameter values; 𝜙=1.

3. Properties

In this section, we study some structural properties of the BHC distribution.

3.1. Expansion for the Density Function

The cdf 𝐹(𝑡) and pdf 𝑓(𝑡) of the beta-G distribution are usually straightforward to compute numerically from the baseline functions 𝐺(𝑡) and 𝑔(𝑡) from (2.1) and (2.2) using statistical software with numerical facilities. However, we provide expansions for these functions in terms of infinite (or finite if both 𝑎 and 𝑏 are integers) power series of 𝐺(𝑡) that can be useful when this function does not have a simple expression.

Expansions for the beta-G cumulative function are given by Cordeiro and Lemonte [16] and follow immediately from (2.1) (for 𝑏>0 real noninteger) as 𝐹(𝑡)=1𝐵(𝑎,𝑏)^∞_𝑟=0𝑤_𝑟𝐺(𝑡)^𝑎+𝑟,(3.1) where 𝑤_𝑟=(−1)^𝑟(𝑎+𝑟)⁻¹^𝑟𝑏−1. If 𝑏 is an integer, the index 𝑟 in (3.1) stops at 𝑏−1. If 𝑎 is an integer, (3.1) gives the beta-G cumulative distribution as a power series of 𝐺(𝑡). Otherwise, if 𝑎 is a real non-integer, we can expand 𝐺(𝑡)^𝑎 as 𝐺(𝑡)^𝑎=^∞_𝑟=0𝑠_𝑟(𝑎)𝐺(𝑡)^𝑟,(3.2) where 𝑠_𝑟(𝑎)=∑^∞𝑗=𝑟(−1)^𝑟+𝑗^𝑎𝑗^𝑗𝑟, and then, 𝐹(𝑡) can be expressed from (3.1) and (3.2) as 𝐹(𝑡)=1𝐵(𝑎,𝑏)^∞_𝑟=0𝑡_𝑟𝐺(𝑡)^𝑟,(3.3) where 𝑡_𝑟=∑^∞𝑚=0𝑤_𝑚𝑠_𝑟(𝑎+𝑚). By simple differentiation, it is immediate from (3.1) and (3.3) that𝑓(𝑡)=𝑔(𝑡)𝐵(𝑎,𝑏)^∞_𝑟=0𝑓(𝑡)=𝑔(𝑡)(𝑎+𝑟)𝑤_𝑟𝐺(𝑡)^{𝑎+𝑟−1},𝐵(𝑎,𝑏)^∞_𝑟=0(𝑟+1)𝑡_𝑟+1𝐺(𝑡)^𝑟,(3.4) which hold if 𝑎 is an integer and 𝑎 is a real noninteger, respectively. Using the expansion arctan(𝑥)=^∞_𝑖=0𝑎_𝑖𝑥^2𝑖+11+𝑥²^𝑖+1,(3.5) where 𝑎_𝑖=2^2𝑖(𝑖!)²/[(2𝑖+1)!],𝐺_𝜙(𝑡) can be expanded as 𝐺_𝜙(𝑡)=𝑡𝜙²+𝑡²^∞_𝑖=0𝑏_𝑖𝜙²𝑡+²𝑡²^𝑖,(3.6) where 𝑏_𝑖=(2𝜙𝑎_𝑖)/𝜋.

By application of an equation from Gradshteyn and Ryzhik [18] for a power series raised to a positive integer 𝑗, we obtain 𝐺_𝜙(𝑡)^𝑗=𝑡𝜙²+𝑡²^𝑗∞_𝑖=0𝑐_𝑗,𝑖𝜙²𝑡+²𝑡²^𝑖,(3.7) where the coefficients 𝑐_𝑗,𝑖 (for 𝑖=1,2,…) can be determined from the recursive equation (𝑐_𝑗,0=𝑏^𝑗0) 𝑐_𝑗,𝑖=𝑖𝑏₀^𝑖−1_𝑚₌₁[(𝑗+1)𝑚−𝑖]𝑏_𝑚𝑐_{𝑗,𝑖−𝑚}.(3.8) The coefficient 𝑐_𝑗,𝑖 follows recursively from 𝑐_𝑗,0,…,𝑐_{𝑗,𝑖−1} and then from 𝑏₀,…,𝑏_𝑖. Here, 𝑐_𝑗,𝑖 can be written explicitly in terms of the quantities 𝑏_𝑚 although it is not necessary for programming numerically our expansions in any algebraic or numerical software. Now, we can rewrite (3.4) as 𝑓(𝑡)=^∞_𝑖_,𝑟=0𝐴_𝑖,𝑟𝑡^{𝑎+𝑟+2𝑖−1}𝜙²+𝑡²^{𝑎+𝑟+𝑖},𝑓(𝑡)=^∞_𝑖_,𝑟=0𝐵_𝑖,𝑟𝑡^𝑟+2𝑖𝜙²+𝑡²^𝑟+𝑖+1,(3.9) where𝐴_𝑖,𝑟=2𝜙(𝑎+𝑟)𝑤^𝑟𝑐_{𝑎+𝑟−1,𝑖}𝜋𝐵(𝑎,𝑏),𝐵^𝑖,𝑟=2𝜙(𝑟+1)𝑡^𝑟+1𝑐_𝑟,𝑖𝜋𝐵(𝑎,𝑏).(3.10) Equations (3.9) are the main results of this section.

3.2. Moments

Here and henceforth, let 𝑇∼BHC(𝜙,𝑎,𝑏). Then, for 𝑎 an integer and 𝑎 a real noninteger, the moments of 𝑇 can be expressed from (3.9) as 𝐸(𝑇^𝑠)=^∞_𝑖_,𝑟=0𝐴_𝑖,𝑟^∞0𝑡^{𝑠+𝑎+𝑟+2𝑖−1}𝜙²+𝑡²^{𝑎+𝑟+𝑖}𝑑𝑡,𝐸(𝑇^𝑠)=^∞_𝑖_,𝑟=0𝐵_𝑖,𝑟^∞0𝑡^{𝑠+𝑟+2𝑖}𝜙²+𝑡²^𝑟+𝑖+1𝑑𝑡,(3.11) respectively. For 0<𝛼<2𝜌, these integrals can be calculated from Prudnikov et al. [19] as ^∞0𝑥^𝛼−1𝑐²+𝑥²^𝜌𝑑𝑥=𝑐^𝛼−2𝜌𝐵(𝛼,2𝜌−𝛼)₂𝐹₁𝛼2,𝜌−𝛼2;𝜌+12;1,(3.12) where₂𝐹₁(𝑝,𝑞;𝑐;𝑧)=^∞_𝑖=0(𝑝)_𝑖(𝑞)_𝑖(𝑐)_𝑖𝑧^𝑖𝑖!(3.13) is the hypergeometric function and (𝑝)_𝑖=𝑝(𝑝+1)⋯(𝑝+𝑖−1) is the ascending factorial (with the convention that (𝑝)₀=1). The function ₂𝐹₁(𝛼/2,𝜌−(𝛼/2);𝜌+(1/2);1) is absolutely convergent, since 𝑐−𝑝−𝑞=1/2>0.

Hence, for 𝑎 a positive integer and 𝑠<𝑎, we can express the moments of 𝑇 as𝐸(𝑇^𝑠)=^∞_𝑖_,𝑟=0𝑃_𝑖,𝑟(𝑠)₂𝐹₁𝑠+𝑎+𝑟+2𝑖2,𝑟+𝑎2−𝑠+1;𝑎+𝑟+𝑖+12;1,(3.14) where 𝑃_𝑖,𝑟(𝑠)=𝜙^{𝑠−𝑟−𝑎}𝐵(𝑠+𝑎+𝑟+2𝑖,𝑟+𝑎−𝑠)𝐴_𝑖,𝑟. The moments of the HC distribution for 𝑠<1 can be computed from (3.14) with 𝑎=𝑏=1.

On the other hand, for 𝑎 a positive real noninteger and 𝑠<1, we can obtain 𝐸(𝑇^𝑠)=^∞_𝑖_,𝑟=0𝑄_𝑖,𝑟(𝑠)₂𝐹₁𝑠+1+𝑟+2𝑖2,𝑟+12−𝑠+1;𝑟+𝑖+23;1,(3.15) where 𝑄_𝑖,𝑟(𝑠)=𝜙^{𝑠−𝑟−1}𝐵(𝑠+𝑟+1+2𝑖,𝑟+1−𝑠)𝐵_𝑖,𝑟. The moments functions (3.14) and (3.15) show that the method of moments will not work for this distribution.

3.3. Generating Function

The mgf 𝑀(−𝑣)=𝐸{exp(−𝑣𝑇)} of 𝑇 can be derived from the following result due to Prudnikov et al. [19] 𝐾_𝑚,𝑛(𝑣;𝜙)=^∞0𝑥^𝑚exp(−𝑣𝑥)𝜙²+𝑥²^𝑛𝑑𝑥=(−1)^{𝑚+𝑛−1}2^𝑛−1(𝑛−1)!𝜕𝜕𝑣^𝑚𝑚𝑣𝜕𝜕𝑣^𝑛−1𝐻(𝑣;𝜙),(3.16) which holds for any 𝑣, where 𝐻(𝑣;𝜙)=𝜙⁻¹[sin(𝜙𝑣)ci(𝜙𝑣)−cos(𝜙𝑣)si(𝜙𝑣)],(3.17) and ci(𝜙𝑣)=−∫^∞𝜙𝑣𝑡⁻¹cos(𝑡)𝑑𝑡 and si(𝜙𝑣)=−∫^∞𝜙𝑣𝑡⁻¹sin(𝑡)𝑑𝑡 are the cosine integral and sine integral, respectively.

For 𝑎 an integer and 𝑎 a real noninteger, the BHC generating function can be determined, from (3.9) and (3.16), as linear combinations of 𝐾_⋅,⋅(𝑣;𝜙) functions𝑀(−𝑣)=^∞_𝑖_,𝑟=0𝑀(−𝑣)=_∞𝐴_𝑖,𝑟𝐾_{𝑎+𝑟+2𝑖−1,𝑎+𝑟+𝑖}(𝑣;𝜙),_𝑖_,𝑟=0𝐵_𝑖,𝑟𝐾_{𝑟+2𝑖,𝑟+𝑖+1}(𝑣;𝜙),(3.18) respectively. Equation (3.18) is the main result of this section.

3.4. Quantile Expansion

The BHC quantile function 𝑡=𝑄(𝑢) is straightforward to be computed from the beta quantile function 𝑄_𝐵(𝑢), which is available in most statistical packages, by 𝑡=𝑄(𝑢)=𝜙tan𝜋𝑄^𝐵(𝑢)2.(3.19) Power series methods are at the heart of many aspects of applied mathematics and statistics. Here, we provide a power series expansion for 𝑄(𝑢) that can be useful to derive some mathematical measures of the new distribution. Further, we propose alternative expressions for the BHC moments on the basis of this expansion.

First, an expansion for the beta quantile function, say 𝑄_𝐵(𝑢), can be found in Wolfram website (http://functions.wolfram.com/06.23.06.0004.01) as 𝑄_𝐵(𝑢)=∑^∞𝑖=0𝑔^𝑖𝑢^𝑖/𝑎, where 𝑔^0=0 and 𝑔^𝑖=𝑞_𝑖[𝑎𝐵(𝑎,𝑏)]^𝑖/𝑎 (for 𝑖≥1) and the quantities 𝑞_𝑖’s (for 𝑖≥2) can be derived from the cubic recursive equation𝑞_𝑖=1×𝑖²+(1𝑎−−𝛿2)_𝑖,2𝑖+(1−𝑎)+^𝑖−1_𝑟=2𝑞_𝑟𝑞_{𝑖+1−𝑟}[𝑟(1−𝑎)(𝑖−𝑟)−𝑟(𝑟−1)_𝑖−1]_{𝑟=1𝑖−𝑟𝑠=1}𝑞_𝑟𝑞_𝑠𝑞_{𝑖+1−𝑟−𝑠}[𝑟(𝑟−𝑎)+𝑠(𝑎+𝑏−2)(𝑖+1−𝑟−𝑠)],(3.20) where 𝛿_𝑖,2=1 if 𝑖=2 and 𝛿_𝑖,2=0 if 𝑖≠2. For example, 𝑞₀=0,𝑞₁=1,𝑞₂=(𝑏−1)/(𝑎+1),𝑞₃=[(𝑏−1)(𝑎²+3𝑎𝑏−𝑎+5𝑏−4)]/[2(𝑎+1)²(𝑎+2)], and so on. We can expand 𝑄(𝑢) (since 𝐸₀=0) as 𝑄(𝑢)=𝜙^∞_𝑘=1𝐸_𝑘𝑄_𝐵(𝑢)^𝑘,(3.21) where 𝐸_2𝑘=0,𝐸_2𝑘−1=(2^2𝑘−1)𝜋^2𝑘−1[2(2𝑘)!]⁻¹𝐵_2𝑘 (for 𝑘=1,2…) and 𝐵_2𝑘 are the Bernoulli numbers. We have 𝐵₂=1/6, 𝐵₄=−1/30,𝐵₆=1/42,𝐵₈=−1/30,…. The beta quantile function can be rewritten as 𝑄_𝐵(𝑢)=𝑢^1/𝑎∑^∞𝑖=0𝑔_𝑖𝑢^𝑖/𝑎 because 𝑔^0=0, where 𝑔_𝑖=𝑔^𝑖+1=𝑞_𝑖+1[𝑎𝐵(𝑎,𝑏)]^{(𝑖+1)/𝑎} for 𝑖=0,1,…. So, 𝑔₀=[𝑎𝐵(𝑎,𝑏)]^1/𝑎,𝑔₁=[(𝑏−1)/(𝑎+1)][𝑎𝐵(𝑎,𝑏)]^2/𝑎, and so on. Now, we obtain 𝑄(𝑢)=𝜙^∞_𝑘=1𝐸_𝑘𝑢^∞1/𝑎_𝑖=0𝑔_𝑖𝑢^𝑖/𝑎^𝑘,(3.22) and then𝑄(𝑢)=𝜙^∞_𝑘=_1,𝑖=0𝐸_𝑘ℎ_𝑘,𝑖𝑢^{(𝑘+𝑖)/𝑎},(3.23) where the constants ℎ_𝑘,𝑖 can be evaluated recursively using (3.8) from the quantities 𝑔_𝑖 by ℎ_𝑘,0=𝑔^𝑘0 and ℎ_𝑘,𝑖=(𝑖𝑔₀)⁻¹∑^𝑖𝑚=1[(𝑘+1)𝑚−𝑖]𝑔_𝑚ℎ_{𝑘,𝑖−𝑚}, for 𝑖=1,2,…. Further, 𝑄(𝑢)=𝜙^∞_𝑝=1𝑁_𝑝𝑢^𝑝/𝑎,(3.24) where 𝑁_𝑝=∑^𝑝𝑘=1𝐸_𝑘ℎ_{𝑘,𝑝−𝑘} for 𝑝=1,2,…. The power series (3.24) for the BHC quantile can be used to obtain some mathematical properties of this distribution. For example, the 𝑠th moment of 𝑇 (for 𝑎 a real noninteger) can be expressed as 𝐸(𝑇^𝑠)=^∞0𝑥^𝑠𝑓(𝑥)𝑑𝑥=¹⁰𝑄(𝑢)^𝑠𝑑𝑢.(3.25) This integral in (0,1) yields an alternative formula for (3.15) as 𝐸(𝑇^𝑠)=𝜙^𝑠¹⁰^∞_𝑝=0𝑀_𝑝𝑢^{(𝑝+1)/𝑎}^𝑠𝑑𝑢=𝜙^𝑠∞_𝑝=0𝐿_𝑠,𝑝¹⁰𝑢^{(𝑝+𝑠)/𝑎}𝑑𝑢=𝑎𝜙^𝑠∞_𝑝=0𝐿_𝑠,𝑝(𝑝+𝑠+𝑎),(3.26) where 𝑀_𝑝=𝑁_𝑝+1=∑^𝑝+1𝑘=1𝐸_𝑘ℎ_{𝑘,𝑝+1−𝑘} and 𝐿_𝑠,𝑝 can be computed from (3.8) by (𝐿_𝑠,0=𝑀^𝑠0) 𝐿_𝑠,𝑝=(𝑝𝑀₀)^𝑝−1_𝑚₌₁[(𝑠+1)𝑚−𝑝]𝑀_𝑚𝐿_{𝑠,𝑝−𝑚}.(3.27)

3.5. Mean Deviations

The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean and median. We can derive the BHC mean deviations about the mean 𝜇=𝐸(𝑇) and about the median 𝑀(𝑀=𝑄(1/2)) from the relations 𝛿₁=2𝜇𝐹(𝜇)−2𝐻(𝜇),𝛿₂=𝐸(𝑇)−2𝐻(𝑀),(3.28) respectively, where 𝜇 can be computed from (3.14) with 𝑠=1 for 𝑎>1, 𝐹(𝜇) and 𝐹(𝑀) are calculated from (2.4) and 𝐻(𝑠)=∫^𝑠0𝑡𝑓(𝑡)𝑑𝑡. After some algebra from (3.24), 𝐻(𝑠) takes the form 𝐻(𝑠)=𝜙^0𝐹(𝑠)^∞_𝑝=1𝑁_𝑝𝑢^𝑝/𝑎𝑑𝑢=𝑎𝜙^∞_𝑝=1𝑁_𝑝𝐹(𝑠)^𝑝/𝑎+1(𝑎+𝑝).(3.29)

An application of the mean deviations is to the Lorenz and Bonferroni curves that are important in fields like economics, reliability, demography, insurance, and medicine. They are defined for a given probability 𝜋 by 𝐿(𝜋)=𝐻(𝑞)/𝜇 and 𝐵(𝜋)=𝐻(𝑞)/(𝜋𝜇), respectively, where 𝑞=𝑄(𝜋) comes from (3.24). In economics, if 𝜋=𝐹(𝑞) is the proportion of units whose income is lower than or equal to 𝑞, 𝐿(𝜋) gives the proportion of total income volume accumulated by the set of units with an income lower than or equal to 𝑞. The Lorenz curve is increasing, and convex and given the mean income, the density function of 𝑇 can be obtained from the curvature of 𝐿(𝜋). In a similar manner, the Bonferroni curve 𝐵(𝜋) gives the ratio between the mean income of this group and the mean income of the population. In summary, 𝐿(𝜋) yields fractions of the total income, while the values of 𝐵(𝜋) refer to relative income levels. The curves 𝐿(𝜋) and 𝐵(𝜋) for the BHC distribution as functions of 𝜋 are readily calculated from (3.29). They are plotted for selected parameter values in Figure 3.

(a)

(b)

Figure 3 

Plots of 𝐿(𝜋) and 𝐵(𝜋) with 𝜙=1 and 𝜇=1.

3.6. Order Statistics and Moments

Order statistics make their appearance in many areas of statistical theory and practice. The density function 𝑓_𝑖∶𝑛(𝑡) of the 𝑖th order statistic, say 𝑇_𝑖∶𝑛, for 𝑖=1,2,…,𝑛, from data values 𝑇₁,…,𝑇_𝑛 having the beta-G distribution can be obtained from (2.2) as 𝑓_𝑖∶𝑛(𝑡)=𝑔(𝑡)𝐺(𝑡)^𝑎−1{1−𝐺(𝑡)}^𝑏−1𝐵(𝑎,𝑏)𝐵(𝑖,𝑛−𝑖+1)^𝑛−𝑖_𝑗=0(−1)^𝑗⎛⎜⎜⎝𝑛−𝑗𝑖⎞⎟⎟⎠𝐹(𝑡)_{𝑖+𝑗−1}.(3.30) From (3.3), (3.7), and (3.8), we can write 𝐹(𝑡)^{𝑖+𝑗−1}=1𝐵(𝑎,𝑏)^{∞𝑖+𝑗−1}_𝑟=0𝑑_{𝑖+𝑗−1,𝑟}𝐺(𝑡)^𝑟,(3.31) where 𝑑_{𝑖+𝑗−1,𝑟}=(𝑟𝑡₀)⁻¹∑^𝑟ℓ=1[(𝑖+𝑗)ℓ−𝑟]𝑡_ℓ𝑑_{𝑖+𝑗−1,𝑟−ℓ} and 𝑑_{𝑖+𝑗−1,0}=𝑡^{0𝑖+𝑗−1}.

Inserting this equation in (3.30), 𝑓_𝑖∶𝑛(𝑡) can be further reduced to 𝑓_𝑖∶𝑛(𝑡)=𝑔(𝑡)^∞_𝑘=0𝑀_𝑖∶𝑛(𝑘)𝐺(𝑡)^𝑘,(3.32) where 𝑀_𝑖∶𝑛(𝑘)=^𝑛−𝑖_𝑗=0(−1)^𝑗^{𝑗𝑛−𝑖}𝐵(𝑎,𝑏)^𝑖+𝑗𝐵(𝑖,𝑛−𝑖+1)^∞_𝑟,𝑚=0(−1)^𝑚⎛⎜⎜⎝𝑏−𝑚1⎞⎟⎟⎠𝑑_{𝑖+𝑗−1,𝑟}𝑠_𝑘(𝑎+𝑟+𝑚−1).(3.33) If 𝑏 is an integer, the index 𝑚 in the above quantity stops at 𝑏−1.

Using (3.7), we obtain 𝑓_𝑖∶𝑛(𝑡)=𝑔_𝜙(𝑡)^∞_𝑘_,𝑝=0𝑐_𝑘,𝑝𝑀_𝑖∶𝑛(𝑘)𝑡^2𝑝+𝑘𝜙²+𝑡²^𝑝+𝑘,(3.34) where 𝑐_𝑘,𝑝 is given by (3.8). By (3.34), we can derive some mathematical properties of 𝑇_𝑖∶𝑛. For example, the 𝑠th moment of 𝑇_𝑖∶𝑛 follows immediately as 𝐸𝑇^{𝑠𝑖∶𝑛}=2𝜋^∞_𝑘,𝑝=0×𝜙^{𝑠−𝑘+2}𝐵(2𝑝+𝑘+𝑠+1,𝑘−𝑠−1)𝑐_𝑘,𝑝𝑀_𝑖∶𝑛(𝑘)₂𝐹₁2𝑝+𝑘2+𝑠+1,𝑘−2𝑠−1;𝑝+𝑘+12;1.(3.35)

L-moments are summary statistics for probability distributions and data samples [20]. They have the advantage that they exist whenever the mean of the distribution exists, even though some higher moments may not exist, and are relatively robust to the effects of outliers. The L-moments can be expressed as linear combinations of the ordered data values 𝜆_𝑟=^𝑟−1_𝑗=0(−1)^{𝑟−1−𝑗}⎛⎜⎜⎝𝑟−𝑗1⎞⎟⎟⎠⎛⎜⎜⎝𝑟−1𝑗+𝑗⎞⎟⎟⎠𝜂_𝑗,(3.36) where 𝜂_𝑗=𝐸{𝑇𝐹(𝑇)^𝑗}=(𝑗+1)⁻¹𝐸(𝑇_{𝑗+1∶𝑗+1}). In particular, 𝜆₁=𝜂₀,𝜆₂=2𝜂₁−𝜂₀,𝜆₃=6𝜂₂−6𝜂₁+𝜂₀, and 𝜆₄=20𝜂₃−30𝜂₂+12𝜂₁−𝜂₀. The L-moments of the BHC distribution can be obtained from the results of this section.

3.7. Entropy

The entropy of a random variable 𝑇 with density function 𝑓(𝑡) is a measure of variation of the uncertainty. Rényi entropy is defined by 𝐼_𝑅(𝜌)=(1−𝜌)⁻¹log{∫𝑓(𝑡)^𝜌𝑑𝑡}, where 𝜌>0 and 𝜌≠1. If a random variable 𝑇 has a BHC distribution, we have𝑓(𝑡)^𝜌=𝐿(𝜌)1+𝑡𝜙²^−𝜌𝐺_𝜙(𝑡)^{(𝑎−1)𝜌}1−𝐺_𝜙(𝑡)^{(𝑏−1)𝜌},(3.37) where 𝐿(𝜌)=2^𝜌[𝜋𝜙𝐵(𝑎,𝑏)]^−𝜌. By expanding the binomial term, we obtain 𝑓(𝑡)^𝜌=𝐿(𝜌)1+𝑡𝜙²^∞−𝜌_𝑗=0𝑅_𝑗𝐺_𝜙(𝑡)^{(𝑎−1)𝜌+𝑗},(3.38) where 𝑅_𝑗=(−1)^𝑗^{𝑗(𝑏−1)𝜌}. By (3.2), we can write 𝑓(𝑡)^𝜌=𝐿(𝜌)1+𝑡𝜙²^∞−𝜌_𝑟=0𝑁_𝑟(𝜌)arctan𝑡𝜙^𝑟,(3.39) where 𝑁_𝑟(𝜌)=^∞_𝑗=0𝑀_𝑗𝑠_𝑟((𝑎−1)𝜌+𝑗)2𝜋^𝑟,(3.40) and 𝑠_𝑟((𝑎−1)𝜌+𝑗) is defined after (3.2). We obtain arctan𝜙𝑡^𝑟=𝜙^𝑟∞_𝑘=0𝑓_𝑟,𝑘𝑡^2𝑘+𝑟𝜙²+𝑡²^𝑘+𝑟,(3.41) where 𝑓_𝑟,0=𝑎^𝑟0,𝑓_𝑟,𝑘=(𝑖𝑎₀)⁻¹∑^𝑘𝑚=1[(𝑟+1)𝑚−𝑘]𝑎_𝑚𝑓_{𝑟,𝑘−𝑚}, and 𝑎_𝑘=2^2𝑘(𝑘!)²/[(2𝑘+1)!]. Thus,^∞0𝑓(𝑡)^𝜌𝑑𝑡=𝐿(𝜌)^∞_𝑟_,𝑘=0𝜙^2𝜌+𝑟𝑁_𝑟(𝜌)𝑓_𝑟,𝑘^∞0𝑡^2𝑘+𝑟𝜙²+𝑡²^{𝑘+𝑟+𝜌}𝑑𝑡.(3.42) Finally, the Rénvy entropy can be determined from ^∞0𝑡^2𝑘+𝑟𝜙²+𝑡²^{𝑘+𝑟+𝜌}𝑑𝑡=𝐵(2𝑘+𝑟+1,𝑟+2𝜌−1)𝜙^{2𝑟+2𝜌−1}𝐹₁2𝑘+𝑟+12,𝜌+𝑟−21;𝑘+𝑟+𝜌+21;1.(3.43)

4. Estimation and Inference

The estimation of the model parameters is investigated by the method of maximum likelihood. Let 𝐭=(𝑡₁,…,𝑡_𝑛)^⊤ be a random sample of size 𝑛 from the BHC distribution with unknown parameter vector 𝜽=(𝜙,𝑎,𝑏)^⊤. The total log-likelihood function for 𝜽 can be written as ℓ(𝜽)=𝑛𝑎log2𝜋−𝑛log(𝜙)−𝑛log{𝐵(𝑎,𝑏)}−^𝑛_𝑖=1+(𝑎−1)_𝑛log𝑤̇_𝑖_𝑖=1log𝑧̇_𝑖+(𝑏−1)^𝑛_𝑖=1log𝑑̇_𝑖,(4.1) where 𝑣̇_𝑖=𝑣̇_𝑖(𝜙)=𝑡_𝑖/𝜙,𝑤̇_𝑖=𝑤̇_𝑖(𝜙)=1+𝑣̇^2𝑖,𝑧̇_𝑖=𝑧̇_𝑖(𝜙)=arctan(𝑣̇_𝑖) and 𝑑̇_𝑖=𝑑̇_𝑖(𝜙)=1−2𝑧̇_𝑖/𝜋, for 𝑖=1,…,𝑛. The maximization of the log-likelihood over three parameters looks easy in practice. The components of the score vector 𝐔_𝜽=(𝑈_𝜙,𝑈_𝑎,𝑈_𝑏)^⊤ are𝑈_𝜙=−𝑛𝜙+𝜙2^3𝑛_𝑖=1𝑡^2𝑖𝑤̇_𝑖−(𝑎−1)𝜙^2𝑛_𝑖=1𝑡_𝑖𝑤̇_𝑖𝑧̇_𝑖+2(𝑏−1)𝜋𝜙^2𝑛_𝑖=1𝑡_𝑖𝑤̇_𝑖𝑑̇_𝑖𝑈_𝑎,=𝑛log2𝜋+𝑛{𝜓(𝑎+𝑏)−𝜓(𝑎)}+^𝑛_𝑖=1𝑈_𝑏log𝑧̇_𝑖,=𝑛{𝜓(𝑎+𝑏)−𝜓(𝑏)}+^𝑛_𝑖=1log𝑑̇_𝑖,(4.2) where 𝜓(⋅) is the digamma function. The maximum likelihood estimates (MLEs) 𝜽=(𝜙,𝑎,̂𝑏)̂^⊤ of 𝜽=(𝜙,𝑎,𝑏)^⊤ are the simultaneous solutions of the equations 𝑈_𝜙=𝑈_𝑎=𝑈_𝑏=0. They can be solved numerically using iterative methods such as a Newton-Raphson type algorithm.

The normal approximation of the estimate 𝜽 can be used for constructing approximate confidence intervals and for testing hypotheses on the parameters 𝜙, 𝑎, and 𝑏. Under standard regularity conditions, we have √𝑛(𝜽−𝜽)∼^𝐴𝒩₃(𝟎,𝐊^𝜽−1), where ∼^𝐴 means approximately distributed and 𝐊_𝜽 is the unit expected information matrix. The asymptotic result 𝐊_𝜽=lim_𝑛→∞𝑛⁻¹𝐉_𝑛(𝜽) holds, where 𝐉_𝑛(𝜽) is the observed information matrix. The average matrix evaluated at 𝜽, say 𝑛⁻¹𝐉_𝑛(𝜽), can estimate 𝐊_𝜽. The elements of the observed information matrix 𝐉_𝑛(𝜽)=−𝜕²ℓ(𝜽)/𝜕𝜽𝜕𝜽^⊤=−{𝑈_𝑖𝑗}, for 𝑖,𝑗=𝜙,𝑎 and 𝑏 are𝑈_𝜙𝜙=𝑛𝜙²−6𝜙^4𝑛_𝑖=1𝑡^2𝑖𝑤̇_𝑖+4𝜙^6𝑛_𝑖=1𝑡^4𝑖𝑤̇^2𝑖+2(𝑎−1)𝜙^3𝑛_𝑖=1𝑡_𝑖𝑤̇_𝑖𝑧̇_𝑖1−𝜙²𝑡^2𝑖𝑤̇_𝑖−𝑡^𝑖−4(𝑏−1)2𝜙𝑤̇_𝑖𝑧̇_𝑖𝜋𝜙^3𝑛_𝑖=1𝑡_𝑖𝑤̇_𝑖𝑑̇_𝑖1−𝜙²𝑡^2𝑖𝑤̇_𝑖+𝑡^𝑖𝑈_𝜙𝑎𝜋𝜙𝑤̇_𝑖𝑑̇_𝑖,=−1𝜙^2𝑛_𝑖=1𝑡_𝑖𝑤̇_𝑖𝑧̇_𝑖,𝑈_𝜙𝑏=2𝜋𝜙^2𝑛_𝑖=1𝑡_𝑖𝑤̇_𝑖𝑑̇_𝑖𝑈_𝑎𝑎,=𝑛𝜓^(𝑎+𝑏)−𝜓^(𝑎),𝑈_𝑎𝑏=𝑛𝜓^(𝑎+𝑏),𝑈_𝑏𝑏=𝑛𝜓^(𝑎+𝑏)−𝜓^(𝑏),(4.3) where 𝜓′(⋅) is the trigamma function. Thus, the multivariate normal 𝒩₃(𝟎,𝐉_𝑛(𝜽)⁻¹) distribution can be used to construct approximate confidence intervals 𝜙±𝑧_𝜂/2×[var(𝜙)]^1/2,𝑎̂±𝑧_𝜂/2×[var(𝑎)̂]^1/2 and 𝑏̂±𝑧_𝜂/2×[var(𝑏)̂]^1/2 for the parameters 𝜙, 𝑎, and 𝑏, respectively, where var(⋅) is the diagonal element of 𝐉_𝑛(𝜽)⁻¹ corresponding to each parameter and 𝑧_𝜂/2 is the quantile 100(1−𝜂/2)% of the standard normal distribution.

We can easily check if the fit using the BHC model is statistically “superior” to “a fit using the HC model for a given data set by computing the likelihood ratio (LR) statistic 𝑤=2{ℓ(𝜙,𝑎,̂𝑏̂)−ℓ(𝜙,1,1)}, where 𝜙, 𝑎̂, and 𝑏̂ are the unrestricted MLEs and 𝜙 is the restricted estimate. The statistic 𝑤 is asymptotically distributed, under the null model, as 𝜒²². Further, the LR test rejects the null hypothesis if 𝑤>𝜉_𝜂, where 𝜉_𝜂 denotes the upper 100𝜂% point of the 𝜒²² distribution.

5. Application

Here, we present an application of the BHC distribution to a real data set. We will compare the fits of the BHC, EHC, and HC distributions. We also consider for the sake of comparison the two-parameter Birnbaum-Saunders (BS), gamma, and Weibull models, and the three-parameter BS and Weibull models. The BHC distribution may be an interesting alternative to these distributions for modeling positive real data sets. The cdf’s of the exponentiated BS (ExpBS), exponentiated Weibull (ExpWeibull), and gamma models are (for 𝑡>0) 𝐹(𝑡)=Φ1𝛼𝑡𝛽−𝛽𝑡^𝛾,𝐹(𝑡)=1−e^{−𝛽𝑡𝛼}^𝛾,𝐹(𝑡)=𝜁(𝛼,𝛽𝑡)Γ(𝛼),(5.1) respectively, where 𝛼>0, 𝛽>0, and 𝛾>0. Here, Φ(⋅) is the cdf of the standard normal distribution and 𝜁(⋅,⋅) is the ordinary incomplete gamma function. If 𝛾=1, we have the two-parameter BS and Weibull models. All the computations were done using the Ox matrix programming language [21] which is freely distributed for academic purposes at http://www.doornik.com. The maximization was performed by the BFGS method with analytical derivatives. For further details about this method, the reader is referred to Nocedal and Wright [22] and Press et al. [23]. We will consider the data set originally due to Bjerkedal [24], which has also been analyzed by Gupta et al. [25]. The data represent the survival times of guinea pigs injected with different doses of tubercle bacilli.

Table 1 lists the MLEs (and the corresponding standard errors in parentheses) of the model parameters and the following statistics: AIC (Akaike information criterion), BIC (Bayesian information criterion), and HQIC (Hannan-Quinn information criterion). These results show that the BHC distribution has the lowest AIC, BIC, and HQIC values in relation to their submodels, and so, it could be chosen as the best model. The LR statistics for testing the hypotheses ℋ₀: EHC against ℋ₁: BHC and ℋ₀: HC against ℋ₁: BHC are 22.9462 and 40.7366, respectively, and all yield 𝑃 values <0.001. Thus, we can reject the null hypotheses in all cases in favor of the BHC distribution at any usual significance level; that is, the BHC model is significantly better than the EHC and HC distributions. In order to assess if the model is appropriate, plots of the estimated density functions are given in Figure 4. They also indicate that the BHC model provides a better fit than the other models.

Table 1 

MLEs (standard errors in parentheses) and the measures AIC, BIC, and HQIC.

Figure 4 

Estimated densities of the BHC, EHC and HC models.

Now, we apply formal goodness-of-fit tests in order to verify which distribution fits better to these data. We consider the Cramér-von Mises (𝑊^∗) and Anderson-Darling (𝐴^∗) statistics described in detail in Chen and Balakrishnan [26]. In general, the smaller the values of these statistics, the better the fit to the data. Let 𝐻(𝑥;𝜽) be the cdf, where the form of 𝐻 is known but 𝜽 (a 𝑘-dimensional parameter vector, say) is unknown. To obtain the statistics 𝑊^∗ and 𝐴^∗, we can proceed as follows: (i) compute 𝑣_𝑖=𝐻(𝑥_𝑖;𝜽), where the 𝑥_𝑖’s are in ascending order, and then 𝑦_𝑖=Φ⁻¹(𝑣_𝑖), where Φ(⋅) is the standard normal cdf and Φ⁻¹(⋅) its inverse; (ii) compute 𝑢_𝑖=Φ{(𝑦_𝑖−𝑦)/𝑠_𝑦}, where 𝑦=𝑛⁻¹∑^𝑛𝑖=1𝑦_𝑖 and 𝑠^2𝑦=(𝑛−1)⁻¹∑^𝑛𝑖=1(𝑦_𝑖−𝑦)²; (iii) calculate 𝑊²=∑^𝑛𝑖=1{𝑢_𝑖−(2𝑖−1)/(2𝑛)}²+1/(12𝑛) and 𝐴²=−𝑛−(1/𝑛)∑^𝑛𝑖=1{(2𝑖−1)log(𝑢_𝑖)+(2𝑛+1−2𝑖)log(1−𝑢_𝑖)}, and then 𝑊^∗=𝑊²(1+0.5/𝑛) and 𝐴^∗=𝐴²(1+0.75/𝑛+2.25/𝑛²). The values of the statistics 𝑊^∗ and 𝐴^∗ for the models are listed in Table 2, thus indicating that the BHC model should be chosen to fit the current data.

Table 2 

Goodness-of-fit tests.

The MLEs (standard errors in parentheses) of the model parameters of the ExpBS, ExpWeibull, BS, gamma, and Weibull models and the statistics 𝑊^∗ and 𝐴^∗ are listed in Table 3. On the basis of these statistics, the ExpWeibull model yields a better fit than the ones of the other distributions. Overall, by comparing the figures in Tables 2 and 3, we conclude that the BHC model outperforms all the models considered in Table 3. So, the proposed distribution can yield a better fit than the classical three- and two-parameter BS, gamma, and Weibull models and therefore may be an interesting alternative to these distributions for modeling positive real data sets. These results illustrate the potentiality of the new distribution and the necessity of additional shape parameters.

Table 3 

MLEs (standard errors in parentheses) and the measures 𝑊^∗ and 𝐴^∗.

6. Concluding Remarks

We introduce a new lifetime model, called the beta half-Cauchy (BHC) distribution, that extends the half-Cauchy (HC) distribution, and study some of its general structural properties. We provide a mathematical treatment of the new distribution including expansions for the density function, moments, generating function, order statistics, quantile function, Rényi entropy, mean deviations, and Lorentz and Bonferroni curves. The model parameters are estimated by maximum likelihood. Our formulas related to the BHC model are manageable, and with the use of modern computer resources with analytic and numerical capabilities, may turn into adequate tools comprising the arsenal of applied statisticians. The usefulness of the proposed model is illustrated in an application to real data using likelihood ratio statistics and formal goodness-of-fit tests. The new model provides consistently better fit than other models available in the literature. We hope that the proposed model may attract wider applications in survival analysis for modeling positive real data sets.

Acknowledgments

The authors gratefully acknowledge grants from CNPq and FAPESP (Brazil). The authors thank an anonymous referee for some comments which improved the original version of the paper.