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Journal of Probability and Statistics
Volume 2011 (2011), Article ID 904705, 18 pages
http://dx.doi.org/10.1155/2011/904705
Research Article

The Beta-Half-Cauchy Distribution

1Departamento de Estatística, Universidade Federal de Pernambuco, 50749-540 Recife, PE, Brazil
2Departamento de Estatística, Universidade de São Paulo, 05311-970 São Paulo, SP, Brazil

Received 28 May 2011; Accepted 13 September 2011

Academic Editor: José María Sarabia

Copyright © 2011 Gauss M. Cordeiro and Artur J. Lemonte. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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,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) is𝐺𝜙2(𝑡)=𝜋𝑡arctan𝜙,𝑡>0,(1.1) where 𝜙>0 is a scale parameter. The probability density function (pdf) corresponding to (1.1) is 𝑔𝜙2(𝑡)=𝑡𝜋𝜙1+𝜙21,𝑡>0.(1.2) For 𝑘<1, the 𝑘th moment comes from (1.2) as 𝜇𝑘=𝜙𝑘sec(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)=𝐼𝐺(𝑡)1(𝑎,𝑏)=𝐵(𝑎,𝑏)0𝐺(𝑡)𝜔𝑎1(1𝜔)𝑏1𝑑𝜔,(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)=0𝑡𝑎1𝑒𝑡d𝑡 is the gamma function, 𝐼𝑦(𝑎,𝑏)=𝐵𝑦(𝑎,𝑏)/𝐵(𝑎,𝑏) is the incomplete beta function ratio, and 𝐵𝑦(𝑎,𝑏)=𝑦0𝜔𝑎1(1𝜔)𝑏1𝑑𝜔 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)𝑎1{1𝐺(𝑡)}𝑏1,(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 as2𝑓(𝑡)=𝑎𝜙𝜋𝑎𝐵𝑡(𝑎,𝑏)1+𝜙21𝑡arctan𝜙𝑎121𝜋𝑡arctan𝜙𝑏1.(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+𝑏1. 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)2(𝑡)=𝑎𝜙𝜋𝑎[]𝐵(𝑎,𝑏)arctan(𝑡/𝜙)𝑎1[]}{1(2/𝜋)arctan(𝑡/𝜙)𝑏11+(𝑡/𝜙)21𝐼(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.

fig1
Figure 1: Plots of the density function (2.3) for some parameter values; 𝜙=1.
fig2
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 1𝐹(𝑡)=𝐵(𝑎,𝑏)𝑟=0𝑤𝑟𝐺(𝑡)𝑎+𝑟,(3.1) where 𝑤𝑟=(1)𝑟(𝑎+𝑟)1𝑟𝑏1. If 𝑏 is an integer, the index 𝑟 in (3.1) stops at 𝑏1. 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=0𝑠𝑟(𝑎)𝐺(𝑡)𝑟,(3.2) where 𝑠𝑟(𝑎)=𝑗=𝑟(1)𝑟+𝑗𝑎𝑗𝑗𝑟, and then, 𝐹(𝑡) can be expressed from (3.1) and (3.2) as 1𝐹(𝑡)=𝐵(𝑎,𝑏)𝑟=0𝑡𝑟𝐺(𝑡)𝑟,(3.3) where 𝑡𝑟=𝑚=0𝑤𝑚𝑠𝑟(𝑎+𝑚). By simple differentiation, it is immediate from (3.1) and (3.3) that𝑓(𝑡)=𝑔(𝑡)𝐵(𝑎,𝑏)𝑟=0(𝑎+𝑟)𝑤𝑟𝐺(𝑡)𝑎+𝑟1,𝑓(𝑡)=𝑔(𝑡)𝐵(𝑎,𝑏)𝑟=0(𝑟+1)𝑡𝑟+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=0𝑎𝑖𝑥2𝑖+11+𝑥2𝑖+1,(3.5) where 𝑎𝑖=22𝑖(𝑖!)2/[(2𝑖+1)!],𝐺𝜙(𝑡) can be expanded as 𝐺𝜙(𝑡𝑡)=𝜙2+𝑡2𝑖=0𝑏𝑖𝑡2𝜙2+𝑡2𝑖,(3.6) where 𝑏𝑖=(2𝜙𝑎𝑖)/𝜋.

By application of an equation from Gradshteyn and Ryzhik [18] for a power series raised to a positive integer 𝑗, we obtain 𝐺𝜙(𝑡)𝑗=𝑡𝜙2+𝑡2𝑗𝑖=0𝑐𝑗,𝑖𝑡2𝜙2+𝑡2𝑖,(3.7) where the coefficients 𝑐𝑗,𝑖 (for 𝑖=1,2,) can be determined from the recursive equation (𝑐𝑗,0=𝑏𝑗0) 𝑐𝑗,𝑖=𝑖𝑏0𝑖1𝑚=1[]𝑏(𝑗+1)𝑚𝑖𝑚𝑐𝑗,𝑖𝑚.(3.8) The coefficient 𝑐𝑗,𝑖 follows recursively from 𝑐𝑗,0,,𝑐𝑗,𝑖1 and then from 𝑏0,,𝑏𝑖. Here, 𝑐𝑗,𝑖 can be written explicitly in terms of the quantities 𝑏𝑚 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=0𝐴𝑖,𝑟𝑡𝑎+𝑟+2𝑖1𝜙2+𝑡2𝑎+𝑟+𝑖,𝑓(𝑡)=𝑖,𝑟=0𝐵𝑖,𝑟𝑡𝑟+2𝑖𝜙2+𝑡2𝑟+𝑖+1,(3.9) where𝐴𝑖,𝑟=2𝜙(𝑎+𝑟)𝑤𝑟𝑐𝑎+𝑟1,𝑖𝜋𝐵(𝑎,𝑏),𝐵𝑖,𝑟=2𝜙(𝑟+1)𝑡𝑟+1𝑐𝑟,𝑖.𝜋𝐵(𝑎,𝑏)(3.10) Equations (3.9) are the main results of this section.

3.2. Moments

Here and henceforth, let 𝑇BHC(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(𝑇𝑠)=𝑖,𝑟=0𝐴𝑖,𝑟0𝑡𝑠+𝑎+𝑟+2𝑖1𝜙2+𝑡2𝑎+𝑟+𝑖𝑑𝑡,𝐸(𝑇𝑠)=𝑖,𝑟=0𝐵𝑖,𝑟0𝑡𝑠+𝑟+2𝑖𝜙2+𝑡2𝑟+𝑖+1𝑑𝑡,(3.11) respectively. For 0<𝛼<2𝜌, these integrals can be calculated from Prudnikov et al. [19] as 0𝑥𝛼1𝑐2+𝑥2𝜌𝑑𝑥=𝑐𝛼2𝜌𝐵(𝛼,2𝜌𝛼)2𝐹1𝛼2𝛼,𝜌21;𝜌+2,;1(3.12) where2𝐹1(𝑝,𝑞;𝑐;𝑧)=𝑖=0(𝑝)𝑖(𝑞)𝑖(𝑐)𝑖𝑧𝑖𝑖!(3.13) is the hypergeometric function and (𝑝)𝑖=𝑝(𝑝+1)(𝑝+𝑖1) 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 𝑐𝑝𝑞=1/2>0.

Hence, for 𝑎 a positive integer and 𝑠<𝑎, we can express the moments of 𝑇 as𝐸(𝑇𝑠)=𝑖,𝑟=0𝑃𝑖,𝑟(𝑠)2𝐹1𝑠+𝑎+𝑟+2𝑖2,𝑟+𝑎𝑠21+1;𝑎+𝑟+𝑖+2,;1(3.14) where 𝑃𝑖,𝑟(𝑠)=𝜙𝑠𝑟𝑎𝐵(𝑠+𝑎+𝑟+2𝑖,𝑟+𝑎𝑠)𝐴𝑖,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(𝑇𝑠)=𝑖,𝑟=0𝑄𝑖,𝑟(𝑠)2𝐹1𝑠+1+𝑟+2𝑖2,𝑟+1𝑠23+1;𝑟+𝑖+2,;1(3.15) where 𝑄𝑖,𝑟(𝑠)=𝜙𝑠𝑟1𝐵(𝑠+𝑟+1+2𝑖,𝑟+1𝑠)𝐵𝑖,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] 𝐾𝑚,𝑛(𝑣;𝜙)=0𝑥𝑚exp(𝑣𝑥)𝜙2+𝑥2𝑛𝑑𝑥=(1)𝑚+𝑛12𝑛1𝜕(𝑛1)!𝑚𝜕𝑣𝑚𝜕𝑣𝜕𝑣𝑛1𝐻(𝑣;𝜙),(3.16) which holds for any 𝑣, where 𝐻(𝑣;𝜙)=𝜙1[],sin(𝜙𝑣)ci(𝜙𝑣)cos(𝜙𝑣)si(𝜙𝑣)(3.17) and ci(𝜙𝑣)=𝜙𝑣𝑡1cos(𝑡)𝑑𝑡 and si(𝜙𝑣)=𝜙𝑣𝑡1sin(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=0𝐴𝑖,𝑟𝐾𝑎+𝑟+2𝑖1,𝑎+𝑟+𝑖(𝑣;𝜙),𝑀(𝑣)=𝑖,𝑟=0𝐵𝑖,𝑟𝐾𝑟+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 𝑄𝐵(𝑢), which is available in most statistical packages, by 𝑡=𝑄(𝑢)=𝜙tan𝜋𝑄𝐵(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 𝑄𝐵(𝑢), can be found in Wolfram website (http://functions.wolfram.com/06.23.06.0004.01) as 𝑄𝐵(𝑢)=𝑖=0𝑔𝑖𝑢𝑖/𝑎, where 𝑔0=0 and 𝑔𝑖=𝑞𝑖[𝑎𝐵(𝑎,𝑏)]𝑖/𝑎 (for 𝑖1) and the quantities 𝑞𝑖’s (for 𝑖2) can be derived from the cubic recursive equation𝑞𝑖=1𝑖2×+(𝑎2)𝑖+(1𝑎)1𝛿𝑖,2𝑖1𝑟=2𝑞𝑟𝑞𝑖+1𝑟[]+𝑟(1𝑎)(𝑖𝑟)𝑟(𝑟1)𝑖1𝑟=1𝑖𝑟𝑠=1𝑞𝑟𝑞𝑠𝑞𝑖+1𝑟𝑠[],𝑟(𝑟𝑎)+𝑠(𝑎+𝑏2)(𝑖+1𝑟𝑠)(3.20) where 𝛿𝑖,2=1 if 𝑖=2 and 𝛿𝑖,2=0 if 𝑖2. For example, 𝑞0=0,𝑞1=1,𝑞2=(𝑏1)/(𝑎+1),𝑞3=[(𝑏1)(𝑎2+3𝑎𝑏𝑎+5𝑏4)]/[2(𝑎+1)2(𝑎+2)], and so on. We can expand 𝑄(𝑢) (since 𝐸0=0) as 𝑄(𝑢)=𝜙𝑘=1𝐸𝑘𝑄𝐵(𝑢)𝑘,(3.21) where 𝐸2𝑘=0,𝐸2𝑘1=(22𝑘1)𝜋2𝑘1[2(2𝑘)!]1𝐵2𝑘 (for 𝑘=1,2) and 𝐵2𝑘 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 𝑄𝐵(𝑢)=𝑢1/𝑎𝑖=0𝑔𝑖𝑢𝑖/𝑎 because 𝑔0=0, where 𝑔𝑖=𝑔𝑖+1=𝑞𝑖+1[𝑎𝐵(𝑎,𝑏)](𝑖+1)/𝑎 for 𝑖=0,1,. So, 𝑔0=[𝑎𝐵(𝑎,𝑏)]1/𝑎,𝑔1=[(𝑏1)/(𝑎+1)][𝑎𝐵(𝑎,𝑏)]2/𝑎, and so on. Now, we obtain 𝑄(𝑢)=𝜙𝑘=1𝐸𝑘𝑢1/𝑎𝑖=0𝑔𝑖𝑢𝑖/𝑎𝑘,(3.22) and then𝑄(𝑢)=𝜙𝑘=1,𝑖=0𝐸𝑘𝑘,𝑖𝑢(𝑘+𝑖)/𝑎,(3.23) where the constants 𝑘,𝑖 can be evaluated recursively using (3.8) from the quantities 𝑔𝑖 by 𝑘,0=𝑔𝑘0 and 𝑘,𝑖=(𝑖𝑔0)1𝑖𝑚=1[(𝑘+1)𝑚𝑖]𝑔𝑚𝑘,𝑖𝑚, for 𝑖=1,2,. Further, 𝑄(𝑢)=𝜙𝑝=1𝑁𝑝𝑢𝑝/𝑎,(3.24) where 𝑁𝑝=𝑝𝑘=1𝐸𝑘𝑘,𝑝𝑘 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(𝑇𝑠)=0𝑥𝑠𝑓(𝑥)𝑑𝑥=10𝑄(𝑢)𝑠𝑑𝑢.(3.25) This integral in (0,1) yields an alternative formula for (3.15) as 𝐸(𝑇𝑠)=𝜙𝑠10𝑝=0𝑀𝑝𝑢(𝑝+1)/𝑎𝑠𝑑𝑢=𝜙𝑠𝑝=0𝐿𝑠,𝑝10𝑢(𝑝+𝑠)/𝑎𝑑𝑢=𝑎𝜙𝑠𝑝=0𝐿𝑠,𝑝,(𝑝+𝑠+𝑎)(3.26) where 𝑀𝑝=𝑁𝑝+1=𝑝+1𝑘=1𝐸𝑘𝑘,𝑝+1𝑘 and 𝐿𝑠,𝑝 can be computed from (3.8) by (𝐿𝑠,0=𝑀𝑠0) 𝐿𝑠,𝑝=(𝑝𝑀0)𝑝1𝑚=1[]𝑀(𝑠+1)𝑚𝑝𝑚𝐿𝑠,𝑝𝑚.(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)=𝜙0𝐹(𝑠)𝑝=1𝑁𝑝𝑢𝑝/𝑎𝑑𝑢=𝑎𝜙𝑝=1𝑁𝑝𝐹(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.

fig3
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 𝑓𝑖𝑛(𝑡) of the 𝑖th order statistic, say 𝑇𝑖𝑛, for 𝑖=1,2,,𝑛, from data values 𝑇1,,𝑇𝑛 having the beta-G distribution can be obtained from (2.2) as 𝑓𝑖𝑛(𝑡)=𝑔(𝑡)𝐺(𝑡)𝑎1{1𝐺(𝑡)}𝑏1𝐵(𝑎,𝑏)𝐵(𝑖,𝑛𝑖+1)𝑛𝑖𝑗=0(1)𝑗𝑗𝑛𝑖𝐹(𝑡)𝑖+𝑗1.(3.30) From (3.3), (3.7), and (3.8), we can write 𝐹(𝑡)𝑖+𝑗1=1𝐵(𝑎,𝑏)𝑖+𝑗1𝑟=0𝑑𝑖+𝑗1,𝑟𝐺(𝑡)𝑟,(3.31) where 𝑑𝑖+𝑗1,𝑟=(𝑟𝑡0)1𝑟=1[(𝑖+𝑗)𝑟]𝑡𝑑𝑖+𝑗1,𝑟 and 𝑑𝑖+𝑗1,0=𝑡0𝑖+𝑗1.

Inserting this equation in (3.30), 𝑓𝑖𝑛(𝑡) can be further reduced to 𝑓𝑖𝑛(𝑡)=𝑔(𝑡)𝑘=0𝑀𝑖𝑛(𝑘)𝐺(𝑡)𝑘,(3.32) where 𝑀𝑖𝑛(𝑘)=𝑛𝑖𝑗=0(1)𝑗𝑗𝑛𝑖𝐵(𝑎,𝑏)𝑖+𝑗𝐵(𝑖,𝑛𝑖+1)𝑟,𝑚=0(1)𝑚𝑚𝑑𝑏1𝑖+𝑗1,𝑟𝑠𝑘(𝑎+𝑟+𝑚1).(3.33) If 𝑏 is an integer, the index 𝑚 in the above quantity stops at 𝑏1.

Using (3.7), we obtain 𝑓𝑖𝑛(𝑡)=𝑔𝜙(𝑡)𝑘,𝑝=0𝑐𝑘,𝑝𝑀𝑖𝑛(𝑡𝑘)2𝑝+𝑘𝜙2+𝑡2𝑝+𝑘,(3.34) where 𝑐𝑘,𝑝 is given by (3.8). By (3.34), we can derive some mathematical properties of 𝑇𝑖𝑛. For example, the 𝑠th moment of 𝑇𝑖𝑛 follows immediately as 𝐸𝑇𝑠𝑖𝑛=2𝜋𝑘,𝑝=0𝜙𝑠𝑘+2𝐵(2𝑝+𝑘+𝑠+1,𝑘𝑠1)𝑐𝑘,𝑝𝑀𝑖𝑛(×𝑘)2𝐹12𝑝+𝑘+𝑠+12,𝑘𝑠121;𝑝+𝑘+2.;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 𝜆𝑟=𝑟1𝑗=0(1)𝑟1𝑗𝑗𝑗𝜂𝑟1𝑟1+𝑗𝑗,(3.36) where 𝜂𝑗=𝐸{𝑇𝐹(𝑇)𝑗}=(𝑗+1)1𝐸(𝑇𝑗+1𝑗+1). In particular, 𝜆1=𝜂0,𝜆2=2𝜂1𝜂0,𝜆3=6𝜂26𝜂1+𝜂0, and 𝜆4=20𝜂330𝜂2+12𝜂1𝜂0. 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 𝐼𝑅(𝜌)=(1𝜌)1log{𝑓(𝑡)𝜌𝑑𝑡}, where 𝜌>0 and 𝜌1. If a random variable 𝑇 has a BHC distribution, we have𝑓(𝑡)𝜌𝑡=𝐿(𝜌)1+𝜙2𝜌𝐺𝜙(𝑡)(𝑎1)𝜌1𝐺𝜙(𝑡)(𝑏1)𝜌,(3.37) where 𝐿(𝜌)=2𝜌[𝜋𝜙𝐵(𝑎,𝑏)]𝜌. By expanding the binomial term, we obtain 𝑓(𝑡)𝜌𝑡=𝐿(𝜌)1+𝜙2𝜌𝑗=0𝑅𝑗𝐺𝜙(𝑡)(𝑎1)𝜌+𝑗,(3.38) where 𝑅𝑗=(1)𝑗𝑗(𝑏1)𝜌. By (3.2), we can write 𝑓(𝑡)𝜌𝑡=𝐿(𝜌)1+𝜙2𝜌𝑟=0𝑁𝑟𝑡(𝜌)arctan𝜙𝑟,(3.39) where 𝑁𝑟(𝜌)=𝑗=0𝑀𝑗𝑠𝑟2((𝑎1)𝜌+𝑗)𝜋𝑟,(3.40) and 𝑠𝑟((𝑎1)𝜌+𝑗) is defined after (3.2). We obtain 𝑡arctan𝜙𝑟=𝜙𝑟𝑘=0𝑓𝑟,𝑘𝑡2𝑘+𝑟𝜙2+𝑡2𝑘+𝑟,(3.41) where 𝑓𝑟,0=𝑎𝑟0,𝑓𝑟,𝑘=(𝑖𝑎0)1𝑘𝑚=1[(𝑟+1)𝑚𝑘]𝑎𝑚𝑓𝑟,𝑘𝑚, and 𝑎𝑘=22𝑘(𝑘!)2/[(2𝑘+1)!]. Thus,0𝑓(𝑡)𝜌𝑑𝑡=𝐿(𝜌)𝑟,𝑘=0𝜙2𝜌+𝑟𝑁𝑟(𝜌)𝑓𝑟,𝑘0𝑡2𝑘+𝑟𝜙2+𝑡2𝑘+𝑟+𝜌𝑑𝑡.(3.42) Finally, the Rénvy entropy can be determined from 0𝑡2𝑘+𝑟𝜙2+𝑡2𝑘+𝑟+𝜌𝑑𝑡=𝐵(2𝑘+𝑟+1,𝑟+2𝜌1)𝜙2𝑟+2𝜌1𝐹12𝑘+𝑟+12,𝜌+𝑟121;𝑘+𝑟+𝜌+2.;1(3.43)

4. Estimation and Inference

The estimation of the model parameters is investigated by the method of maximum likelihood. Let 𝐭=(𝑡1,,𝑡𝑛) 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 2(𝜽)=𝑛𝑎log𝜋𝑛log(𝜙)𝑛log{𝐵(𝑎,𝑏)}𝑛𝑖=1̇𝑤log𝑖+(𝑎1)𝑛𝑖=1loġ𝑧𝑖+(𝑏1)𝑛𝑖=1̇𝑑log𝑖,(4.1) where ̇𝑣𝑖=̇𝑣𝑖(𝜙)=𝑡𝑖̇𝑤/𝜙,𝑖=̇𝑤𝑖̇𝑣(𝜙)=1+2𝑖,̇𝑧𝑖=̇𝑧𝑖̇𝑣(𝜙)=arctan(𝑖) and ̇𝑑𝑖=̇𝑑𝑖(𝜙)=12̇𝑧𝑖/𝜋, for 𝑖=1,,𝑛. The maximization of the log-likelihood over three parameters looks easy in practice. The components of the score vector 𝐔𝜽=(𝑈𝜙,𝑈𝑎,𝑈𝑏) are𝑈𝜙𝑛=𝜙+2𝜙3𝑛𝑖=1𝑡2𝑖̇𝑤𝑖(𝑎1)𝜙2𝑛𝑖=1𝑡𝑖̇𝑤𝑖̇𝑧𝑖+2(𝑏1)𝜋𝜙2𝑛𝑖=1𝑡𝑖̇𝑤𝑖̇𝑑𝑖,𝑈𝑎2=𝑛log𝜋+𝑛{𝜓(𝑎+𝑏)𝜓(𝑎)}+𝑛𝑖=1loġ𝑧𝑖,𝑈𝑏=𝑛{𝜓(𝑎+𝑏)𝜓(𝑏)}+𝑛𝑖=1̇𝑑log𝑖,(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 𝑈𝜙=𝑈𝑎=𝑈𝑏=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(𝜽𝜽)𝐴𝒩3(𝟎,𝐊𝜽1), where 𝐴 means approximately distributed and 𝐊𝜽 is the unit expected information matrix. The asymptotic result 𝐊𝜽=lim𝑛𝑛1𝐉𝑛(𝜽) holds, where 𝐉𝑛(𝜽) is the observed information matrix. The average matrix evaluated at 𝜽, say 𝑛1𝐉𝑛(𝜽), can estimate 𝐊𝜽. The elements of the observed information matrix 𝐉𝑛(𝜽)=𝜕2(𝜽)/𝜕𝜽𝜕𝜽={𝑈𝑖𝑗}, for 𝑖,𝑗=𝜙,𝑎 and 𝑏 are𝑈𝜙𝜙=𝑛𝜙26𝜙4𝑛𝑖=1𝑡2𝑖̇𝑤𝑖+4𝜙6𝑛𝑖=1𝑡4𝑖̇𝑤2𝑖+2(𝑎1)𝜙3𝑛𝑖=1𝑡𝑖̇𝑤𝑖̇𝑧𝑖𝑡12𝑖𝜙2̇𝑤𝑖𝑡𝑖̇𝑤2𝜙𝑖̇𝑧𝑖4(𝑏1)𝜋𝜙3𝑛𝑖=1𝑡𝑖̇𝑤𝑖̇𝑑𝑖𝑡12𝑖𝜙2̇𝑤𝑖+𝑡𝑖̇𝑤𝜋𝜙𝑖̇𝑑𝑖,𝑈𝜙𝑎1=𝜙2𝑛𝑖=1𝑡𝑖̇𝑤𝑖̇𝑧𝑖,𝑈𝜙𝑏=2𝜋𝜙2𝑛𝑖=1𝑡𝑖̇𝑤𝑖̇𝑑𝑖,𝑈𝑎𝑎𝜓=𝑛(𝑎+𝑏)𝜓(𝑎),𝑈𝑎𝑏=𝑛𝜓(𝑎+𝑏),𝑈𝑏𝑏𝜓=𝑛(𝑎+𝑏)𝜓,(𝑏)(4.3) where 𝜓() is the trigamma function. Thus, the multivariate normal 𝒩3(𝟎,𝐉𝑛(𝜽)1) distribution can be used to construct approximate confidence intervals 𝜙±𝑧𝜂/2×[var(𝜙)]1/2,̂𝑎±𝑧𝜂/2×[var(̂𝑎)]1/2 and ̂𝑏±𝑧𝜂/2̂×[var(𝑏)]1/2 for the parameters 𝜙, 𝑎, and 𝑏, respectively, where var() is the diagonal element of 𝐉𝑛(𝜽)1 corresponding to each parameter and 𝑧𝜂/2 is the quantile 100(1𝜂/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{(𝜙,̂𝑎,𝑏)(𝜙,1,1)}, where 𝜙, ̂𝑎, and ̂𝑏 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>𝜉𝜂, where 𝜉𝜂 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) 1𝐹(𝑡)=Φ𝛼𝑡𝛽𝛽𝑡𝛾,𝐹(𝑡)=1e𝛽𝑡𝛼𝛾,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 0: EHC against 1: BHC and 0: HC against 1: 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.

tab1
Table 1: MLEs (standard errors in parentheses) and the measures AIC, BIC, and HQIC.
904705.fig.004
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 𝑣𝑖=𝐻(𝑥𝑖;𝜽), where the 𝑥𝑖’s are in ascending order, and then 𝑦𝑖=Φ1(𝑣𝑖), where Φ() is the standard normal cdf and Φ1() its inverse; (ii) compute 𝑢𝑖=Φ{(𝑦𝑖𝑦)/𝑠𝑦}, where 𝑦=𝑛1𝑛𝑖=1𝑦𝑖 and 𝑠2𝑦=(𝑛1)1𝑛𝑖=1(𝑦𝑖𝑦)2; (iii) calculate 𝑊2=𝑛𝑖=1{𝑢𝑖(2𝑖1)/(2𝑛)}2+1/(12𝑛) and 𝐴2=𝑛(1/𝑛)𝑛𝑖=1{(2𝑖1)log(𝑢𝑖)+(2𝑛+12𝑖)log(1𝑢𝑖)}, 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.

tab2
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.

tab3
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.

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