Characterizations of the Weibull-X and Burr XII Negative Binomial Families of Distributions

Transcription

1 Interntionl Journl of Sttistics nd Probbility; Vol. 4, No. 2; 2015 ISSN E-ISSN Published by Cndin Center of Science nd Eduction Chrcteriztions of the Weibull-X nd Burr XII Negtive Binomil Fmilies of Distributions M. Ahsnullh 1, A. Alztreh 2, I. Ghosh 3 & G. G. Hmedni 4 1 Deprtment of Mngement Sciences, Rider University, New Jersey, USA 2 Deprtment of Mthemtics, Nzrbyev University, Astn, KZ 3 Deprtment of Mthemtics nd Sttistics, Austin Pey Stte University, Tennessee, USA 4 Deprtment of Mthemtics, Sttistics nd Computer Science, Mrquette University, Wisconsin, USA Correspondence: Aymn Alztreh, Deprtment of Mthemtics, Nzrbyev University, Astn, KZ. Tel: E-mil: ymn.lztreh@nu.edu.kz Received: Mrch 18, 2015 Accepted: April 11, 2015 Online Published: April 23, 2015 doi: /ijsp.v4n2p113 URL: Abstrct In this pper, we estblish certin chrcteriztions of the Weibull-X fmily of distributions proposed by Alztreh et l. (2013) s well s of the Burr XII Negtive Binomil distribution, introduced by Rmos et l. (2015). These chrcteriztions re bsed on two truncted moments, hzrd rte function nd conditionl expecttion of functions of rndom vribles. 1. Introduction In designing stochstic model for prticulr modeling problem, n investigtor will be vitlly interested to know if their model fits the requirements of specific underlying probbility distribution.to this end, the investigtor will rely on the chrcteriztions of the selected distribution. Generlly speking, the problem of chrcterizing distribution is n importnt problem in vrious fields nd hs recently ttrcted the ttention of mny reserchers. Consequently, vrious chrcteriztion results hve been reported in the literture. These chrcteriztions hve been estblished in mny different directions. There exists plethor of studies deling with the chrcteriztions of the Weibull distribution in the literture. Scholz (1990) gives chrcteriztion in terms of the quntiles nd Kgn et l. (1999) proposes chrcteriztion bsed on the Fisher informtion nd censored experiments. For dditionl chrcteriztion results we refer the interested reder to Nigm nd El-Hwry (1996), Mohie El-Din et l. (1991) nd Chudhuri nd Chndr (1990). Mukherjee nd Roy (1987) del with four miscellneous chrcteriztions of two-prmeter Weibull distribution. Khn nd Beg (1987), Khn nd Abu-Slih (1988) study chrcteriztion of Weibull distribution through some conditionl moments. Ali (1997) shows tht Weibull distribution cn be chrcterized by the product moments of ny two order sttistics. For comprehensive list of Weibull distribution chrcteriztions, see Murthy et l. (2004) nd the references therein. This vst rry of Weibull distribution chrcteriztions justifies somewht the importnce of the chrcteriztions of the Weibull-X Fmily (WXF) of distributions, simply becuse of the fct tht WXF ugments the Weibull distribution in terms of its flexibility to model vrious observed phenomen. In this pper, we consider severl different pproches to chrcterize the WXF s well s the Burr XII Negtive Binomil (BXIINB) distributions. These chrcteriztions re bsed on: (i) simple reltionship between two truncted moments; (ii) conditionl expecttions of single function of the rndom vrible; (iii) hzrd function; (iv) other chrcteriztions. Needless to sy tht, these pproches re mrkedly different from those mentioned bove. We like to mention tht the chrcteriztion (i) which is expressed in terms of the rtio of truncted moments is stble in the sense of wek convergence. It lso serves s bridge between first order differentil eqution nd probbility nd it does not require closed form for the distribution function. The rest of the pper is orgnized in the following wy. In section 2, we present the WXF nd BXIINB fmilies of distributions. Section 3 dels with the chrcteriztions (i) (iv) mentioned bove. In section 4, we provide some concluding remrks. 113

2 2. The WXF nd BXIINB Fmilies of Distributions Let r(t) be the probbility density function (pd f ) of rndom vrible T [, b] nd let G(x) be the cumultive distribution function (cd f ). Assume the link function W( ) : (0, 1) [, b] stisfies the following conditions: (i) W( ) is differentible nd monotoniclly non-decresing, (ii) W(0) nd W(1) b. (1) Alztreh et l. (2013) defined the cdf of the T-X fmily of distributions by where W(x) stisfies the conditions in (1). F(x) = W(G(x)) r(t)dt, (2) If T (0, ), X is continuous rndom vrible nd W(x) = log(1 x), then the pd f corresponding to (2) is given by g(x) f (x) = 1 G(x) r( log [ 1 G(x) ] ) = hg (x) r ( H g (x) ), (3) g(x) where h g (x) = 1 G(x) nd H g(x) = log[1 G(x)] re the hzrd nd cumultive hzrd rte functions corresponding to the bseline pd f g(x) = d dxg (x), respectively. Let T hve Weibull distribution with prmeters c > 0 nd > 0, then from (3), the pdf of the Weibull-X fmily is given by f (x) = c [ ] h Hg (x) c 1 [ ] g(x) e Hg(x) c ; x S uppg, (4) where S uppg denotes the support of the rndom vrible X with cd f G. WOLG we ssume tht S uppg = (0, ). The cd f nd hzrd rte function corresponding to (4) re, respectively, given by F(x) = 1 e, (5) nd h f (x) = c [ ] h Hg (x) c 1 g(x). (6) For certin generl properties of the WXF, we refer the interested reder to Alztreh et l. (2013; 2014). Remrk 2.1. Recently, Ndrjh nd Roch (2014) hve developed the necessry code for the WXF including the density, distribution, quntile functions s well s rndom genertion. The R pckge nmed, Newdistns is freely vilble t Remrk 2.2. The WXF of distributions is closed under minimiztion, i.e., if X i, i = 1, 2 re independent nd follow WXF (c, i ), then min {X 1, X 2 } WXF. Proof. For ny x (0, ), P (min {X 1, X 2 } > x) = P (X 1 > x, X 2 > x ) = P (X 1 > x ) P (X 2 > x ) = exp ( [ H g (x) ( c 1 + )) c 2, which estblishes the fct tht min {X 1, X 2 } WXF. 114

3 Note: If 1 = 2 =, i.e. if X 1 nd X 2 re i.i.d.,then min {X 1, X 2 } WXF ( c, 2 1/c ). In modern times, there hs been lot of work in developing new clss of lifetime distributions vi compounding of some discrete distributions with n pproprite lifetime distributions. One such distribution rising from Burr XII nd Negtive Binomil by compounding hs been studied in detil by Rmos et l (2015). The pprent wide pplicbility of such models demnds further investigtion in terms of its chrcteriztion. The pd f nd cd f of the BXIINB proposed by Rmos et l. (2015) re given, respectively, by nd f (x) = sβck c ( x c ] k 1 { [ ( x ) c ] k } s 1 [ [1 (1 β) s 1 ] x c β 1 +, (7) ) F (x) = where, s, c, k ll positive nd β (0, 1) re prmeters. 3. Chrcteriztion Results { (1 β) s 1 β [ 1 + ( ) x c ] k } s [ (1 β) s ] 1, x > 0, (8) In this section, we present chrcteriztions of WXF nd BXIINB distributions vi severl subsections: (i) bsed on simple reltionship between two truncted moments; (ii) in terms of conditionl expecttions of single function of the rndom vrible; (iii) bsed on the hzrd function nd (iv) other chrcteriztions. 3.1 Chrcteriztions Bsed on Two Truncted Moments In this subsection we present chrcteriztions of WXF nd BXIINB distributions in terms of simple reltionship between two truncted moments. Our chrcteriztion results presented here will employ n interesting result due to Glänzel (1987) (Theorem 3.1 below). The dvntge of the chrcteriztions given here is tht, cd f F need not require to hve closed form nd is given in terms of n integrl whose integrnd depends on the solution of first order differentil eqution, which cn serve s bridge between probbility nd differentil eqution. Theorem 3.1. Let (Ω,, P) be given probbility spce nd let H = [, b] be n intervl for some < b ( =, b = might s well be llowed). Let X : Ω H be continuous rndom vrible with the distribution function F nd let q 1 nd q 2 be two rel functions defined on H such tht E [ q 1 (X) X x ] = E [ q 2 (X) X x ] η (x), x H, is defined with some rel function η. Assume tht q 1, q 2 C 1 (H), η C 2 (H) nd F is twice continuously differentible nd strictly monotone function on the set H. Finlly, ssume tht the eqution q 2 η = q 1 hs no rel solution in the interior of H. Then F is uniquely determined by the functions q 1, q 2 nd η, prticulrly F (x) = x C η (u) η (u) q 2 (u) q 1 (u) exp ( s (u)) du, where the function s is solution of the differentil eqution s = η q 2 df = 1. H η q 2 q 1 nd C is constnt, chosen to mke We like to mention tht this kind of chrcteriztion bsed on the rtio of truncted moments is stble in the sense of wek convergence, in prticulr, let us ssume tht there is sequence {X n } of rndom vribles with distribution functions {F n } such tht the functions q 1,n, q 2,n nd η n (n N) stisfy the conditions of Theorem 3.1 nd let q 1,n q 1, q 2,n q 2 for some continuously differentible rel functions q 1 nd q 2. Let, finlly, X be rndom vrible with distribution F. Under the condition tht q 1,n (X) nd q 2,n (X) re uniformly integrble nd the fmily {F n } is reltively compct, the sequence X n converges to X in distribution if nd only if η n converges to η, where η (x) = E [ q 1 (X) X x ] E [ q 2 (X) X x ]. 115

4 Remrk 3.2. () In Theorem 3.1, the intervl H need not be closed since the condition is only on the interior of H. (b) Clerly, Theorem 3.1 cn be stted in terms of two functions q 1 nd η by tking q 2 (x) 1, which will reduce the condition given in Theorem 3.1 to E [ q 1 (X) X x ] = η (x). However, dding n extr function will give lot more flexibility, s fr s its ppliction is concerned. Proposition 3.3. Let X : Ω (0, ) be continuous rndom vrible nd let q 1 (x) = q 2 (x) [ H g ] (x) c nd [ ] Hg(x) c q 2 (x) = 2e for x > 0. The pd f of X is (4) if nd only if the function η defined in Theorem 3.1 hs the form Proof. Let X hve pdf (4), then η (x) = 1 [ ] 2 + Hg (x) c, x > 0. nd nd finlly (1 F (x)) E [ q 2 (X) X x ] = e 2 (1 F (x)) E [ q 1 (X) X x ] = ( [ Hg (x), x > 0, ) e 2, x > 0, Conversely, if η is given s bove, then nd hence η (x) q 2 (x) q 1 (x) = e > 0, for x > 0. s η (x) q 2 (x) (x) = η (x) q 2 (x) q 1 (x) = 2c [ ] h Hg (x) c 1 g (x), x > 0, [ ] Hg (x) c s (x) = 2, x > 0. Now, in view of Theorem 3.1, X hs pdf (4). Corollry 3.4. Let X : Ω (0, ) be continuous rndom vrible nd let q 2 (x) be s in Proposition 3.3. The pdf of X is (4) if nd only if there exist functions q 1 nd η defined in Theorem 3.1 stisfying the differentil eqution η (x) q 2 (x) η (x) q 2 (x) q 1 (x) = 2c [ ] h Hg (x) c 1 g (x), x > 0. Remrk 3.5. (c) The generl solution of the differentil eqution in Corollry 3.4 is η (x) = e 2 c h g (x) [ Hg (x) 1 [ ] e Hg(x) c q 1 (x) dx + D, for x > 0, where D is constnt. One set of pproprite functions is given in Proposition 3.3 with D = 0. (d) Clerly there re other triplets of functions (q 1, q 2, η) stisfying the conditions of Theorem 3.1 We presented one such triplet in Proposition

5 { Proposition 3.6. Let X : Ω (0, ) be continuous rndom vrible nd let q 1 (x) = 1 β [ 1 + ( ) x c ] k } s+1 nd q 2 (x) = q 1 (x) [ 1 + ( x ) c ] 1 for x > 0. The pd f of X is (7) if nd only if the function η defined in Theorem 3.1 hs the form η (x) = k + 1 k [ ( x c ] 1 +, x > 0. ) Remrk 3.7. A Corollry nd Remrks similr to Corollry 3.4 nd Remrks 3.5 cn be stted for BXIINB s well. 3.2 Chrcteriztions Bsed on Conditionl Expecttion of Function of the Rndom Vrible In this subsection we employ single function ψ of X nd stte chrcteriztion results in terms of ψ (X). Proposition 3.8. Let X : Ω (, b) be continuous rndom vrible with cd f F. Let ψ (x) be differentible function on (, b) with lim x + ψ (x) = δ > 1 nd lim x b ψ (x) =. Then if nd only if E [ (ψ (X)) δ X x ] = δ (ψ (x)) δ 1, x (, b), (9) Proof. From (9), we hve ψ (x) = δ [ 1 + (F (x)) 1 δ 1 ] 1, x (, b). (10) x (ψ (u)) δ f (u) du = δ (ψ (x)) δ 1 F (x). Tking derivtives from both sides of the bove eqution, we rrive t from which we hve (ψ (X)) δ f (x) = δ { (δ 1) ψ (x) (ψ (x)) δ 2 F (x) + (ψ (x)) δ 1 f (x) }, { } f (x) = (δ 1) ψ (x) F (x) ψ (x) + ψ (x). ψ (x) δ Integrting both sides of this eqution from x to b nd using the condition lim x b ψ (x) =, we obtin (10). Proposition 3.9. Let X : Ω (, b) be continuous rndom vrible with cdf F. Let ψ 1 (x) be differentible function on (, b) with lim x + ψ 1 (x) = δ/2 > 1/2 nd lim x b ψ 1 (x) = δ. Then if nd only if E [ (ψ 1 (X)) δ X x ] = δ (ψ 1 (x)) δ 1, x (, b), (11) ψ 1 (x) = δ [ 1 + (1 F (x)) 1 δ 1 ] 1, x (, b). (12) Proof. Is similr to tht of Proposition 3.8. ( [ ] Remrk (e) Tking, e.g., (, b) = (0, ) nd ψ (x) = δ e Hg(x) c ) 1 1 δ 1, Proposition 3.8 gives [ [ ] chrcteriztion of WXF distribution. ( f ) Tking, e.g., (, b) = (0, ) nd ψ 1 (x) = δ 1 + e 1 Hg(x) c ] 1 δ 1, 117

6 Proposition 3.9 gives chrcteriztion of WXF distribution. (g) Similr conclusions cn be drwn for BXIINB distribution. 3.3 Chrcteriztions Bsed on Hzrd Function It is obvious tht the hzrd function of twice differentible distribution function stisfies the first order differentil eqution ζ F (x) ζ F (x) ζ F (x) = p (x), where p (x) is n pproprite integrble function. Although this differentil eqution hs n obvious form since f (x) f (x) = ζ F (x) ζ F (x) ζ F (x), (13) for mny univrite continuous distributions (13) seems to be the only differentil eqution in terms of the hzrd function. The gol of the chrcteriztion bsed on hzrd function is to estblish differentil eqution in terms of hzrd function, which hs s simple form s possible nd is not of the trivil form (13). For some generl fmilies of distributions this my not be possible. Here we present chrcteriztion of WXF distribution in terms of the hzrd function nd one for BXIINB for s = 1. Proposition Let X : Ω (0, ) be continuous rndom vrible with twice differentible cd f F. The rndom vrible X hs pd f (4) if nd only if its hzrd function ζ F (x) stisfies the differentil eqution ζ F (x) g (x) (g (x)) 1 ζ F (x) = c ( ) [ ] h g (x) Hg (x) c 2 { c 1 H 2 g (x) }, x > 0. (14) 1 G (x) Proof. If X hs pd f (4), then clerly (14) holds. Now, if (14) holds, then or d { (g (x)) 1 ζ F (x) } = d ( ) [ ] c 1 Hg (x) c 1 dx dx 1 G (x), 2 f (x) 1 F (x) = ζ F (x) = c [ ] h Hg (x) c 1 2 g (x). Integrting both sides of the bove eqution from 0 to x, we rrive t from which we hve [ Hg (x) log (1 F (x)) = 1 F (x) = e [ ] Hg(x) c 1. Proposition Let X : Ω (0, ) be continuous rndom vrible with twice differentible cd f F. The rndom vrible X hs pd f (7) for s = 1, if nd only if its hzrd function ζ F (x) stisfies the differentil eqution ζ F (x) (c 1) x 1 ζ F (x) = ck c x c 1 d [ 1 + dx ( x c ] 1 { [ ( x ) c ] k } 1 1 β 1 + ), x >

7 3.4 Other Chrcteriztions The first three chrcteriztions given below, hve ppered in Hmedni s (unpublished s of now) work. We stte them here for the ske of completeness nd pply them to obtin chrcteriztions of WXF nd BXIINB distributions. Proposition Let X : Ω (0, ) be continuous rndom vrible with cd f F nd pd f f. Let ψ (x) nd q (x) be two differentible functions on (0, ) such tht dt =. Then implies 0 q (t) ψ(t) q(t) E [ ψ (X) X x ] = q (x), x > 0, { x q } (t) F (x) = 1 exp 0 ψ (t) q (t) dt, x 0. Remrk (h) Tking, e.g., ψ (t) = 2q (x) nd q (x) = exp { [ H g ] (x) c }, Proposition 3.13 gives chrcteriztion of (4). (i) Clerly there re other pirs of functions (ψ (t), q (x)) which stisfy conditions of Proposition We used the one in (h) for the simplicity. Proposition Let X : Ω (0, ) be continuous rndom vrible with cd f F nd pd f f. Let ψ (x) nd q (x) be two differentible functions on (0, ) such tht ( ) ψ (t) q (t) q(t) > 0 nd ψ (t) q (t) q(t) dt =. Then implies E [ ψ (X) X x ] = ψ (x) q (x), x > 0, { x F (x) = 1 exp 0 ψ (t) q } (t) dt, x 0. q (t) Remrk ( j) Tking, e.g., ψ (t) = 2q (x) nd q (x) = exp { [ H g ] (x) c }, Proposition 3.15 gives chrcteriztion of (4). (k) Clerly there re other pirs of functions (ψ (t), q (x)) which stisfy conditions of Proposition We used the one in ( j) for the simplicity. Proposition Let X : Ω (, b) be continuous rndom vrible with cd f F nd pd f differentible function on (, b) such tht lim x + ψ (x) = 0. Then, for 0 < δ < 1 0 f. Let ψ (x) be if nd only if E [ ψ (X) X x ] = δ + (1 δ) ψ (x), x (, b), ψ (x) = 1 (1 F (x)) δ 1 δ, x (, b). Remrk Tking, e.g., (, b) = (0, ) nd ψ (x) = 1 of (4). Similr observtion cn be mde for BXIINB. ( e ) δ 1 δ, Proposition 3.17 gives chrcteriztion Kmps(1995) defined generlized order sttistics (gos) in items of their joint pd f. Order sttistics, record vlues nd progressive type II order sttistics re specil cses of gos. The rndom vribles X(1, n, m, k), X(2, n, m, k),..., X(n.n.m.k) from n bsolutely continuous distribution function F(x) with pd f f (x) re clled gos if their joint pd f f 1,2,...,n (x 1, x 2,..., x n ) is given s follows. f 1,2,...,n (x 1, x 2,..., x n ) = kπ n 1 j=1 j(1 F(x j )) m f (x j )(1 F(x n ) k 1 f (x n ), < x i <, i = 1, 2,..., n, where j = k + (n j)(m + 1), for ll j, 1 j n 1, k nd n re positive integers nd m 1.The mrginl pd f f r,n,m,k (x) of X(r, n, m, k) is given by f r,n,m,k (x) = c r Γ(r) (1 F(x)) r 1 (g m (F(x))) r 1, < x <, 119

8 where c r = Π r j=1 j nd g m (x) = { 1 m+1 (1 (1 x)m+1, 1 log(1 x), m = 1, 0 < x < 1. The conditionl pd f f r,n,m,k (x) of X(r + 1, n, m, k) X(rm, n, m, k) = x is given by f r,n,m,k (x) = r+1(1 F(y)) r+1 1 (1 F(x)) r+1 }. f (y), < x < y <. For vrious chrcteriztions of distributions by regressions of generlized order sttistics, see Beg et l. (2013). The following proposition gives chrcteriztions of generl clss of distributions of which WXF is member bsed on regression property of X(r + 1, n, m, k) X(r, n, m, k) = x. Proposition Let X : Ω (0, ) be continuous rndom vrible with strictly incresing cd f F nd corresponding pd f f. Let K (x) be strictly incresing function with lim x 0 + K (x) = 0 nd lim x K (x) =. Then, F (x) = 1 e K(x) if nd only if E [K (X (r + 1, n, m, k)) X (r, n, m, k) = x ] = K (x) + 1 r+1. (15) Proof. The conditionl pd f of X (r + 1, n, m, k) X (r, n, m, k) = x is given by f r+1,r,n,m,k (y x) = r+1 (1 F(y)) r+1 1 (1 F(x)) r+1 It is now esy to show tht for F (x) = 1 e K(x), eqution (15) holds. Conversely, suppose tht (15) holds, then for x > 0 we hve f (y). or x r+1 K(y) (1 F(y)) r+1 1 (1 F(x)) r+1 f (y)dy = K (x) + 1 r+1, x r+1 K(y)(1 F(y)) r+1 1 f (y)dy = (1 F(x)) r+1 Differentiting both sides of the bove eqution with respect to x,we obtin ( K (x) + 1 ). r+1 r+1 K(x)(1 F(x)) r+1 1 f (x) = ( r+1 f (x) (1 F(x)) r+1 1 K (x) + 1 ) r+1 +(1 F(x)) r+1 K (x). Upon simplifiction of the lst eqution, we rrive t f (x) 1 F(x)) = K (x). (16) Integrting both sides of eqution (16) from 0 to x nd using the condition lim x 0 + K (x) = 0, we obtin F (x) = 1 e K(x). Remrk Tking K (x) = [ H g ] (x) c, Proposition 3.19 provides chrcteriztion of WXF distribution. 120

9 4. Summry nd Conclusions Chrcteriztion of probbility distributions hve ppered from time to time in literture, most of the theorems del with chrcteristic functions or re bsed on independence of pir of suitble sttistics. Lukcs (1956) hs given n extensive bibliogrphy of such theorems. In modeling vrious type of lifetime dt, pplictions in the re of relibility, Weibull distribution plys n importnt role. However to ugment its flexibility in modeling ny type of dt, reserchers hve developed Weibull- X fmily of distributions, where X is ny bsolutely continuous rndom vrible. Similrly, the construction of the Burr XII negtive binomil rndom vrible (Rmos et l., 2015) ws motivted by the fct tht this new model subsumes s specil cses, some importnt lifetime distributions, such s log-logistic, Weibull, Preto (type II) nd Burr XII distributions. With the dvent of such n rmory of new fmilies of distributions, by compounding distribution with bseline lifetime distributions, it becomes necessry to develop new wys of chrcterizing such flexible fmilies of distributions. In this pper, two such flexible fmilies of distributions, nmely the Weibull-X fmily nd the Burr XII negtive binomil distribution hve been considered. It is our sincere hope tht these chrcteriztion results presented in this pper will motivte reserchers to find new venues for chrcteriztion nd construction of such flexible fmilies in more thn one dimension (bivrite nd multivrite models). References Ali, M. A. (1997). Chrcteriztion of Weibull nd exponentil distributions. Journl of Sttisticl Studies, 17, Alztreh, A., Lee, C., & Fmoye, F. (2013). A new method for generting fmilies of continuous distributions. Metron, 71, Alztreh, A., & Ghosh, I. (2014). On the Weibull-X fmily of distributions. Theory nd Applictions. Accepted in Journl of Sttisticl Chudhuri, A., & Chndr, N. K. (1990). On testimting the Weibull shpe prmeter. Communictions in Sttistics-Computtion nd Simultion, 19, Glänzel, W. (1987). A chrcteriztion theorem bsed on truncted moments nd its ppliction to some distribution fmilies, Mthemticl Sttistics nd Probbility Theory (Bd Ttzmnnsdorf, 1986), Vol. B, Reidel, Dordrecht, Glnzel, W., & Hmedni, G. G. (2001). Chrcteriztions of univrite continuous distributions. Studi Scientirum Mthemticrum Hungric, 37, Hmedni, G. G. (2002). Chrcteriztions of univrite continuous distributions. Studi Scientirum Mthemticrum Hungric, 39, Hmedni, G. G. (2006). Chrcteriztions of univrite continuous distributions. Studi Scientirum Mthemticrum Hungric, 43, Hmedni, G. G. (2014). Chrcteriztions of distributions of rtio of certin independent rndom vribles. Journl of Applied Mthemtics, Sttistics nd Informtion, 9, Kmps, U. (1995). A Concept of generlized order sttistics, Teubner, Stuttgrt, Germny. Kgn, A., & Gertsbkh, I. (1999) Chrcteriztion of the Weibull distribution by properties of the Fisher informtion under type-i censoring. Sttistics nd Probbility Letters, 42, Khn, A. H., & Beg, M. I. (1987). Chrcteriztion of the Weibull distribution by conditionl vrince. Snkhy, Series A, 49, Khn, A. H., & Abu-Slih, M. S. (1988). Chrcteriztion of the Weibull nd inverse Weibull distributions through conditionl moments. Journl of Inform Optim Sci., 9, Lukcs, E. (1956). Chrcteriztion of Popultions by Properties of Suitble Sttistics, Proc. Third Berkeley Symp. on Mth. Sttist. nd Prob., Vol. 2 (Univ. of Clif. Press), Mukherjee, S. P., & Roy, D. (1987). Filure rte trnsform of the Weibull vrite nd its properties. Communictions in Sttistics- Theory Methods, 16,

10 Mohie El-Din, M. M., Mhmoud, M. A. W., & Youssef, S. E. A. (1991). Moments of order sttistics from prbolic nd skewed distributions nd chrcteriztion of Weibull distribution. Communictions in Sttistics- Computtion nd Simultion, 20, Murty, D. N. P, Xie, M., & Jing, R. (2004). Weibull Models, Wiley Series in Probbility nd Sttistics, John Wiley, New York. Ndrjh, S., & Roch, Ri. (2014). An R pckge for new fmilies of distributions. Journl of Sttisticl Softwre. Nigm, E. M., & EL-Hwry, H. M. (1996). On order Sttistics from Weibull Distribution. ISSR. Ciro University, 40, Rmos, M. W. A., Percontini, A., Cordeiro, G. M., & d Silv, R. V. (2015). The Burr XII Negtive Binomil Distribution with Applictions to Lifetime Dt. Interntionl Journl of Sttistics nd Probbility, 4, Scholz, F. W. (1990). Chrcteriztion of the Weibull distribution. Computtionl Sttistics nd Dt Anlysis, 10, Copyrights Copyright for this rticle is retined by the uthor(s), with first publiction rights grnted to the journl. This is n open-ccess rticle distributed under the terms nd conditions of the Cretive Commons Attribution license ( 122