In a recent edition of SORT, Bidram and Nekoukhou proposed a novel class of distributions and derived its mathematical properties. Several of the mathematical properties are expressed as single infinite sums or double infinite sums. Here, we show that many of these properties can be expressed in terms of known special functions, functions for which in-built routines are widely available

Nadarajah, Saralees

Pogány, Tibor K.

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Statistics & Operations Research TransactionsSORT 39 (2) July-December 2015, 273-280Statistics &Operations ResearchTransactions© Institut d’Estadı´stica de Catalunyasort@idescat.catISSN: 1696-2281eISSN: 2013-8830www.idescat.cat/sort/A note on “Double bounded Kumaraswamy-powerseries class of distributions”Tibor K. Poga´ny1 and Saralees Nadarajah2AbstractIn a recent edition of SORT, Bidram and Nekoukhou proposed a novel class of distributions andderived its mathematical properties. Several of the mathematical properties are expressed assingle infinite sums or double infinite sums. Here, we show that many of these properties canbe expressed in terms of known special functions, functions for which in-built routines are widelyavailable.MSC: 11G55, 33C20, 33E20, 60E10.Keywords: Double bounded Kumaraswamy-power series class of distributions, Fox Wright gen-eralized hypergeometric function, Generalized hypergeometric function.1. IntroductionBidram and Nekoukhou (2013), referred to as BN from now, present a novel class ofdistributions referred to as double bounded Kumaraswamy-power series distributions.They derive various mathematical properties of the distributions, including their den-sity functions, survival functions, hazard rate functions, quantiles, moment generatingfunctions, moments, order statistic properties and stress strength parameter. They alsodiscuss maximum likelihood estimation of the parameters of the distributions and pro-vide a real data application.Several of the expressions given in BN involve single infinite sums or double infinitesums. This is the case with the moment generating functions given in BN, Table 2; themoments given in BN, Table 2; the density of the ith order statistic given in BN, page221; the rth moment of the ith order statistic given in BN, page 221; the stress-strength1 Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, CroatiaInstitute of Applied Mathematics, Obuda University, 1034 Budapest, Hungary.2 School of Mathematics, University of Manchester, Manchester M13 9PL, UKReceived: November 2014Accepted: June 2015274 A note on “Double bounded Kumaraswamy-power series class of distributions”parameter given in BN, page 222; and others. This is not very convenient for practicalimplementation of the mathematical properties. The aim of this note is to show thatmany of the infinite sums and so the mathematical properties given in BN can be reducedto known special functions, functions for which in-built routines are widely available.LetN= {1,2,3, . . .},N0 =N∪{0}, Z−0 = {0,−1,−2, . . .}, R the set of real numbers,R+ the set of positive real numbers and C the set of complex numbers.The closed form expressions in Section 2 involve several special functions. Firstis the gamma function defined by Γ(a) =∫∞0ta−1e−tdt for a ∈ R+. The second isthe polylogarithm function defined by Liν(z) = ∑n≥1n−ν zn for |z| < 1. The third is thegeneralized hypergeometric function pFq[·] defined bypFq[z] = pFq[a1, . . . ,apb1, . . . ,bq;z]=∞∑k=0(a1)k · · · (ap)k(b1)k · · · (bq)kzkk! , (1)where (λ)µ denotes the Pochhammer symbol defined by(λ)µ :=Γ(λ+µ)Γ(λ)=1,(µ= 0; λ ∈ C\{0}),λ(λ+1) · · ·(λ+n−1),(µ= n ∈ N; λ ∈ C) (2)with the convention that (0)0 := 1. The Gauss hypergeometric function 2F1(a,b : c;z) isthe particular case of (1) for p = 2, q = 1. In the case a,b ∈ Z−0 are negative integers,2F1(a,b : c;z) becomes a polynomial PN(z) of degree deg(PN) = N = min(−a,−b).The fourth is the Fox Wright generalized hypergeometric function pΨ∗q[·] with pnumerator parameters a1, . . . ,ap and q denominator parameters b1, . . . ,bq, defined by(Kilbas et al., 2006, page 56)pΨ∗q[(a1,ρ1) , . . . ,(ap,ρp)(b1,σ1) , . . . ,(bq,σq);z]=∞∑n=0p∏j=1(a j)ρ jnq∏j=1(b j)σ jnznn!(3)for a j ∈ C, j = 1,2, . . . , p, bk ∈ C, k = 1,2, . . . ,q, ρ j ∈ R+, j = 1,2, . . . , p and σk ∈ R+,k = 1,2, . . . ,q. The series in (3) converges in the whole complex z-plane when∆ := 1+q∑j=1σ j −p∑j=1ρ j > 0.Tibor K. Poga´ny and Saralees Nadarajah 275If ∆ = 0, then the series in (3) converges for |z|< ∇, where∇ :=(p∏j=1ρ−ρ jj)(q∏j=1σσ jj).The particular case of (3) for ρ1 = · · ·=ρp = 1 andσ1 = · · ·=σq = 1 is the generalizedhypergeometric function in (1).In-built routines for computing these special functions are widely available inpackages like Maple, Matlab and Mathematica. Gamma[z] in Mathematica computes thegamma function, PolyLog[ν,z] in Mathematica computes the polylogarithm function,HypergeometricPFQ[{a1,. . . ,ap}, {b1,. . . ,bq},z] in Mathematica computesthe generalized hypergeometric function, and so on. The routines allow for arbitraryprecision, so computational accuracy is not an issue.2. Closed form expressionsThe closed form expressions are given by Propositions 2.1 to 2.3. Proposition 2.1expresses Fi:n(x), the cumulative distribution function of the ith order statistic given inBN, page 217, equation (12), in terms of the Gauss hypergeometric function. Proposition2.2 expresses the moments given in BN, page 220, Table 2 in terms of the Fox Wrightgeneralized hypergeometric function. These moments are to any real order, the onesgiven in BN were for integer orders only. Proposition 2.3 expresses R, the stress-strengthparameter given in BN, page 222, in terms of the polylogarithm function.Proposition 2.1 For all 1 ≤ i ≤ n and for all x ∈ (0,1),Fi:n(x) =(ni)Ai 2F1 (−n+ i, i; i+1;A) = Pn(A)is a polynomial in A, whereA = 1−C(θ (1− xa)b)C(θ) .Moreover, Fn:n(x) = An.Proof: Follows by notingFi:n(x) =1B(i,n− i+1)n−i∑k=0(n− ik)(−1)kk+ i Ak+i276 A note on “Double bounded Kumaraswamy-power series class of distributions”=AiB(i,n− i+1)n−i∑k=0(−n+ i)kΓ(k+ i)Γ(k+ i+1)Akk!=Aii B(i,n− i+1)n−i∑k=0(−n+ i)k(i)k(i+1)kAkk!=Ai Γ(n+1)Γ(i+1) Γ(i,n− i+1) 2F1(−n+ i, i; i+1;A)=Ai n!i! (n− i)! 2F1(−n+ i, i; i+1;A).The hypergeometric function reduces to 1 when i = n, so Fn:n(x) = An.Proposition 2.2 Let XKG, XKP, XKL and XKB be random variables following, respec-tively, the Kumaraswamy geometric, Kumaraswamy Poisson, Kumaraswamy logarith-mic and Kumaraswamy binomial distributions defined in BN. Then, for all real r >−aand b > 0, we haveE(X rKG)= b(1−θ)B(1+ ra,b)2Ψ∗1[(1+b,b), (1,1)(1+b+ ra,b) ;θ], (4)E(X rKP)=b θeθ −1B(1+ra,b)1Ψ∗1[(b,b)(1+b+ ra,b) ;θ], (5)E(X rKL)=−bθlog(1−θ)B(1+ra,b)2Ψ∗1[(b,b), (1,1)(1+b+ ra,b) ;θ](6)andE(X rKB)=−bmθ(1+θ)m−1B(1+ra,b)2Ψ∗1[(b,b), (1−m,1)(1+b+ ra,b) ;−θ]. (7)Each of these expressions is valid for all |θ |< 1.Tibor K. Poga´ny and Saralees Nadarajah 277Proof: (4) follows by noting thatE(X rKG)= b(1−θ)∑n≥1n B(1+ra,nb)θ n−1= b(1−θ)Γ(1+ra)∑n≥1n Γ(nb) θ n−1Γ(1+ra+nb)= (1−θ)Γ(1+ra)∑n≥1Γ(nb+1) θ n−1Γ(1+ra+nb)= (1−θ)Γ(1+ra)∑m≥0Γ(1+b+mb) θmΓ(1+b+ ra+mb)= (1−θ)Γ(1+ra)Γ(1+b)Γ(1+b+ ra) ∑m≥0(1+b)mb (1)m(1+b+ ra)mbθmm!and that the infinite sum in the last step corresponds to a Fox Wright generalizedhypergeometric function with ∆ = 0, ∇ = 1.(5) follows by noting thatE(X rKP)=beθ −1 ∑n≥1 B(1+ra,nb) θ n(n−1)!=b Γ(1+ra)θeθ −1 ∑m≥0Γ(b+mb)Γ(1+b+ ra+mb) θmm!=b Γ(b) Γ(1+ ra)θ(eθ −1)Γ(1+ra+b) ∑m≥0(b)mb(1+b+ ra)mbθmm!and that the infinite sum in the last step corresponds to a Fox Wright generalizedhypergeometric function with ∆ = 1.The proof of (6) is similar to the proof of (4).278 A note on “Double bounded Kumaraswamy-power series class of distributions”(7) follows by noting thatE(X rKB)=bθ(1+θ)m−1 ∑n≥1 n(mn)B(1+ra,nb)θ n−1=bθm(1+θ)m−1Γ(1+ra)∑n−1≥0(−1)n−1(1−m)n−1Γ(b+(n−1)b)Γ(1+b+ ra+(n−1)b) θ n−1(n−1)!=bθm(1+θ)m−1Γ(b)Γ(1+ra)Γ(1+b+ ra) ∑k≥0(−1)k(1−m)k(b)kb(1+b+ ra)kbθ kk!and that the infinite sum in the last step corresponds to a Fox Wright generalizedhypergeometric function with ∆ = 0, ∇ = 1.Proposition 2.3 For all |θ1|< 1 and |θ2|< 1,R(θ1,θ2) = ∑k≥0∑j≥0(k+1)θ k1θj2(k+ j+1)2(k+ j+2)=1(θ2−θ1)2[Li2 (θ1)−Li2 (θ2)+θ1−θ2+(2−θ2/θ1−θ2) log(1−θ1)− (1−θ2) log(1−θ2)]. (8)Proof: The double series for R(θ1,θ2) converges for all |θ1| < 1, |θ2| < 1, where it isterm-wise differentiable with respect both θ1, θ2. To obtain a closed form expression forR, we defineR(θ1,θ2) = ∑k≥0∑j≥0θ k1θj2(k+ j+1)2(k+ j+2).By the differentiability property,R =ddθ1(θ1 ·R(θ1,θ2)) . (9)Tibor K. Poga´ny and Saralees Nadarajah 279By partial fractions,R(θ1,θ2) = ∑k≥0∑j≥0(1k+ j+2 −1k+ j+1 +1(k+ j+1)2)θ k1 θj2 =: S1−S2 +S3. (10)By legitimate integration-summation order exchange and the definition of gammafunction,S1 =∫∞0e−2t ∑k≥0(θ1e−t)k ∑j≥0(θ2e−t) j dt = ∫ ∞0e−2t dt(1−θ1e−t)(1−θ2e−t) ,S2 =∫∞0e−t ∑k≥0(θ1e−t)k ∑j≥0(θ2e−t) j dt = ∫ ∞0e−t dt(1−θ1e−t)(1−θ2e−t) ,S3 =∫∞0te−t ∑k≥0(θ1e−t)k ∑j≥0(θ2e−t) j dt = ∫ ∞0t e−t dt(1−θ1e−t)(1−θ2e−t) .Routine but lengthy calculations show thatS1 =θ1 log(1−θ2)−θ2 log(1−θ1)θ1θ2 (θ1 −θ2),S2 =log(1−θ2)− log(1−θ1)θ1 −θ2.Using the fact∫∞0t e−t dt1−ae−t=1aLi2(a),we can reduceS3 =1θ2 −θ1∫∞0t e−t dt1−θ1e−t−1θ2−θ1∫∞0t e−t dt1−θ2e−t==1θ2 −θ1[1θ1Li2 (θ1)−1θ2Li2 (θ2)].280 A note on “Double bounded Kumaraswamy-power series class of distributions”Collecting S j, j = 1,2,3, we obtain by virtue of (10) thatR(θ1,θ2) =θ2 (1−θ1) log(1−θ1)−θ1 (1−θ2) log(1−θ2)θ1θ2 (θ2 −θ1)+1θ2−θ1[1θ1Li2 (θ1)−1θ2Li2 (θ2)]=(1−θ1) log(1−θ1)+Li2 (θ1)θ1 (θ2 −θ1)−(1−θ2) log(1−θ2)+Li2 (θ2)θ2 (θ2−θ1).The result follows by applying (9).AcknowledgmentsThe authors would like to thank the Editor and the two referees for careful reading andcomments which greatly improved this letter.ReferencesBidram, H. and Nekoukhou, V. (2013). Double bounded Kumaraswamy-power series class of distributions.SORT, 37, 211–230.Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J. (2006). Theory and Applications of Fractional DifferentialEquations, North-Holland Mathematical Studies, volume 204, Elsevier (North-Holland) SciencePublishers, Amsterdam, London and New York.

A note on "Double bounded Kumaraswamy-power series class of distributions"

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A note on "Double bounded Kumaraswamy-power series class of distributions"

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