Moment Generating Functions of Exponential-Truncated Negative Binomial Distribution based on Ordered Random Variables

Size: px

Start display at page:

Download "Moment Generating Functions of Exponential-Truncated Negative Binomial Distribution based on Ordered Random Variables"

Anne Willis
6 years ago
Views:

J. Stat. Appl. Pro. 3, No. 3, 413-423 2015 413 Jornal of Statistics Applications & Probability An International Jornal http://d.doi.org/10.

1 J. Stat. Appl. Pro. 3, No. 3, Jornal of Statistics Applications & Probability An International Jornal oment Generating Fnctions of Eponential-Trncated Negative Binomial Distribtion based on Ordered Random Variables Devendra Kmar Department of Statistics, Amity Institte of Applied Sciences, Amity University, Noida , India Received: 29 Apr. 2014, Revised: 29 Oct. 2014, Accepted: 30 Oct Pblished online: 1 Nov Abstract: In this paper, we established eplicit epressions and some recrrence relations for marginal and joint moment generating fnctions of generalized order statistics from eponential-trncated negative binomial distribtion. The reslts for record vales and order statistics are dedced from the reslt. Frther, a characterizing reslt of this distribtion on sing the conditional epectation of generalized order statistics is discssed. Keywords: Generalized order statistics; order statistics; record vales; eponential-trncated negative binomial distribtion; marginal and joint moment generating fnctions; recrrence relations and characterization. AS Sbject Classification: 62G30, 62E10 1 Introdction A random variable X is said to have eponential-trncated negative binomial distribtion if its probability density fnction pd f is of the form f= and the corresponding srvival fnction is F= 1 αλ α e λ 1 α [1 1 αe λ ] +1, >0, α, > 0 and λ > α 1 α [{1 1 αe λ } 1], >0, α, > 0 and λ > Some distribtions arise as special cases of the eponential-trncated negative binomial distribtion: i For = 1, we obtain the arshall-olkin eponential distribtion with scale parameter λ and shape parameter α. ii For = 1 and α = 2, we obtain the half-logistic distribtion with scale parameter λ. iii When α 0, eponential-trncated negative binomial distribtion redces to the eponential distribtion with scale parameter α. iv When 0 and 0 < α < 2, eponential-trncated negative binomial distribtion redces to the Eponential-Logarithmic distribtion with scale parameter λ and shape parameter 1 α, see Tahmasbi and Rezaei [16]. For more details and some applications of this distribtion one may refer to Nadarajah et al. [11]. Kamps [4] introdced and etensively stdied the generalized order statistics gos. The order statistics, seqential order statistics, Stigler s order statistics, record vales are special cases of gos. Sppose X1,n,m,k,...,Xn,m,m,k are n gos from an absoltely continos cmlative distribtion fnctioncd f F with the corresponding pd f f. Their joint pd f is n 1 k j=1 n 1 γ j i=1 Corresponding athor devendrastats@gmail.com [1 F i ] m f i [1 F n ] k 1 f n 1.3 Natral Sciences Pblishing Cor.

2 414 D. Kmar: oment Generating Fnctions of Eponential-Trncated... for F 1 0+< n > F 1 1, m 1, γ r = k+n rm+1>0, r = 1,2,...,n 1, k 1 and n is a positive integer. Choosing the parameters appropriately, models sch as ordinary order statistics γ i = n i+1; i=1,2,...,n i.e m 1 = m 2 =... = m n 1 = 0, k = 1, k th record vales γ i = k i.e m 1 = m 2 =... = m n 1 = 1, k N, seqential order statistics γ i = n i+1α; α 1,α 2,...,α n > 0, order statistics with non-integral sample size γ i = α i+1; α > 0, Pfeifer s record vales γ i = β i ; β 1,β 2,...,β n > 0 and progressive type II censored order statistics m i N, k N are obtained Kamps [4], Kamps and Cramer [5]. For simplicity we shall assme m 1 = m 2 =...=m n 1 = m. The marginal pd f of the r th gos, Xr, n, m, k, 1 r n, is and the joint pd f of Xr,n,m,k and Xs,n,m,k, 1 r< s n, is f Xr,n,m,k = C! [ F] γ fg m F 1.4 C s 1 f Xr,n,m,k,Xs,n,m,k,y=!s! [ F] m fg m F [h m Fy h m F] s [ Fy] γ s 1 fy, <y, 1.5 C = r i=1 γ i, F=1 F, γ i = k+n im+1, { h m = m m+1, m 1 ln1, m= 1 and g m =h m h m 1, [0,1. Ahsanllah and Raqab [1], Raqab and Ahsanllah [12], [13] have established recrrence relations for moment generating fnctions of record vales from Pareto and Gmble, power fnction and etreme vale distribtions. Recrrence relations for marginal and joint moment generating fnctions of gos from power fnction distribtion, Gompertz distribtion, Erlang-trncated eponential distribtion and etended type II generalized logistic distribtion are derived by Saran and Pandey [14], Khan et al. [6], Klshrestha et al. [7] and Kmar [10], respectively. Saran and Pandey [15] and Kmar [9] have established recrrence relations for marginal and joint moment generating fnctions of lower generalized order statistics from power fnction distribtion and arshall-olkin etended logistic distribtion respectively. Al-Hssaini et al. [2], [3] have established recrrence relations for conditional and joint moment generating fnctions of gos based on mied poplation. Kmar [8] have established eplicit epressions and some recrrence relations for moment generating fnctions of record vales from generalized logistic. The aim of this note is to gave eact epressions and some recrrence relations for marginal and joint moment generating fnctions of gos from eponential-trncated negative binomial distribtion. Reslts for order statistics and record vales are dedced as special cases and a characterization of this distribtion is obtained by sing the conditional epectation of fnction of gos. 2 Relations for marginal moment generating fnctions Note that for eponential-trncated negative binomial distribtion defined in 1.1 [ F = = {1 αe λ } 1] f. 2.1 The relation in 2.1 will be eploited in this paper to derive some recrrence relations for the moment generating fnctions of gos from the eponential-trncated negative binomial distribtion. Let s denote the marginal moment generating fnctions of Xr,n,m,k by Xr,n,m,k t and its j th derivative by j Xr,n,m,k t and the joint moment generating fnctions of Xr,n,m,k and Xs,n,m,k by Xr,n,m,kXs,n,m,kt 1,t 2 and Natral Sciences Pblishing Cor.

3 J. Stat. Appl. Pro. 3, No. 3, / i, j itsi, j th partial derivatives with respect to t 1 and t 2, respectively by Xr,n,m,kXs,n,m,k t 1,t 2. For the eponential-trncated negative binomial distribtion as given in 1.2, the moment generating fnctions of Xr,n,m,k is given as Xr,n,m,k t= e t f Xr,n,m,k d Frther, on sing the binomial epansion, we can rewrite 2.2 as = C e t [ F] γ fg m Fd. 2.2! Xr,n,m,k t= C!m+1 =0 1 Now letting z= F in 2.3, we get e t [ F] γr 1 fd. 2.3 Since Xr,n,m,k t= 1 αt/λ C!m+1 r q=0 =0 1 t/λp p/ q 1 α qb k α m+1 + n r++ q m+1,1, 2.4 b a=0 Ba,b is the complete beta fnction. Therefore, { αα+1...α+p 1, p>0 α p = 1,. 1 a b a Xr,n,m,k t= 1 αt/λ C m+1 r =1 α t/λ q=0 Γ Γ Ba+k,c= Bk,c+b 2.5 q=0 k+n rm+1+q m+1 k+nm+1+q m+1 t/λ p p/ q 1 α t/λ p p/ q 1 α α α q 1 r a=1 q 1+ q γa. 2.6 Special Cases i Ptting m = 0, k = 1, in 2.6, the eplicit formla for marginal moment generating fnctions of order statistics from the eponential-trncated negative binomial distribtion can be obtained as for r=1 Xr:n t= 1 αt/λ n! n r! q=0 X1:n t=n1 α t/λ t/λ p p/ q q=0 1 α α q Γn r+1+q, Γn+1+q t/λ p p/ q 1 α q. n+q α ii Setting m = 1 in 2.6, we get the eplicit epression for marginal moment generating fnctions of k th pper record vales from eponential-trncated negative binomial distribtion can be obtained as Natral Sciences Pblishing Cor.

4 416 D. Kmar: oment Generating Fnctions of Eponential-Trncated... Xr,n, 1,k t= q=0 t/λ p p/ q 1 α q 1 α t/λ α r, t q k A recrrence relation for marginal moment generating fnctions for gos from 2.1 can be obtained in the following theorem. Theorem 2.1. For the distribtion given in 1.1 and for 2 r n, n 2 k=1,2,..., 1 t j j t= γ Xr,n,m,k X,n,m,k t+ j j 1 r γ Xr,n,m,k t r Proof. From 1.4, we have 1 +1 γ r α 1 =2 { t j j 1 t+ λ 1+ j }. Xr,n,m,k Xr,n,m,k t+ λ Xr,n,m,k t= C e t [ F] γ fg m Fd. 2.8! Integrating by parts treating[ F] γ f for integration and rest of the integrand for differentiation, we get Xr,n,m,k t= X,n,m,k t+ tc γ r! e t [ F] γ r g m Fd, the constant of integration vanishes since the integral considered in 2.8 is a definite integral. On sing 2.1, we obtain 1 +1 γ r Xr,n,m,k t= X,n,m,k t+ t γ r Xr,n,m,k t = Differentiating both the sides of 2.9 j times with respect to t, we get 1 α 1 Xr,n,m,k t+ λ j j t= Xr,n,m,k t+ t j X,n,m,k γ Xr,n,m,k t+ j j 1 r γ Xr,n,m,k t r t +1 γ r =2 j +1 γ r = α 1 j Xr,n,m,k t+ λ 1 1 α 1 j 1 t+ λ 1. Xr,n,m,k The recrrence relation in eqation 2.7 is derived simply by rewriting the above eqation. By differentiating both sides of eqation 2.7 with respect to t and then setting t = 0, we obtain the recrrence relations for single moments of gos from eponential-trncated negative binomial distribtion in the form E[X j r,n,m,k]=e[x j,n,m,k]+ j γ r E[X j r,n,m,k] j +1 γ r = α 1 E[φXr,n,m,k], 2.10 φ= j 1 e λ 1. Remark 2.1: Ptting m=0, k= 1 in 2.7 and 2.10, we can get the relations for marginal moment generating fnctions and moments of order statistics for eponential-trncated negative binomial distribtion in the form t 1 n r+1 j X r:n t= j X :n t+ j n r+1 j 1 X r:n t Natral Sciences Pblishing Cor.

5 J. Stat. Appl. Pro. 3, No. 3, / and +1 1 n r α 1 =2 { } t j X r:n t+ λ 1+ j j 1 t+ λ 1 E[Xr:n]=E[X j j :n ]+ j j 1 E[X n r+1 r:n ] X r:n 1 n r+1 1 α 1 E[φX r:n ]. +1 = Remark 2.2: Setting m = 1 and k 1, in 2.7 and 2.10, relations for k th record vales can be obtained as and For k= 1 and { 1 t k t j Z r k j Z r k 1 +1 k =2 t= j t+ j Z k k j 1 Z r k 1 α 1 t } t+ λ 1+ j j 1 t+ λ 1 Z r k E[Z r k j ]=E[Z k j ]+ j k E[Zk r j 1 ] 1 +1 k = t j X Ur t= j X U t+ j j 1 X Ur t 1 +1 k = α 1 { t j X Ur t+ λ 1+ j j 1 X Ur t+ λ =2 E[X j Ur ]=E[X j j 1 ]+ je[x U Ur ] 1 α 1 E[φX Ur ]. 1 α 1 E[φZ r ]. Remark 2.3: i Ptting = 1 in 2.6 and 2.7, the reslts for marginal moment generating fnctions of gos are dedced for arshal-olkin eponential distribtion. ii Setting = 1, α = 2 in 2.6 and 2.7, we obtain the marginal moment generating fnctions of gos for the half-logistic distribtion. iii When α 1 in 2.6 and 2.7, we can get the reslts for marginal moment generating fnctions of gos for the eponential distribtion. iv If 1 and 0<α < 2 in 2.6 and 2.7, then we obtain relations for marginal moment generating fnctions of gos for the eponential-logarithmic distribtion. } 3 Relations for joint moment generating fnctions For eponential-trncated negative binomial distribtion, the joint moment generating fnctions of Xr, n, m, k and Xs,n,m,k is given as Xr,n,m,k,Xs,n,m,k t 1,t 2 = e t 1+t 2 y f Xr,n,m,kXs,n,m,k,yddy. Natral Sciences Pblishing Cor.

6 418 D. Kmar: oment Generating Fnctions of Eponential-Trncated... On sing 1.3 and binomial epansion, we have C s 1 Xr,n,m,k,Xs,n,m,k t 1,t 2 =!s!m+1 s 2 By setting z= Fy in 3.2, we obtain G=1 α t 2/λ =0 s v=0 1 +v s e v t1 [ F] s r+ vm+1 1 fgd, 3.1 G= e t2y [ Fy] γs v 1 fydy. 3.2 q=0 t 2 /λ p p/ q 1 α α q [ F] γs v+q γ s v + q. On sbstitting the above epression of G in 3.1, and simplifying the reslting eqation, we get Xr,n,m,k,Xs,n,m,k t 1,t 2 = 1 α α s v=0 On sing relation 2.5 in 3.3, we get q=0 l=0 w=0 q+w =0 1 α t 1+t 2 /λ C s 1!s!m+1 s t 2 /λ p p/ q t 1 /λ l l/ w l!w! 1 1 v s v Xr,n,m,k,Xs,n,m,k t 1,t 2 = q=0 l=0 w=0 k B m+1 Which after simplification yields k q+w B + n r++ m+1 m+1,1 k B m+1 + n s+v+ q m+1, α t 1+t 2 /λ C s 1!s!m+1 s t 2 /λ p p/ q t 1 /λ l l/ w l!w! + n r+ q+w m+1,r B Xr,n,m,k,Xs,n,m,k t 1,t 2 =1 α t 1+t 2 /λ t 2/λ p p/ q t 1 /λ l l/ w l!w! r a=1 1 α α q+w k m+1 + n s+ q m+1,s r q=0 l=0 w=0 1 α α q+w q+w γ a s b=r+1 1+ q γb Special Cases i Ptting m = 0, k = 1 in 3.4, the eplicit formla for joint moment generating fnctions of order statistics for the eponential-trncated negative binomial distribtion can be obtained as Xr:n X s:n t 1,t 2 = 1 αt 1+t 2 /λ n! n s! q=0 l=0 w=0 t 2 /λ p p/ q Natral Sciences Pblishing Cor.

7 J. Stat. Appl. Pro. 3, No. 3, / t 1/λ l l/ w l!w! 1 α α q+w Γn r+1+q+wγn s+1+q. Γn+1+q+wΓn r+1+q ii Setting m = 1 in 3.4, we dedce the eplicit epression for joint moment generating fnctions of k th pper record vales for eponential-trncated negative binomial distribtion in the form XUr:k X Us:k t 1,t 2 = q=0 l=0 w=0 t 2 /λ p p/ q t 1 /λ l l/ w l!w! 1 α q+w 1 α t1+t2/λ α r s r. 1+ q+w k aking se of 2.1, we can derive the recrrence relations for joint moment generating fnctions of gos from q k Theorem 3.1. For the distribtion given in 1.1 and for 1 r<s n, n 2 and k=1, t 2 γ s i, j Xr,n,m,kXs,n,m,k t 1,t 2 = j Xr,n,m,k,Xs 1,n,m,k t 1,t 2 + j i, j 1 γ Xr,n,m,kXs,n,m,k t 1,t 2 1 s γ s [ i, j t 2 +1 = Xr,n,m,kXs,n,m,k t 1,t 2 + λ 1 1 α 1 ] i, j 1 + j Xr,n,m,kXs,n,m,k t 1,t 2 + λ Proof: Using 1.5, the joint moment generating fnctions of Xr, n, m, k and Xs, n, m, k is given by Xr,n,m,k,Xs,n,m,k t 1,t 2 = C s 1!s! [ F] m fg m FId 3.6 I= e t 1+t 2 y [h m Fy h m F] s [ Fy] γs 1 fydy. Solving the integral in I by parts and sbstitting the reslting epression in 3.6, we get Xr,n,m,kXs,n,m,k t 1,t 2 = Xr,n,m,kXs 1,n,m,k t 1,t 2 t 2 C s 1 + γ s!s! e t 1+t 2 y [ F] m f g m F[h m Fy h m F] s [ Fy] γ s dyd, the constant of integration vanishes since the integral in I is a definite integral. On sing the relation 2.1, we obtain Xr,n,m,kXs,n,m,k t 1,t 2 = Xr,n,m,kXs 1,n,m,k t 1,t 2 + t 2 γ Xr,n,m,kXs,n,m,k t 1,t 2 t +1 2 s γ s = α 1 Xr,n,m,kXs,n,m,k t 1,t 2 + λ Differentiating both the sides of 3.7 i times with respect to t 1 and then j times with respect to t 2, we get i, j i, j Xr,n,m,kXs,n,m,k t 1,t 2 = Xr,n,m,kXs 1,n,m,k t 1,t 2 Natral Sciences Pblishing Cor.

8 420 D. Kmar: oment Generating Fnctions of Eponential-Trncated... t +1 2 γ s + t 2 i, j γ Xr,n,m,kXs,n,m,k t 1,t 2 + j i, j 1 s γ Xr,n,m,kXs,n,m,k t 1,t 2 s =2 j +1 γ s = α 1 i, j Xr,n,m,kXs,n,m,k t 1,t 2 + λ 1 1 α 1 i, j 1 Xr,n,m,kXs,n,m,k t 1,t 2 + λ 1, which, when rewritten gives the recrrence relation in 3.5. One can also note that Theorem 2.1 can be dedced from Theorem 3.1 by letting t 1 tends to zero. By differentiating both sides of eqation 3.5 with respect to t 1, t 2 and then setting t 1 = t 2 = 0, we obtain the recrrence relations for prodct moments of gos from eponential-trncated negative binomial distribtion in the form E[X i r,n,m,kx j s,n,m,k]=e[x i r,n,m,kx j s 1,n,m,k] + j E[X i r,n,m,kx j 1 s,n,m,k] j +1 γ s γ s = α 1 E[φXr,n,m,kXs,n,m,k], 3.8 φ,y= i y j 1 e λ 1y. Remark 3.1 Ptting m = 0, k = 1 in 3.5 and 3.8, we obtain the recrrence relations for joint moment generating fnctions and single moments of order statistics for eponential-trncated negative binomial distribtion as and t 2 i, j i, j 1 X n s+1 r,s:n t 1,t 2 = X r,s 1:n t 1,t 2 j j i, X n s+1 r,s:n t 1,t 2 n s+1 1 α 1[ i, j t 2 X r,s:n t 1,t 2 + λ 1+ j i, j E[X 1 n s+1 r,s:n]=e[x +1 =2 i, j r,s 1:n ]+ j n s+1 +1 =2 i, j 1 X r,s:n ] t 1,t 2 + λ 1 E[Xi, j 1 r,s:n ] α 1 E[φX r,s:n ]. Remark 3.2 Sbstitting m= 1 and k 1, in 3.5 and 3.8, we get recrrence relation for joint moment generating fnctions and prodct moments of k th pper record vales for eponential-trncated negative binomial distribtion. Remark 3.3: i Ptting = 1 in 3.4 and 3.5, the joint moment generating fnctions gos are dedced for arshall-olkin eponential distribtion. ii Setting = 1, α = 2 in 3.4 and 3.5, we obtain the relations for joint moment generating fnctions of gos for the half-logistic distribtion. iii When α 1 in 3.4 and 3.5, we can get the relations for joint moment generating fnctions of for the eponential distribtion. iv If 1 and 0<α < 2 in 3.4 and 3.5, we can get the relations for joint moment generating fnctions of gos for the eponential distribtion. Natral Sciences Pblishing Cor.

9 J. Stat. Appl. Pro. 3, No. 3, / Characterization Let Xr,n,m,k, r= 1,2,...,n be gos, then from a continos poplation with cd f F and pd f f, then the conditional pd f of Xs,n,m,k given Xr,n,m,k=, 1 r< s n, in view of 1.4 and 1.5, is C s 1 f Xs,n,m,k Xr,n,m,k y = s!c [h mfy h m F] s [Fy] γ s 1 [ F] γ r+1 fy. <y 4.1 Theorem 4.1: Let X be a non-negative random variable having an absoltely continos distribtion fnction F with F0=0 and 0<F<1 for all >0, then if and only if Proof. From 4.1, we have E[e txs,n,m,k Xr,n,m,k=]=1 α t/λ [1 {1 1 αe λ } ] q s r j=1 q=0 t/λ p p/ q γr+ j, 4.2 γ r+ j + q F= α 1 α [{1 1 αe λ } 1], >0, α, > 0 and λ > 0 E[e txs,n,m,k Xr,n,m,k=]= e ty[ 1 By setting = Fy from 1.2 in 4.3, we obtain F Fy F E[e txs,n,m,k Xr,n,m,k=]= q=0 C s 1 s!c m+1 s m+1 ] s Fy F γs 1 fy dy. 4.3 F 1 α t/λ C s 1 s!c m+1 s t/λ p p/ q [1 {1 1 αe λ } ] q 1 γs+q 1 1 m+1 s d Again by setting t = m+1 in 4.4 and simplifying the reslting epression, we derive the relation given in 4.2. To prove sfficient part, we have from 4.1 and 4.2 H r =1 α t/λ C s 1 s!c m+1 s e ty [ F m+1 Fy m+1 ] s q=0 [ Fy] γ s 1 fydy=[ F] γ r+1 H r, 4.5 t/λ p p/ q [1 {1 1 αe λ } ] q s r Differentiating 4.5 both sides with respect to and rearranging the terms, we get C s 1 [ F] m f s r 2!C m+1 s r 2 j=1 e ty [ F m+1 Fy m+1 ] s r 2 γr+ j. γ r+ j + q Natral Sciences Pblishing Cor.

10 422 D. Kmar: oment Generating Fnctions of Eponential-Trncated... or Therefore, which proves that [ Fy] γ s 1 fydy=h r [ F] γ r+1 γ r+1 H r [ F] γ r+1 1 f γ r+1 H r+1 [ F] γ r+2+m f = H r [ F] γ r+1 + γ r+1 H r [ F] γ r+1 1 f. f F = H r γ r+1 [H r+1 H r ] = 1 αλ e λ [1 1 αe λ ] +1 [{1 1 αe λ } 1] F= α 1 α [{1 1 αe λ } 1], >0, α, > 0 and λ > 0 Remark 4.1. For m = 0, k = 1 and m = 1, k = 1, we obtain the characterization reslts of the eponential-trncated negative binomial distribtion based on order statistics and record vales, respectively. 5 Conclsions In this stdy some eplicit epressions and recrrence relations for marginal and joint moment generating fnctions of gos from the eponential-trncated negative binomial distribtion have been established. Frther, conditional epectation of gos has been tilized to obtain a characterization of the eponential-trncated negative binomial distribtion. Acknowledgment The athor is gratefl to the learned referees for giving valable comments that led to an improvement in the presentation of the stdy. References [1]Ahsanllah,. and Raqab,.Z., Recrrence relations for the moment generating fnctions of record vales from Pareto and Gmble distribtions, Stoch. odel. Appl., , [2]Al-Hssaini E.K., Ahmad, A.A. and Al-Kashif,.A., Recrrence relations for moment and conditional moment generating fnctions of generalized order statistics, etrika, , [3]Al-Hssaini E.K., Ahmad, A.A. and Al-Kashif,.A., Recrrence relations for joint moment generating fnctions of generalized order statistics based on mied poplation, J. Stat. Theory Appl., , [4]Kamps, U., A concept of generalized Order Statistics, J. Stat. Plann. Inference, , [5]Kamps, U. and Cramer, E., On distribtion of generalized order statistics, Statistics, , [6]Khan, R.U., Zia, B. and Athar, H., On moment generating fnctions of generalized order statistics from Gompertz distribtion and its characterization, J. Stat. Theory Appl., , [7]Klshrestha A., Khan, R.U. and Kmar, D., On moment generating fnctions of generalized order statistics from Erlang-trncated eponential distribtion, Open J. Statist., , [8]Kmar, D., Recrrence relations for marginal and joint moment generating fnctions of generalized logistic distribtion based on k th lower record vales and its characterization, ProbStats Foram, , [9]Kmar, D., Relations for marginal and joint moment generating fnctions of arshall-olkin etended logistic distribtion based on lower generalized order atistics and characterization, Amer. J. ath. anag. Sci., , [10]Kmar, D., On moment generating fnction of generalized order statistics from etended type II generalized Logistic distribtion, J. Stat. Theory Appl., , [11]Nadarajah, S., Jayakmar, K. and Ristic,.., A new family of lifetime models, J. Statist. Comp. Siml., , Natral Sciences Pblishing Cor.

11 J. Stat. Appl. Pro. 3, No. 3, / [12]Raqab,.Z. and Ahsanllah,., Relations for marginal and joint moment generating fnctions of record vales from power fnction distribtion, J. Appl. Statist. Sci., , [13]Raqab,.Z. and Ahsanllah,., On moment generating fnction of records from etreme vale distribtion, Pakistan J. Statist., , [14]Saran, J. and Pandey, A., Recrrence relations for marginal and joint moment generating fnctions of generalized order statistics from power fnction distribtion, etron, LXI 2003, [15]Saran, J. and Pandey, A., Recrrence relations for marginal and joint moment generating fnctions of dal generalized order statistics from power fnction distribtion, Pak. J. Statist., , [16]Tahmasbi, R. and Rezaei, S., A two-parameter lifetime distribtion with decreasing failre rate, Comptational Statistics and Data Analysis, , Natral Sciences Pblishing Cor.

The Complementary Exponential-Geometric Distribution Based On Generalized Order Statistics

Applied Mathematics E-Notes, 152015, 287-303 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ The Complementary Exponential-Geometric Distribution Based On Generalized