Marquette Univerity e-publication@marquette Mathematic, Statitic and Computer Science Faculty Reearch and Publication Mathematic, Statitic and Computer Science, Department of 6-1-2014 Beta Burr XII OR Five Parameter Beta Lomax Ditribution: Remark and Characterization Z. Javanhiri Ferdowi Univerity of Mahhad Mehdi Maadooliat Marquette Univerity, mehdi.maadooliat@marquette.edu Publihed verion. Journal of Statitical Theory and Application, Vol. 13, No. 2 ( June 2014): 105-110. DOI. 2014 The Author. Ued with permiion.
Journal of Statitical Theory and Application, Vol. 13, No. 2 (June 2014), 105-110 Beta Burr XII OR Five Parameter Beta Lomax Ditribution: Remark and Characterization Z. Javanhiri Department of Statitic, Ferdowi Univerity of Mahhad P. O. Box 91775-1159, Mahhad, Iran zo ja15@um.ac.ir M. Maadooliat Department of Mathematic, Statitic and Computer Science Marquette Univerity, Milwaukee, WI 53201-1881, USA mehdi@mc.mu.edu The ditribution taken up in two recently publihed paper are compared and certain characterization of them are preented. Thee characterization are baed on: (i) a imple relationhip between two truncated moment; (ii) truncated moment of certain function of the n th order tatitic; (iii) truncated moment of certain function of the random variable. Keyword: Beta Burr XII; Beta Lomax; Characterization. 1. Introduction In a paper entitled On Five Parameter Beta Lomax Ditribution by Rajab et al. [publihed, March, 2014], the author preent the pdf (probability denity function) and cdf (cumulative ditribution function) of a ditribution which they call Five Parameter Beta Lomax a follow: k f (x) = f (x;a,b, µ,,k) = B(a,b) { 1 [ 1 + [ 1 + ( )] x µ (kb+1) ( )] } x µ k a 1, x > µ, (1.1) Correponding author. E-mail: mehdi@mc.mu.edu. Publihed by Atlanti Pre Copyright: the author 105
Z. Javanhiri and M. Maadooliat and k F (x) = F (x;a,b, µ,,k) = B(a,b) { 1 a 1 i=0 [ 1 + ( ) ( 1) i a 1 1 i kb + ik ( )] } x µ (kb+ik), (1.2) where a, b,, k are all poitive parameter, µ i a hift parameter and B(α, β) = 1 0 xα 1 (1 x) β 1 dx. The author tate that their ditribution i new and invetigate ome of it propertie. Thi ditribution, however, i a pecial cae of the one introduced by Paranaíba et al. (2011), which they (Paranaíba et al.) call it The beta Burr XII ditribution with the pdf and cdf given, repectively, by and ck ) c ] (kb+1) f (x) = f (x;a,b,c,,k) = c B(a,b) xc 1 1 + { ) c ] } k a 1 1 1 +, x > 0, (1.3) F (x) = F (x;a,b,c,,k) = I 1 [1+( x ) c k (a,b) ] = 1 1 [1+( x ) c ] k ω a 1 (1 ω) b 1 dω, (1.4) B(a,b) 0 x 0, where a, b, c,, k are all poitive parameter. Letting c = 1 and introducing a hift parameter µ, (1.3) reduce to (1.1). Both paper have certain formula for Reliability and Hazard function of their repective ditribution. Rajab et al. obtain an expreion for the r th moment of the random variable with pdf (1.1) for the pecial cae of a = 1. Thi pecial cae, clearly, make the computation quite eay and traightforward. On the other hand, Paranaíba et al. preent (a) everal important propertie of the random variable with pdf (1.3); (b) derive it moment generating function; (c) dicu other meaure, uch a: mean deviation, Bonferroni and Lorenz curve; (d) order tatitic and moment; (e) maximum likelihood etimation; ( f ) imulation tudy; (g) a Bayeian analyi; and (h) an application of their ditribution. In the next ection we preent certain characterization of Beta Burr XII (BBXII) ditribution. Clearly, thee characterization will hold if the hift parameter µ i introduced a well. 2. Characterization In deigning a tochatic model for a particular modeling problem, an invetigator will be vitally intereted to know if their model fit the requirement of a pecific underlying probability ditribution. To thi end, the invetigator will rely on the characterization of the elected ditribution. Generally peaking, the problem of characterizing a ditribution i an important problem in variou Publihed by Atlanti Pre Copyright: the author 106
Beta Burr XII: Remark and Characterization field and ha recently attracted the attention of many reearcher. Conequently, variou characterization reult have been reported in the literature. Thee characterization have been etablihed in many different direction. The preent work deal with the characterization of (BBXII) ditribution. Thee characterization are baed on: (i) a imple relationhip between two truncated moment; (ii) truncated moment of certain function of the n th order tatitic; (iii) truncated moment of certain function of the random variable. 2.1. Characterization baed on two truncated moment In thi ubection we preent characterization of (BBXII) ditribution in term of a imple relationhip between two truncated moment. We like to mention here the work of Glänzel (1987, 1990), Glänzel and Hamedani (2001) and Hamedani (2002, 2006, 2010), among other, in thi direction. A in Hamedani work, our characterization reult preented here will employ the reult due to Glänzel (1987) (Theorem 2.1 below). Following Hamedani, we like to mention that the advantage of thi characterization i two fold: it relate the cdf or pdf of a ditribution to the olution of a firt order differential equation and doe not require that the cdf to have a imple format. Theorem 2.1. Let (Ω,F,P) be a given probability pace and let H = [a,b] be an interval for ome a < b (a =, b = might a well be allowed). Let X : Ω H be a continuou random variable with the ditribution function F and let g and h be two real function defined on H uch that E[g(X) X x] = E[h(X) X x]η (x), x H, i defined with ome real function η. Aume that g, h C 1 (H), η C 2 (H) and F i twice continuouly differentiable and trictly monotone function on the et H. Finally, aume that the equation hη = g ha no real olution in the interior of H. Then F i uniquely determined by the function g, h and η, particularly x F (x) = C η (u) a η (u)h(u) g(u) exp( (u)) du, where the function i a olution of the differential equation = η h η h g and C i a contant, choen to make H df = 1. Again following Glänzel and Hamedani (2001), we like to mention that thi kind of characterization baed on the ratio of truncated moment i table in the ene of weak convergence, in particular, let u aume that there i a equence {X n } of random variable with ditribution function {F n } uch that the function g n, h n and η n (n N) atify the condition of Theorem 2.1 and let g n g, h n h for ome continuouly differentiable real function g and h. Let, finally, X be a random variable with ditribution F. Under the condition that g n (X) and h n (X) are uniformly integrable and the family {F n } i relatively compact, the equence X n converge to X in ditribution if and only if η n converge to η, where η (x) = E [g(x) X x] E [h(x) X x]. Thi tability theorem make ure that the convergence of ditribution function i reflected by correponding convergence of the function g, h and η, repectively. It guarantee, for intance, the Publihed by Atlanti Pre Copyright: the author 107
Z. Javanhiri and M. Maadooliat convergence of characterization of the Wald ditribution to that of the Lévy-Smirnov ditribution if α. A further conequence of the tability property of Theorem 2.1 i the application of thi theorem to pecial tak in tatitical practice uch a the etimation of the parameter of dicrete ditribution. For uch purpoe, the function g, h and, pecially, η hould be a imple a poible. Since the function triplet i not uniquely determined it i often poible to chooe η a a linear function. Therefore, it i worth analyzing ome pecial cae which help to find new characterization reflecting the relationhip between individual continuou univariate ditribution and appropriate in other area of tatitic. Propoition 2.1. Let X : Ω (0, ) be a continuou random variable and let h(x) = { 1 [ 1 + ( ) x c ] } k 1 a [1 ( + x ) c ] { kb and g(x) = 1 [ 1 + ( ) x c ] } k 1 a, for x (0, ). The pdf of X i (1.3) if and only if the function η defined in Theorem 2.1 ha the form ) c ] kb η (x) = 2 1 + for x > 0. Proof. Let X have denity (1.3), then (1 F (x)) E[h(X) X x] = 1 ) c ] 2kb 1 +, x > 0, 2bB(α,β) and (1 F (x)) E[g(X) X x] = 1 ) c ] kb 1 +, x > 0, bb(α,β) and finally { ) c ] } k 1 a η (x)h(x) g(x) = 1 1 + > 0 for x > 0. Converely, if η i given a above, then (x) = η (x)h(x) η (x) h(x) g(x) = 2bck ) c ] 1 c x c 1 1 +, x > 0, and hence ) c ] } 2bk (x) = ln{ 1 +, x > 0. Now, in view of Theorem 2.1, X ha denity (1.3). Corollary 2.1. Let X : Ω (0, ) be a continuou random variable and let h(x) be a in Propoition 2.1. The pdf of X i (1.3) if and only if there exit function g and η defined in Theorem 2.1 Publihed by Atlanti Pre Copyright: the author 108
Beta Burr XII: Remark and Characterization atifying the differential equation η (x)h(x) η (x)h(x) g(x) = 2bck ) c ] 1 c x c 1 1 +, x > 0. Remark 2.1. (a) The general olution of the differential equation in Corollary 2.1 i ) c ] [ 2bk η (x) = 1 + g(x) 2bck ) c ] bk 1 c x c 1 1 + { ) c ] } ] k a 1 1 1 + dx + D, for x > 0, where D i a contant. One et of appropriate function i given in Propoition 2.1 with D = 0. (b) Clearly there are other triplet of function (h, g, η) atifying the condition of Theorem 2.1. We preented one uch triplet in Propoition 2.1. 2.2. Characterization baed on truncated moment of certain function of the n th order tatitic We preent here a characterization of (BBXII) ditribution baed on ome function of the n th order tatitic. Our characterization will be a conequence of the following propoition, which i imilar to the one appeared in Hamedani (2010). Propoition 2.2. Let X : Ω (0, ) be a continuou random variable with cdf F. Let ψ (x) and q(x) be two differentiable function on (0, ) uch that lim x 0 ψ (x)[f (x)] n = 0 and q (t) 0 [ψ(t) q(t)] dt =. Then E [ψ (X n:n ) X n:n < t] = q(t), t > 0, (2.1) implie { q } (t) F (x) = exp x n[ψ (t) q(t)] dt, x 0. (2.2) { } 1 [1+( x Taking, e.g., ψ (x) = 2 ) c ] k n 0 ω a 1 (1 ω) b 1 dω and q(x) = 1 2ψ (x), (2.2) will reult in (1.4). Clearly, there are other choice for thee function a well. Remark 2.2. We like to point out that Propoition 2.2 hold true (with of coure appropriate modification) if we replace X n:n with the bae random variable X. 2.3. Characterization baed on ingle truncated moment of certain function of the random variable In thi ubection we employ a ingle function ψ 1 of X and characterize the ditribution of X in term of the truncated moment of ψ 1 (X). The following propoition i imilar to the one which ha Publihed by Atlanti Pre Copyright: the author 109
Z. Javanhiri and M. Maadooliat already appeared in a Technical Report (Hamedani, 2013), o we will jut tate it here which can be ued to characterize (BBXII) ditribution. Propoition 2.3. Let X : Ω (0, ) be a continuou random variable with cdf F. Let ψ 1 (x) be a differentiable function on (0, ) with lim x ψ 1 (x) = 1. Then for δ 1 1, if and only if E [ψ 1 (X) X < x] = δ 1 ψ 1 (x), x > 0, ψ 1 (x) = (F (x)) 1 δ 1 1. x 0. Remark 2.3. For a uitable choice of ψ 1 (x), Propoition 2.3 provide a characterization of (BBXII) ditribution. Acknowledgment The author would like to thank an anonymou referee for their comment. We are pecially grateful to the Editor, Profeor Hamedani, for hi uggetion regarding the important reference, hi guidance in the preentation of the reult and for allowing u to quote a few tatement from hi work. Reference [1] Glänzel, W., A characterization theorem baed on truncated moment and it application to ome ditribution familie, Mathematical Statitic and Probability Theory (Bad Tatzmanndorf, 1986), Vol. B, Reidel, Dordrecht, (1987) 75 84. [2] Glänzel, W., Some conequence of a characterization theorem baed on truncated moment, Statitic, 21 (1990) 613 618. [3] Glänzel, W. and Hamedani, G.G., Characterization of univariate continuou ditribution, Studia Sci. Math. Hungar., 37 (2001) 83 118. [4] Hamedani, G.G., Characterization of univariate continuou ditribution. II, Studia Sci. Math. Hungar., 39 (2002) 407 424. [5] Hamedani, G.G., Characterization of univariate continuou ditribution. III, Studia Sci. Math. Hungar., 43 (2006) 361 385. [6] Hamedani, G.G., Characterization of continuou univariate ditribution baed on the truncated moment of function of order tatitic, Studia Sci. Math. Hungar., 47 (2010) 462 484. [7] Hamedani, G.G., On certain generalized gamma convolution ditribution II, Technical Report No. 484, MSCS, Marquette Univerity, (2013). [8] Paranaíba, P.F., Ortega, E.M.M., Cordeiro, G.M. and Pecim, R.R., The beta Burr XII ditribution with application to lifetime data, Computational Statitic and Data Analyi, 55 (2011) 1118-36. [9] Rajab, M., Aleem, M., Nawaz, T. and Daniyal, M., On five parameter beta Lomax ditribution, J. of Statitic, 20 (2013) 102-118. Publihed by Atlanti Pre Copyright: the author 110