The gamma half-cauchy distribution: properties and applications

Transcription

1 Hacettepe Journal of Mathematics and Statistics Volume 45 4) 206), The gamma half-cauchy distribution: properties and applications Ayman Alzaatreh, M. Mansoor, M. H. Tahir, Ÿ, M. Zubair and Shakir Ali Ghazali Abstract A new distribution, namely, the Gamma-Half-Cauchy distribution is proposed. Various properties of the Gamma-Half-Cauchy distribution are studied in detail such as limiting behavior, moments, mean deviations and Shannon entropy. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is obtained. Two data sets are used to illustrate the applications of Gamma-Half-Cauchy distribution. Keywords: Folded Cauchy distribution, half-cauchy distribution, gamma distribution, Shannon entropy AMS Classication: 60E05; 62E0; 62N05. Received : Accepted : Doi : /HJMS Department of Mathematics, Nazarbayev University, Astana-00000, Kazakhstan, ayman.alzaatreh@nu.edu.kz Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 6300, Pakistan, mansoor.abbasi43@gmail.com Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 6300, Pakistan, mtahir.stat@gmail.com, mht@iub.edu.pk Ÿ Corresponding Author. Department of Statistics, Government Degree College Khairpur Tamewali, Pakistan, zubair.stat@yahoo.com Department of Statistics, The Islamia University of Bahawalpur, Bahawalpur 6300, Pakistan, shakir.ghazali@iub.edu.pk

2 44. Introduction Half-Cauchy distribution is the folded standard Cauchy distribution around the origin so that positive values are observed. Modeling with half or folded) distributions has been proposed and ve folded distributions have been reported so far in literature, namely, the students' t, normal, normal-slash, logistic and Cauchy. These folded distributions have been used in Bayesian paradigm when a proper prior is necessary. Although some applications of the half Cauchy distribution exist in the literature, but the fact that the nite moments of order greater than or equal to one do not exist, the central limit theorem does not hold. This fact reduces the applicability of this distribution in modeling real life scenarios. A random variable X has the half-cauchy HC) distribution with scale parameter σ > 0, if its cumulative distribution function cdf) is given by.) F x) = 2 π tan x/σ), x > 0. The probability density function pdf) corresponding to.) is.2) fx) = 2 [ + x/σ) 2 ]. π σ Henceforth, we denote by X HCσ), the random variable having the HC density in.2) with parameters σ. As a heavy tailed distribution, the HC distribution has been used as an alternative to exponential distribution to model dispersal distances [8] as the former predicts more frequent long distance dispersed events than the later. 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. Few generalizations of the HC distribution exist in the literature, namely, beta-half- Cauchy BHC) by Cordeiro and Lemonte [9], Kumaraswamy-half-Cauchy KHC) by Ghosh [] and Marshall-Olkin half-cauchy MOHC) by Jacob and Kayakumar [3]. In this paper, we propose a new generalization of the HC distribution using the technique dened by Alzaatreh et al. [7]. Let rt) be the probability density function pdf) of a random variable T [a, b] for a < b and let F x) be the cumulative distribution distribution function cdf) of a random variable X such that the link function W ) : [0, ] [a, b] satises the following conditions: i) W ) is dierentiable and monotonically non-decreasing, and.3) ii) W 0) a and W ) b. The T-X family of distributions dened by Alzaatreh et al. [7] as.4) Gx) = W [F x)] a rt) dt. If T 0, ), X is a continuous random variable and W [F x)] = log[ F x)], then the pdf corresponding to.4) is given by.5) gx) = fx) F x) r log [ F x) ]) ) = h f x)r H f x), where h f x) and H f x) are, respectively, the hazard and cumulative hazard function corresponding to fx). For more details about the T-X family, one is refer to Alzaatreh et al. [3, 6], Alzaatreh and Ghosh [5] and Lee et al. [4]. If a random variable T follows the gamma distribution with parameters α and β, rt) = β α Γα) ) t α e t/β, t 0. Then from.5),the pdf of Gamma-X family of

3 45 distributions is given by.6) gx) = β α Γα) fx) log [ F x) ]) α [ ] F x) β. The cdf corresponding to.6) is.7) Gx) = Γα) γ α, β log [ F x) ]), where γα, t) = t 0 uα e u du is the incomplete gamma function. Several properties of gamma-x family have been studied in literature. For more details see Alzaatreh et al. [3,?, 6, 4, 8]. The paper is unfolded as follows. In Section 2, we dene a new generalization of the HC distribution, namely, Gamma-half-Cauchy GHC) distribution. In Section 3, some properties of the GHC are investigated. The density of the order statistics is obtained in Section 4. In Section 5, the model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. In Section 6, we explore the usefulness of the proposed distribution by means of two real data sets. Finally, Section 7 oers some concluding remarks. 2. The gamma-half Cauchy GHC) distribution From.),.2),.6) and.7), it follows that the pdf and cdf of the GHC are given by 2 [ gx) = + x/σ) 2 ] log [ 2 π tan x/σ) ]) α π σ Γα) β α 2.) and [ 2 π tan x/σ) ] β 2.2) Gx) = Γα) γ α, β log [ 2 π tan x/σ) ]), respectively. Henceforth, a random variable having pdf in 2.2) is denoted by X GHCα, β, σ). Special cases of GHC distribution: i) If α = β = in 2.2), the GHC distribution reduces to the HC distribution with parameter σ. ii) If α = in 2.2), the GHC distribution reduces to the exponentiated HC distribution with parameters β and σ. iii) If α = n + and β = in 2.2), the density of GHC reduces to the density of the nth upper record of the HC distribution. Note that the special case in ii) does not exist in the literature and it is considered another generalization of the HC distribution. The survival function sf), Sx), hazard rate function hrf), hx), and cumulative hazard rate function chrf), Hx), of X are, respectively, given by Sx) = Γα) γ α, β log [ 2 π tan x/σ) ]), 2 log [ 2 π tan x/σ) ]) α [ 2 π tan x/σ) ] β hx) = π σ β α + x/σ) 2 ) Γα) γ α, β log [ 2 π tan x/σ) ])}

4 46 and [ Hx) = log Γα) γ α, β log [ 2 π tan x/σ) ])]. 2.. Asymptotic behavior of the pdf. The limit of the pdf of X as x is 0. Further, the limits of the pdf of X as x 0 + are given by, if α < lim gx) = 2, if α =, x 0 + π σ β 0, if α >. In Figures and 2, various graphs of the density when σ = and for dierent values of α and β are displayed. Figure indicates that the GHC distribution is well-suited for right-skewed data. For xed α, the density is always reversed-j shaped. For xed α >, the peakedness increases as β decreases. Also, Figure 2 shows that the hazard function of the GHC distribution has DFR decreasing failure) or UBT upside down bathtub) properties. gx) α = β = α = β = 2 α = 0.5 β = 0.5 α = 0.5 β = 5 α = 0.2 β = 5 gx) α = 2 β = α = 2 β = 0.5 α = 20 β = 0. α = 30 β = 0.05 α = 40 β = x x a) b) Figure. Plots of the GHC densities for various values of α and β. 3. Properties of the GHC distribution In this section, we provide some properties of the GHC distribution. Some proofs are omitted in case of trivial results. The following Lemma gives the relation between GHC and gamma distributions. 3.. Transformation. 3.. Lemma. If a random variable ) Y follows the gamma distribution with parameters α π and β, then X = σ cot 2 e Y GHCα, β, σ).

5 47 hx) α = β =.5 α = β = 2 α = 0.4 β = 0.9 α = 0.5 β = 5 α = 0. β = 5 hx) α = 2 β = α = 2 β = 0.5 α = 20 β = 0. α = 40 β = 0.05 α = 30 β = x x a) b) Figure 2. Plots of GHC hazard rates for various values of α and β Mode Lemma. The mode of GHC distribution is the solution of kx) = 0, where 3.) ] kx) = x σ + π [ 2π tan x/σ) } α log [ 2π tan x/σ)] β +. Proof. Setting g x) is equivalent to, g 4 [ + x/σ)] 2 x) = log [ 2π tan x/σ) ] ) α πσ 2 Γα)β α 3.2) where 3.3) [ ] 2π tan β x/σ) kx), kx) = x/σ) + π [ 2π tan x/σ) ] } α log [ 2π tan x/σ)] β +. Hence the critical values of gx) is the solution of kx) = 0. Note that equation implies the following; when α = β =, the mode of GHC is at x = 0 which is the mode of HC distribution. When α <, implies that x < 0 and as x 0 +, kx). Also, when α =, x = 0 is a modal point and as x 0 +, kx) 2. π σ β Hence, when α, GHC has a unique mode at x = Quantile function. The following Lemma gives the quantile function for the GHC distribution Lemma. The mode of GHC distribution is given by 3.4) Qλ) = σ cot 0.5 π e β γ )) α, λ Γα). Proof. Follows by inverting equation 2..

6 Shannon entropy Theorem. The Shannon entropy for the GHC distribution is given by 3.5) η X = 3 log0.5 π) + αβ ) + log β Γα) ) + α) ψα) 2 w k + 2kβ) α, where ψ ) is the digamma function and w k = )k π) 2k B 2k 2k 2k)!. Proof. Based on Alzaatreh et al. [8], the Shannon entropy for the gamma-x family is given by ))} 3.6) η X = E log f F e T + α β) + log β Γα) ) + α) ψα), where T Gammaα, β). We rst need to nd E log f F e ))} T, where fx) and F x) are the pdf and cdf of HC distribution. It follows immediately that log f F e )) T = log0.5 π) + 2 log sin0.5 π e T ) ) and hence by using the series expansion for log sin0.5 π e T ) ) see [2]) as 3.7) log sin0.5 π e T ) ) ) k π) 2k B 2k = log0.5 π) T + e 2kT, 2k 2k)! k= }} w k where B 2k is the Bernoulli number. Therefore, ))} 3.8) E log f F e T = 3 log0.5 π) 2ET ) + 2 k= w k E e 2kT ). The results in 3.5) followed by substituting 3.8) in 3.6) and noting that ET ) = αβ and E e ) 2kT = + 2kβ) α Moments. By using the Lemma 3., the rth moments of GHC distribution can be written as 3.9) EX r σ r r ) = cot0.5 π e )) u u α e u/β du. β α Γα) 0 A series expansion for cot0.5 π e u ) can be obtained from [2] as follows 3.0) cot0.5 π e u ) = v k e 2k )u, k=0 where v k = 2 )k π) 2k B 2k. 2k)! Hence, ) r cot0.5 π e u ) = k,...,k r=0 v k,...,k r e 2sr )u, where v k,...,k r = v k= v k=2... v k=r and s r = k + k k r. Therefore, from 3.9) we get 3.) EX r ) = σ r k,...,k r=0 v k,...,k r 2β sr β + ) α Theorem. Let X GHCα, β, σ), then EX r ) exists i β < r. k=

7 49 Proof. The rth moment of GHC can be obtained from 3.2) EX r ) = 0 x r gx) dx + where gx) is dened in 2.2). x r gx) dx, Without loss of generality assume σ =. From 3.2), the existence of EX r ) equivalent to the existence of x r gx) dx. Now, 3.3) where 3.4) x r gx) dx = I = π β α Γα) I, x r + x 2 log [ 0.5π tan x) ]} α [ 0.5π tan x) ] β dx. Consider the following inequality Abramowitz and Stegun []) 3.5) x < log x) < x, x <, x 0. x Now, for α, one can use the right hand-side of the inequality in 3.5) to show that 3.6) I < x r [ 0.5π tan x) ] α [ 0.5π tan x) ] β α dx. + x } 2 } τ x) Let τ 2x) = x β +α+r 2 τ, then lim x) x = τ 2 0.5π ) β α. Therefore, τ x) x) exists i τ 2x) exists i > α + r. Since α, this implies that > r. If α <, the β β left hand side of the inequality in 3.5) implies that 3.7) I < x r + x 2 [ 0.5π tan x) ] α [ 0.5π tan x) ] β dx. Similarly, one can show the right hand side of the integrand in exists i > r. This ends β the proof Mean deviations. The mean deviations about the mean δ X) = E X µ ) ) and about the median δ 2X) = E X M ) ) of X can be expressed as 3.8) δ X) = 2µ F µ ) 2m µ ) and δ 2X) = 2µ 2m M), respectively, where µ = Ex) can be obtained from 3.) by setting r = and M is the median of the GHC which can be calculated from Lemma 3.3 as 3.9) M = σ cot 0.5 π e β γ )) α, 0.5 Γα). Further, F µ ) can easily be computed from the 2.) and m z) = z 0 rst incomplete moment of X) can be computed from 3.20) m z) = z 0 cot0.5 π e u ) u α e u/β du. The result immediately follows from 3.0) as 3.2) m z) = σ v k + 2β k β) α γ α, z ) Γα) β + 2β k β). k=0 x fx) dx the

8 Mean residual life function. Let X be a random variable with cdf F such that EX) <. The mean residual life MRL) function ξx) of X is dened by ξx) = EX x X > x). It plays a major rule in many elds such as industrial reliability, life insurance and biomedical science. The following theorem provides an expansion for the MRL for the GHC distribution Theorem. Let X be a random variable which follows the GHCα, β, σ) such that β <, then the MRL function is given by σ Γα, 2k + β ) x) 3.22) ξx) = v k x Γα) Sx) 2β k β + ) α k=0 where Γx, a) = t a e t dt is the upper incomplete gamma function and v x k is dened in 3.0) and Sx) is survival function of GHC dened in section 2. Proof. From Lemma 3.) EX X > x) = σ β α Γα)Sx) y cot 0.5πe y) y α e y/β dy. On using the expansion in 3.0), one can get the result in 3.22) Reliability estimation. The reliability parameter R is dened as R = P X > Y ), where X and Y are independent random variables. Many applications of the reliability parameter have appeared in the literature such as the area of classical stress-strength model and the breakdown of a system having two components. If X and Y are two continuous random variables with cdfs F x) and F 2y) and their pdfs f x) and f 2y) respectively. Then, the reliability parameter R can be written as 3.23) R = P X > Y ) = F 2x) f x) dx Theorem. Suppose that X and Y are two independent GHC random variables with parameters α, β and α 2, β 2, and xed scale parameter σ. Then ) α +k β ) k Γα + α 2 + k) 3.24) R =. Γα ) Γα 2) k! Γα 2 + k) k=0 β 2 Proof. On using the following series expansion from [] ) k x k+α 3.25) γα, x) = k! k + α), k=0 and then substituting u = log [ 2π tan x/σ) ], 3.23) reduces to 3.26) R = β α Γα ) Γα 2) k=0 ) k k! α 2 + k) β α 2+k 2 The result in 3.24) follows immediately from 3.26). 0 u α +α 2 +k e u/β du Mixture representation of GHC density Theorem. The GHC distribution is the linear combination of innite exponentiated- HC densities 3.27) gx) = w i,j h α+k+i, σ) x), k=0

9 5 where h α+k+i, σ) x) is the exponentiated-hc density with power parameter α + k + i and ) ) ) k k + α k w i,j = β ) j+k+i p j,k k j i α j )α + k + i) Γα) β. α j=0 i=0 Proof. Based on the formula given at functions.wolfram.com/ ElementaryFunctions/Log/06/0/04/03/, we can write log [ 2 π tan x/σ) ]) α = ) k 3.28) α ) k=0 k + α k j=0 ) j+k k j) pj,k [ 2 ] α+k. α j ) π tan x/σ) Here, the constants p j,k for j 0 and k ) can be determined recursively by p j,k = k m= [k mj + )] c m p j,k m, where p j,0 = and c k = ) k+ k + ). Now, using the generalized binomial series expansion 3.29) [ 2 π tan x/σ) ] β = i=0 ) i β i ) [ 2 ] i, π tan x/σ) where a i) = aa ) a i + )/i!. The result 3.27) follows immediately by substituting 3.28) and 3.29) in 2.2). Note that the second summation in w i,j is nite whenever β is a natural number. 4. Order statistics Order statistics make their appearance in many areas of statistical theory and practice. Suppose X,..., X n is a random sample from the GHC distribution. Let X i:n denote the ith order statistic. Then, the pdf of X i:n can be expressed as f i:nx) = j=0 = n! i )!n i)! fx) F x)i F x)} n i n! n i ) j n i i )!n i)! j j=0 ) fx) F x) j+i. Inserting 2.) and 2.2) in the last equation and after some algebra, we obtain n i ) j Γn + ) i + j) 2 [ f i:nx) = + x/σ) 2 ] Γi Γj + )Γn i j + ) π σ Γα) β α log [ 2 π tan x/σ) ]) α [ 2 π tan x/σ) γ α, β log [ 2 π tan x/σ) ]) j+i }. Γα) ] β Hence, 4.) f i:nx) = n i j=0 η j f α,β,j+i) x),

10 52 where η j = ) j Γn + ) i + j) Γi) Γj + ) Γn i j + ) and f α,β,j+i) x) is the exponentiated-ghc density with parameters α, β, i + j) ). Equation 4.) is the main result of this section. It reveals that the pdf of the GHC order statistics is a linear combination of exponentiated-ghc densities. So, several mathematical quantities of these order statistics like ordinary and incomplete moments, factorial moments, mgf, mean deviations and several others can be derived from those quantities of the GHC distribution. 5. Estimation and information matrix In this section, the method of maximum likelihood estimation is used to estimate the GHC distribution parameters. The maximum likelihood estimates MLEs) enjoy desirable properties that can be used when constructing condence intervals and regions and deliver simple approximations that work well in nite samples. The resulting approximation for the MLEs in distribution theory is easily handled either analytically or numerically. Let x,..., x n be a sample of size n from the GHC distribution given by 2.2). The log-likelihood function for the vector of parameters Θ = α, β, σ) can be expressed as [ l = n log +α ) + 2 π σ Γα) β α ] n log [ + x i/σ) 2] n log log [ 2 π tan x ]) i/σ) β ) n log [ 2 π tan x i/σ) ] The components of the score vector JΘ) are given by n J α = n ψα) n log β + log log [ 2 π tan x ]) i/σ), n J β = n α β β 2 log [ 2 π tan x ] i/σ), J σ = n σ + 2σ 3 n 2α ) π σ 2 +2 x 2 i n β ) π σ 2 [ + xi/σ) 2] xi tan x [ i/σ) 2 π tan x ] } i/σ) log [ 2 π tan x ] i/σ) } x i tan x [ i/σ) 2 π tan x ]. i/σ) n Setting these equations to zero and solving them simultaneously yield the maximum likelihood estimates MLEs) of the model parameters. Numerical methods can be used to obtain the MLE Θ. For example, the Newton-Raphson iterative technique could be applied to solve the likelihood equations and obtain Θ numerically. For interval estimation of the parameters, we require the 3 3 observed information matrix JΘ) = J rs} for r, s = α, β, σ) given in Appendix A. The observed information matrix can be determined numerically from standard maximization routines, which provide the observed

11 53 information matrix as part of their output; e.g., one can use the R functions optim or nlm, the Ox function MaxBFGS, the SAS procedure NLMixed, among others, to compute JΘ) numerically. Under standard regularity conditions, the multivariate normal N 30, J Θ) ) distribution can be used to construct approximate condence intervals for the model parameters. Here, J Θ) is the total observed information matrix evaluated at Θ. Then, the 00 γ)% condence intervals for α, β and σ are given by ˆα ± z γ /2 varˆα), ˆβ ± z γ /2 var ˆβ) and ˆσ ± z γ /2 varˆσ), respectively, where the var )'s denote the diagonal elements of J Θ) corresponding to the model parameters, and z γ /2 is the quantile γ /2) of the standard normal distribution. The likelihood ratio LR) statistic can be used to check if the GHC distribution is strictly superior to the HC distribution for a given data set. The test of H 0 : α = β = versus H : H 0 is not true is equivalent to compare the GHC and HC distributions and the statistic w = 2 log λ = 2l α, β, σ) l,, σ)}, where α, β and σ are the MLEs under H and σ is the MLE under H 0, is asymptotically follows chi-square distribution with 2 degrees of freedom. Similarly, the test of H 0 : α = versus H : α is equivalent to compare the GHC and exponentiated HC distributions with the statistic w = 2l α, β, σ) l, β, σ)}, where α, β and σ are the MLEs under H and β and σ are the MLEs under H 0. In this case w is asymptotically follows chi-square distribution with degrees of freedom. 5.. Simulation study. We evaluate the performance of the maximum likelihood method for estimating the GHC parameters using Monte Carlo simulation for a total of twenty four parameter combinations and the process is repeated 200 times. Two dierent sample sizes n = 00 and 300 are considered. The MLEs and the standard deviations of the parameter estimates are listed in Table. The MLEs of α, β and σ are determined by solving the nonlinear equations UΘ) = 0. From Table, we note that the ML method performs well for estimating the model parameters. Also, as the sample size increases, the biases and the standard deviations of the MLEs decrease as expected.

12 54 Table : MLEs and standard deviations for various parameter values. Sample size Actual values Estimated values Standard deviations n α β σ α β σ α β σ Applications In this section, we provide two applications to real data to illustrate the importance of the GHC distribution. The model parameters are estimated by the method of maximum likelihood and three well-recognized goodness-of-t statistics are calculated to compare the GHC distribution with other competing models. The rst data set represents the annual food discharge rates for the 39 years ) at Floyd River located in James, Iowa, USA. The Floyd River data were reported by Mudholkar and Hutson [5] and Akinsete et al. [2]. The second data set consists of the waiting times between 65 consecutive eruptions of the Kiama Blowhole da Silva et al. [0]; Pinho et al. [7]). The Kiama Blowhole is a tourist attraction located nearly 20km to the south of Sydney. The swelling of the ocean pushes the water through a hole bellow a cli. The water then erupts through an exit usually drenching whoever is nearby. The times between eruptions of a 340 hours period starting from July 2th of 998 were recorded using a digital watch. Both data sets are reported in Appendix B. We tted the GHC model to the three data sets and compared it with other models: the BHC, KHC, EHC and HC. The measures of goodness-of-t statistics including the log-likelihood function evaluated at the MLEs log ˆl), Akaike information criterion AIC) and Kolmogrov-Smirnov K-S) are computed to compare the tted models. In general, the smaller the values of these statistics, the better the t to the data. The required computations are carried out using the R-software.

13 55 Table 2: The statistics log ˆl, AIC and K-S for the data sets, 2 and 3. Distribution α β σ log ˆl AIC K-S K-S p-value Data set GHC ) 0.758) ) BHC ) ) ) KHC ) 0.265) ) EHC ) ) HC ) Data set 2 GHC ) 0.346).959) BHC ) ) ) KHC ) 0.325) 2.476) EHC ) ) HC )

14 56 Table 2 lists the MLEs and their corresponding standard errors in parentheses) of the model parameters for data sets and 2. The numerical values of the model selection statistics log ˆl, AIC and K-S, and p-values are listed in Table 2. In general, the results from Table 2 indicate that the GHC distribution provides the best t among the BHC, KHC, EHC and HC models. The histogram of the data sets and 2, and the estimated pdfs and cdfs of the GHC distribution and its competitive models are displayed in Figures 3 and 4. These Figures support the results in Table 2. To compare the GHC distribution with its sub-models, EHC and HC distributions, the LR test is used for both data sets and 2. When comparing the ts between GHC and EHC HC) for data, w = w = ) with p-value= p-value=0.0376). For data 2, w =.0436 w = ) with p-value= p-value=0.008). These values suggest that GHC performs signicantly better for both data sets when comparing it with the sub-models EHC and HC distributions. 7. Concluding remarks In this paper, we propose a generalization of half-cauchy distribution called the gamma-half-cauchy distribution. We study some properties of gamma-half Cauchy distribution including quantile function, moments, mean deviations and Shannon entropy. The maximum likelihood method is used for estimating the model parameters and the observed information matrix is analytically derived. We t the gamma-half-cauchy to two real data sets to demonstrate its usefulness. The new model provides consistently better t than other competing models. Density GHC BHC KHC EHC HC cdf GHC BHC KHC EHC HC x x a) Estimated pdfs b) Estimated cdfs Figure 3. Plots of the estimated pdfs and cdfs of the GHC, BHC, KHC and EHC models for data set. Appendix A The observed information matrix for the parameter vector Θ = α, β, σ) is given by J JΘ) = 2 lθ) αα J αβ J ασ Θ Θ = J ββ J βσ, J σσ

15 57 Density GHC BHC KHC EHC HC cdf GHC BHC KHC EHC HC a) Estimated pdfs x b) Estimated cdfs x Figure 4. Plots of the estimated pdfs and cdfs of the GHC, BHC, KHC and EHC models for data set 2. whose elements are J αα = n ψ α), J αβ = n β, J ασ = 2 π σ 2 n xi tan x [ i/σ) 2 π tan x ] } i/σ) log [ 2 π tan x ] i/σ) n log [ 2 π tan x ] i/σ), J ββ = nα β + 2 β 3 J βσ = 2 π σ 2 β 2 } n x i tan x [ i/σ) 2 π tan x ] i/σ),

16 58 J σσ = 2 σ + n 4 x 4 i 2 σ 6 [ + x i/σ) 2 ] 2 ) n β } 6 x 2 i σ 4 [ + x i/σ) 2 ] 4 x i tan x i/σ) π σ 3[ 2 π tan x i/σ) ] } 4 x 2 i tan x i/σ) 2 + π 2 σ [ 4 2 π tan x ]} x 2 i tan x i/σ) i/σ) π σ [ 4 2 π tan x ]} i/σ) α ) n 4 x i tan x i/σ) π σ 3 log [ 2 π tan x ] ) i/σ) 2 π tan x i/σ) 4 x 2 i tan x i/σ) 2 + π 2 σ 4 log [ 2 π tan x ]} )} 2 2 i/σ) 2 π tan x i/σ) 4 x 2 i tan x i/σ) 2 + π 2 σ 4 log [ 2 π tan x ] )} 2 i/σ) 2 π tan x i/σ) } 2 x 2 i tan x i/σ) + π σ 4 log [ 2 π tan x ] ), i/σ) 2 π tan x i/σ) where ψα) = ψα) α log Γα) α = Γα) Γα) is the trigamma function. is the polygamma function and ψ α) = 2 log Γα) α) 2 = Appendix B The rst data set are: 460, 4050, 3570, 2060, 300, 390, 720, 6280, 360, 7440, 5320, 400, 3240, 270, 4520, 4840, 8320, 3900, 7500, 6250, 2260, 38, 330, 970, 920, 500, 2870, 20600, 380, 726, 7500, 770, 2000, 829, 7300, 4740, 3400, 2940, The second data set were reported by professor Jim Irish and can be obtained at The data are: 83, 5, 87, 60, 28, 95, 8, 27, 5, 0, 8, 6, 29, 54, 9, 8, 7, 55, 0, 35,47, 77, 36, 7, 2, 36, 8, 40, 0, 7, 34, 27, 28, 56, 8, 25, 68, 46, 89, 8, 73, 69, 9, 37, 0, 82, 29, 8, 60, 6, 6, 8, 69, 25, 8, 26,, 83,, 42, 7, 4, 9, 2. Acknowledgments The authors would like to thank the Editor and the two referees for careful reading and for comments which greatly improved the paper. References [] Abramowitz, M. and Stegun, A.I. Handbook of Mathematical Functions Dover, New York 968) [2] Akinsete, A., Famoye, F. and Lee, C. The beta-pareto distribution, Statistics 42, , [3] Alzaatreh, A., Famoye, F. and Lee, C. Gamma-Pareto distribution and its applications, Journal Modern Applied Statistical Methods, 7894, 202.

17 59 [4] Alzaatreh, A., Famoye, F. and Lee, C. The gamma-normal family of distribution: Properties and applications, Computational Statistics and Data Analysis 69,6780, 204. [5] Alzaatreh, A. and Ghosh, I. A study of the gamma-pareto IV) distribution and its applications, Communications in StatisticsTheory and Methods 45, , 206. [6] Alzaatreh, A., Ghosh, I. and Said, H. On the gamma-logistic distribution, Journal Modern Applied Statistical Methods 3, 5570, 204. [7] Alzaatreh, A., Lee, C. and Famoye, F. A new method for generating families of distributions, Metron 7, 6379, 203. [8] Alzaatreh, A., Lee, C. and Famoye, F. T-normal family of distribution: A new approach to generalize the normal distribution, Journal of Statistical Distribution and Applications, Article 6, 204. [9] Cordeiro, G.M. and Lemonte, A.J. The beta-half Cauchy distribution, Journal of probability and Statistics Article ID , 8 pages, 20. [0] da Silva, R.V., de Andrade, T.A.N., Maciel, D.B.M., Campos, R.P.S. and Cordeiro, G.M. A new lifetime model: The gamma extended Fréchet distribution, Journal of Statistical Theory and Applications 2, 3954, 203. [] Ghosh, I. The Kumaraswamy-half Cauchy distribution: Properties and applications, Journal of Statistical Theory and Applications 3, 2234, 204. [2] Gradshteyn, I.S. and Ryzhik, I.M. Table of Integrals, Series and Products, Sixth edition Academic Press, San Diego, 2000). [3] Jacob, E. and Jayakumar, K. On half Cauchy distribution and process, International Journal of Statistics and Mathematics 3, 778, 203. [4] Lee, C., Famoye, F. and Alzaatreh, A. Methods for generating families of continuous distribution in the recent decades, Wiley Inerdisciplinary Reviews: Computational Statistics 5, 29238, 203. [5] Mudholkar, G.S. and Hutson, A.D. The exponentiated Weibull family: Some properties and a ood data application, Communications in StatisticsTheory and Methods 25, , 996. [6] Paradis, E., Baillie, S.R. and Sutherland, W.J. Modeling large-scale dispersal distances, Ecological Modeling 5, , [7] Pinho, L.G., Cordeiro, G.M. and Nobre, J. The Harris extended exponential distribution, Communications in StatisticsTheory and Methods 44, , 205. [8] Shaw, M.W. Simulation of population expansion and spatial pattern when individual dispersal distributions do not decline exponentially with distance, Proceedings Royal Socociety of London B25, , 995.

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