Tamsui Oxford Journal of Information and Mathematical Sciences 29(2) (2013) Aletheia University

Transcription

1 Tamsui Oxford Journal of Information and Mathematical Sciences Aletheia University Relations for Marginal and Joint Moment Generating Functions of Extended Type I Generalized Logistic Distribution based on Lower Generalized Order Statistics and Characterization Devendra Kumar Department of Statistics, Amity Institute of Applied Sciences, Amity University, Noida , India Received March 20, 2012, Accepted November 23, Abstract In this study we give exact expressions and some recurrence relations for marginal and joint moment generating functions of lower generalized order statistics from extended type I generalized logistic distribution. The results for order statistics and lower record values are deduced from the relations derived. Further two characterization Theorems of this distribution has been considered on using conditional expectation and recurrence relations for marginal moment generating functions of the lower generalized order statistics is presented. Keywords and Phrases: Lower generalized order statistics, Order statistics, Lower record values, Extended type I generalized logistic distribution, Marginal and joint moment generating function, Recurrence relations and characterization Mathematics Subject Classification. Primary 62G30, 62E10. devendrastats@gmail.com

2 220 Devendra Kumar 1. Introduction Kamps [18] introduced the concept of generalized order statistics gos. It is know that order statistics, upper record values and sequential order statistics are special cases of gos. In this paper we will consider the lower generalized order statistics lgos. Which is given as Let n N, k 1, m R, be the parameters such that γ r = k + n rm + 1> 0, for all 1 r n. Then 1, n, m, k,..., n, n, m, k are n lgos from an absolutely continuous distribution function df F x with the corresponding probability density function pdf fx if their joint pdf has the form k n 1 n 1 γ j [F x i ] m i fx i [F x n ] k 1 fx n 1.1 j=1 i=1 for F 1 1 > x 1 x 2... x n > F 1 0. The marginal pdf of the r th lgos, r, n, m, k is f r,n,m,kx = C r 1 r 1! [F x]γr 1 fxg r 1 m F x 1.2 and the joint pdf of r, n, m, k and s, n, m, k, 1 r < s n is expressed from 1.1 as f r,n,m,k, s,n,m,kx, y = C s 1 r 1!s r 1! [F x]m fxgm r 1 F x where and [h m F y h m F x] s r 1 [F y] γs 1 fy, x > y, 1.3 C r 1 = r γ i, γ i = k + n im + 1, i=1 h m x = { 1 m+1 xm+1, m 1 lnx, m= 1 g m x = h m x h m 1, x [0, 1.

3 Relations for Marginal and Joint Moment Generating Functions 221 We shall also take 0, n, m, k = 0. If m = 0, k = 1, then r, n, m, k reduced to the n r + 1 th order statistics, n r+1:n from the sample 1, 2,..., n and when m = 1, then r, n, m, k reduced to the r th lower k record value Pawlas and Szynal, [14]. The work of Burkschat et al. [11] may also refer for lower generalized order statistics. Saran and Pandey [7] and Khan et al. [16] have established recurrence relations for marginal and joint moment generating functions of lgos from power function and generalized exponential distributions. Ahsanullah and Raqab [10], Raqab and Ahsanullah [12, 13] and Kumar [3] have established recurrence relations for moment generating functions of record values from Pareto, Gumble, power function, extreme value and generalized logistic distributions. Recurrence relations for marginal and joint moment generating functions of from power function and Erlange-truncated exponential distribution are derived by Saran and Singh [8] and Kumar et al. [4]. Al-Hussaini et al. [5, 6] have established recurrence relations for conditional and joint moment generating functions of gos based on mixed population, respectively. Khan et al. [15] have established explicit expressions and some recurrence relations for moment generating function of generalized order statistics from Gompertz distribution. Kamps [19] investigated the importance of recurrence relations of order statistics in characterization. In the present study, we have established exact expressions and some recurrence relations for marginal and joint moment generating functions of lgos from extended type I generalized logistic distribution. Results for order statistics and lower record values are deduced as special cases and a characterization of extended type I generalized logistic distribution has been obtained on using conditional expectation and recurrence relation for marginal moment generating functions. A random variable is said to have extended type I generalized logistic distribution if its pdf is of the form fx = and the corresponding df is αα e x, < x <, α, > e x α+1 α, F x = < x <, α, > e x

4 222 Devendra Kumar When α = = 1, we have the ordinary logistic distribution and when = 1, we have the type I generalized logistic distribution. For more details on this distribution and its applications one may refer to Olapade [1, 2]. 2. Relations for Marginal Moment Generating Function Note that for extended type I generalized logistic distribution defined in 1.4 αf x = 1 + e x fx. 2.1 The relation in 2.1 will be used to derive some recurrence relations for the moment generating function of lgos from extended type I generalized logistic distribution. Let us denote the marginal moment generating functions of r, n, m, k by M r,n,m,kt and its j th derivative by M j r,n,m,k t. We shall first establish the explicit expressions for M r,n,m,kt. Using 1.2, we obtain when m 1 where M r,n,m,kt = I j a, b = C r 1 r 1! e tx [F x] γr 1 fxgm r 1 F xdx = C r 1 r 1! I jγ r 1, r e tx [F x] a fxg b mf xdx 2.3 Further, on using the binomial expansion, we can rewrite 2.3 as I j a, b = 1 m + 1 b b b 1 u e tx [F x] a+m+1u fxdx 2.4 u u=0 Making the substitution z = [F x] 1/α, 2.4 reduces to I j a, b = α t m + 1 b b b 1 u u u=0 1 1 z t z αa+m+1u+1+t 1 dz. 0

5 Relations for Marginal and Joint Moment Generating Functions 223 On using the Maclaurine series expansion 1 z t = t p z p, p! where t p = { tt+1...t+p 1, p=1,2,... 1, and integrating the resulting expression, we obtain α b I j a, b = t m + 1 b 1 u b t p u p![αa + m + 1u t + p], t 0 u=0 and when m = 1 that I j a, b = 0 b b 0, as 1 u = 0. u Since I j a, b is of the form 0 at m = 1, therefore we have 0 α b b I j a, b = 1 u t p t m + 1 b u p![αa + m + 1u t + p]. u=0 u= Differentiating numerator and denominator of 2.6 b times with respect to m, we get I j a, b = α t u=0 b b 1 u+b u On applying L Hospital rule, we have lim I ja, b = αb+1 m 1 t t p p![αa t + p] b+1 t p α b u b p![αa + m + 1u t + p] b+1. b b 1 u+b u u=0 u b, b > 0. But for all integers n 0 and for all real numbers x, we have Ruiz [17] n n 1 i x i n = n!. 2.7 i i=0 Therefore, b b 1 u+b u u=0 u b = b!.

6 224 Devendra Kumar Hence lim I ja, b = b!αb+1 m 1 t t p. 2.8 p![αa t + p] b+1 Now substituting the above expressions for I j γ r 1, r 1 in 2.2 and simplifying, we obtain when m 1 that M r,n,m,kt = and when m = 1 that αc r 1 t r 1!m + 1 r 1 r 1 1 u r 1 u u=0 t p p!αγ r u + t + p. 2.9 M r,n, 1,kt = αkr t t p, t p!αk + t + p r Applying D Alembert s Ratio test for convergence, it can easily be seen that M r,n, 1,kt exist t, < t < and is analytic in t. Differentiating 2.9 and 2.10 and evaluating at t = 0, we get the mean of the r th lgos. Special cases i Putting m = 0, k = 1 in 2.9, the explicit formula for marginal moment generating functions of order statistics of the extended type I generalized logistic distribution can be obtained as M n r+1:n t = αc r:n t r 1 u=0 1 u r 1 u t p p![αn r u + t + p]. That is where M r:n t = αc r:n t n r u=0 1 u n r u t p p![αr + u + t + p], C r:n = n! r 1!n r!.

7 Relations for Marginal and Joint Moment Generating Functions 225 ii Setting k = 1 in 2.10, we get the explicit expression for marginal moment generating function of lower record values from extended type I generalized logistic distribution can be obtained as M Lr t = αr t t p p![α + t + p] r, t 0. A recurrence relation for marginal moment generating function for lgos from df 1.5 can be obtained in the following theorem. Theorem 2.1. For the distribution given in 1.5 and for 2 r n, n 2 and k = 1, t M j j αγ r,n,m,kt = M r 1,n,m,k t j M j 1 r αγ r,n,m,k t r { tm j j 1 αγ r,n,m,kt jm r,n,m,k. t r Proof. From 1.2, We have M r,n,m,kt = C r 1 r 1! e tx [F x] γr 1 fxg r 1 m F xdx Integrating by parts treating [F x] γr 1 fx for integration and rest of the integrand for differentiation, we get M r,n,m,kt = M r 1,n,m,kt tc r 1 e tx [F x] γr gm r 1 F xdx, γ r r 1! the constant of integration vanishes since the integral considered in 2.12 is a definite integral. On using 2.1, we obtain M r,n,m,kt = M r 1,n,m,kt tc r 1 γ r r 1! { 1 + e e tx [F x] γr 1 x fx α g r 1 m F xdx = M r 1,n,m,kt t αγ r { M r,n,m,kt M r,n,m,kt

8 226 Devendra Kumar Differentiating both the sides of 2.13 j times with respect to t, we get M j j r,n,m,kt = M r 1,n,m,k t t M j αγ r,n,m,k t r j M j 1 t αγ r,n,m,kt M j j r αγ r,n,m,kt + 1 M j 1 r αγ r,n,m,kt + 1. r The recurrence relation in equation 2.11 is derived simply by rewriting the above equation. By differentiating both sides of equation 2.11 with respect to t and then setting t = 0, we obtain the recurrence relations for moments of lgos from extended type I generalized logistic distribution in the form where E[ j r, n, m, k] = E[ j r 1, n, m, k] j { E[ j 1 r, n, m, k] + E[φ r, n, m, k], 2.14 αγ r φx = x j 1 e x. Remark 2.1. Putting m = 0, k = 1 in 2.11, we obtain the recurrence relation for marginal moment generating function of order statistics for extended type I generalized logistic distribution in the form 1 + t M j αn r + 1 n r+1:n t = M j n r+2:n t j αn r + 1 M j 1 n r+1:n t { tm j αn r + 1 n r+1:n t jm j 1 n r+1:n t + 1. Replacing n r + 1 by r 1, we have M j r:n t = 1 + t M j αr 1 r 1:n t + j αr 1 M j r 1:n t { + tm j αr 1 r 1:n t jm j 1 r 1:n t + 1.

9 Relations for Marginal and Joint Moment Generating Functions 227 Remark 2.2. Setting m = 1 and k 1, in Theorem 2.1, we get a recurrence relation for marginal moment generating function of lower k record values for extended type I generalized logistic distribution in the form 1 + t αk M j j r,n, 1,kt = M r 1,n, 1,k t j αk M j 1 { tm j j 1 αk r,n, 1,kt jm r 1,n, 1,k. t + 1 r,n, 1,k t 3. Relations for Joint Moment Generating Function On using 1.3 and binomial expansion, the explicit expression for the joint moment generating of lgos r, n, m, k and s, n, m, k 1 r < s n, can be obtained when m 1 as M r,n,m,k, s,n,m,kt 1, t 2 = where Ix = x r 1 s r 1 a=o b=0 C s 1 r 1!s r 1!m + 1 s 2 1 a+b r 1 a s r 1 b e t 1x [F x] s r+am+1 1 fxixdx, x > y 3.1 e t 2y [F y] γ s b 1 fydy. 3.2 By setting z = [F y] 1/α in 3.2, we obtain Ix = α t 2 p [F x] γ s b+p+t 2 /α. t 2 p!αγ s b + p + t 2 On substituting the above expression of Ix in 3.1, we find that M r,n,m,k, s,n,m,kt 1, t 2 = αc s 1 t 2 r 1!s r 1!m + 1 s 2

10 228 Devendra Kumar = r 1 a=o s r 1 b=0 t 2 p p!αγ s b + p + t 2 1 a+b r 1 a α 2 C s 1 t 1+t 2r 1!s r 1!m + 1 s 2 s r 1 b and for s = r + 1 s r 1 b e t 1x [F x] γ r a 1+p+t 2 /α fxdx r 1 q=0 a=o s r 1 b=0 1 a+b r 1 a t 2 p t 1 q p!q!αγ s b + p + t 2 αγ r a + p + q + t 1 + t M r,n,m,k, r+1,n,m,kt 1, t 2 = α 2 C r t 1+t 2r 1!m + 1 r 1 r 1 q=0 a=o 1 a r 1 a t 2 p t 1 q p!q!αγ r+1 + p + t 2 αγ r a + p + q + t 1 + t At m = 1, 3.3 is of the form 0 as r 1 r 1 0 a=0 1a = 0. a Therefore applying L Hospital rule and using 2.7, we have M r,n, 1,k, s,n, 1,kt 1, t 2 = αks t 1+t 2 and for s = r + 1 q=0 M r,n, 1,k, r+1,n, 1,kt 1, t 2 = αkr+1 t 1+t 2 t 2 p t 1 q p!q!αk + p + t 2 s r αk + p + q + t 1 + t 2 r 3.5 q=0 t 2 p t 1 q p!q!αk + p + t 2 αk + p + q + t 1 + t 2 r. 3.6 Differentiating 3.3, 3.4, 3.5 and 3.6 and evaluating at t 1 = t 2 = 0, we get the mean of the r th lgos.

11 Relations for Marginal and Joint Moment Generating Functions 229 Special cases i Putting m = 0, k = 1, in 3.3, the explicit formula for joint moment generating functions of order statistics for extended type I generalized logistic distribution can be obtained as M n r+1,n s+1:n t 1, t 2 = α2 C r,s:n t 1+t 2 r 1 q=0 a=o s r 1 b=0 1 a+b r 1 a s r 1 b That is t 2 p t 1 q p!q![αn s b + p + t 2 ][αn r a + p + q + t 1 + t 2 ]. M r,s:n t 1, t 2 = α2 C r,s:n t 1+t 2 r 1 q=0 a=o s r 1 b=0 1 a+b r 1 a s r 1 b where C r,s:n = t 2 p t 1 q p!q![αr + b + p + t 2 ][αs + a + p + q + t 1 + t 2 ], n! r 1!s r 1!n s!. ii Setting k = 1 in 3.5, we get the explicit expression for joint moment generating function of lower record value for extended type I generalized logistic distribution can be obtained as M Lr, Ls t 1, t 2 = αs t 1+t 2 q=0 t 2 p t 1 q p!q!α + p + t 2 s r α + p + q + t 1 + t 2 r. Making use of 2.1, we can derive the recurrence relations for joint moment generating function of lgos from 1.5. Theorem 3.1. For the distribution given in 1.5 and for 1 r < s n, n 2 and k = 1, t 2 M i,j αγ r,n,m,k, s,n,m,k t 1, t 2 = M i,j r,n,m,k, s 1,n,m,k t 1, t 2 s

12 230 Devendra Kumar j αγ s M i,j 1 r,n,m,k, s 1,n,m,k t 1, t 2 αγ s { t 2 M i,j r,n,m,k, s,n,m,k t 1, t jm i,j 1 r,n,m,k, s,n,m,k t 1, t Proof. Using 1.3, the joint moment generating function of r, n, m, k and s, n, m, k is given by M r,n,m,k, s,n,m,kt 1, t 2 where = C s 1 r 1!s r 1! [F x] m fxgm r 1 F xgxdx 3.8 Gx = x e t 1x+t 2 y [h m F y h m F x] s r 1 [F y] γs 1 fydy. Solving the integral in Gx by parts and substituting the resulting expression in 3.8, we get M r,n,m,k, s,n,m,kt 1, t 2 = M r,n,m,k, s 1,n,m,kt 1, t 2 t 2 C s 1 γ s r 1!s r 1! x e t 1x+t 2 y [F x] m fx gm r 1 F x[h m F y h m F x] s r 1 [F y] γs dydx, the constant of integration vanishes since the integral in Gx is a definite integral. On using the relation 2.1, we obtain M r,n,m,k, s,n,m,kt 1, t 2 = M r,n,m,k, s 1,n,m,kt 1, t 2 t 2 αγ s M r,n,m,k, s,n,m,kt 1, t 2 t 2 αγ s M r,n,m,k, s,n,m,kt 1, t

13 Relations for Marginal and Joint Moment Generating Functions 231 Differentiating both the sides of 3.9 i times with respect to t 1 and then j times with respect to t 2, we get M i,j r,n,m,k, s,n,m,k t 1, t 2 = M i,j r,n,m,k, s 1,n,m,k t 1, t 2 t 2 αγ s M i,j r,n,m,k, s,n,m,k t 1, t 2 j αγ s M i,j 1 r,n,m,k, s,n,m,k t 1, t 2 t 2 αγ s M i,j r,n,m,k, s,n,m,k t 1, t j αγ s M i,j 1 r,n,m,k, s,n,m,k t 1, t 2 + 1, which, when rewritten gives the recurrence relation in 3.7. One can also note that Theorem 2.1 can be deduced from Theorem 3.1 by letting t 1 tends to zero. By differentiating both sides of equation 3.7 with respect to t 1, t 2 and then setting t 1 = t 2 = 0, we obtain the recurrence relations for product moments of lgos from extended type I generalized logistic distribution in the form E[ i r, n, m, k, j s, n, m, k] = E[ i r, n, m, k, j s 1, n, m, k] j { E[ i r, n, m, k, j 1 s, n, m, k] αγ s +E[φ r, n, m, k, s, n, m, k], 3.10 where φx, y = x i y j 1 e y. Remark 3.1. Putting m = 0, k = 1 in 3.7, we obtain the recurrence relations for joint moment generating function of order statistics for extended type I generalized logistic distribution in the form t M i,j αn s + 1 n r+1,n s+1:n t 1, t 2 = M i,j n r+1,n s+2:n t 1, t 2 j αn s + 1 M i,j 1 n r+1,n s+1:n t 1, t 2

14 232 Devendra Kumar { t 2 M i,j αn s + 1 n r+1,n s+1:n t 1, t jm i,j 1 n r+1,n s+1:n t 1, t That is M i,j r,s:n t 1, t 2 = 1 + t 1 M i,j αr 1 r 1,s:n t 1, t 2 + i αr 1 M i 1,j r 1,s:n t 1, t 2 { + t 1 M i,j αr 1 r 1,s:n t 1 + 1, t 2 + im i 1,j r 1,s:n t 1, t 2. Remark 3.2. Substituting m = 1 and k 1, in Theorem 3.1, we get recurrence relation for joint moment generating function of lower k record values for extended type I generalized logistic distribution in the form 1 + t 2 M i,j αk r,n, 1,k, s,n, 1,k t 1, t 2 = M i,j r,n, 1,k, s 1,n, 1,k t 1, t 2 j αk M i,j 1 r,n, 1,k, s,n, 1,k t 1, t 2 { t 2 M i,j αk r,n, 1,k, s,n, 1,k t 1, t jm i,j 1 r,n, 1,k, s,n, 1,k t 1, t Characterization Let r, n, m, k, r = 1, 2,..., n be lgos from a continuous population with df F x and pdf fx, then the conditional pdf of s, n, m, k given r, n, m, k = x, 1 r < s n, in view of 1.2 and 1.3, is f s,n,m,k r,n,m,ky x = C s 1 s r 1!C r 1 [F x] m γr+1 [h m F y h m F x] s r 1 [F y] γs 1 fy. 4.1 Theorem 4.1. Let be an absolutely continuous random variable df F x and pdf fx on the support, then E[e t s,n,m,k l, n, m, k = x]

15 Relations for Marginal and Joint Moment Generating Functions 233 = if and only if 1 t t p t p! + e x p s l j=1 γ l+j γ l+j p/α α, F x = < x <, α, > 0. + e x Proof. From 4.1, we have E[e t s,n,m,k r, n, m, k = x] = By setting u = F y x e [1 ty = F x +e x F y F x E[e t s,n,m,k r, n, m, k = x] = = 1 0, l = r, r C s 1 s r 1!C r 1 m + 1 s r 1 m+1 ] s r 1 F y γs 1 fy dy. 4.3 F x F x +e y α from 1.5 in 4.3, we obtain C s 1 s r 1!C r 1 m + 1 s r 1 [ + e x u 1/α ] t u γs 1 1 u m+1 s r 1 du C s 1 s r 1!C r 1 m + 1 s r Again by setting z = u m+1 in 4.4, we get 1 t t p t p! + e x u γs p/α 1 1 u m+1 s r 1 du. 4.4 E[e t s,n,m,k r, n, m, k = x] = = 1 t t p t p! C s 1 C r 1 m + 1 s r + e x p t t p t p! C s 1 s r 1!C r 1 m + 1 s r z αk p αm+1 +n s 1 1 z s r 1 dz p αk p + e x p Γ + n s αm+1 αk p Γ + n r αm+1

16 234 Devendra Kumar = C s 1 C r 1 1 t t p t p! + e x p 1 s r j=1 γ r+j p/α and hence the result given in 4.2. To prove sufficient part, we have from 4.1 and 4.2 C s 1 s r 1!C r 1 m + 1 s r 1 e t 2y [F x m+1 F y m+1 ] s r 1 where H r x = [F y] γs 1 fydy = [F x] γ r+1 H r x, t t p t p! + e x p s r j=1 γ r+j γ r+j p/α. Differentiating 4.5 both sides with respect to x, we get C s 1 [F x] m fx s r 2!C r 1 m + 1 s r 2 Therefore, e t 2y [F x m+1 F y m+1 ] s r 2 [F y] γs 1 fydy = H rx[f x] γ r+1 + γ r+1 H r x[f x] γ r+1 1 fx γ r+1 H r+1 x[f x] γ r+2+m fx fx F x = which proves that = H rx[f x] γ r+1 + γ r+1 H r x[f x] γ r+1 1 fx. H rx γ r+1 [H r+1 x H r x] = α 1 + e x α, F x = < x <, α, > 0. + e x Theorem 4.2. Let be an absolutely continuous distribution function F x with F 0 = 0 and 0 < F x < 1 for all x, then M r,n,m,kt M r 1,n,m,kt

17 if and only if Relations for Marginal and Joint Moment Generating Functions 235 t { M αγ r,n,m,kt + M r 1,n,m,kt r α, F x = < x <, α, > 0. + e x Proof. The necessary part follows immediately from equation On the other hand if the recurrence relation in equation 4.6 is satisfied, then on using equation 1.2, we have C r 1 r 1! e tx [F x] γr 1 fxgm r 1 F xdx = r 1C r 1 γ r r 1! tc r 1 αγ r r 1! tc r 1 αγ r r 1! e tx [F x] γr+m fxgm r 2 F xdx e tx [F x] γr 1 fxgm r 1 F xdx e t+1x [F x] γr 1 fxg r 1 m F xdx. 4.7 Integrating the first integral on the right hand side of equation 4.7, by parts, we get tc r 1 γ r r 1! e tx [F x] γr 1 gm r 1 F x {F x 1 ex fx α α fx dx = Now applying a generalization of the Müntz-Szász Theorem Hwang and Lin, [9] to equation 4.8, we get fx F x = which proves that α 1 + e x α, F x = < x <, α, > 0. + e x

18 236 Devendra Kumar 5. Conclusion This paper deals with the lower generalized order statistics from the extended type I generalized logistic distribution. Some explicit expressions and recurrence relations for marginal and joint moment generating functions are derived. Characterizing results of the extended type I generalized logistic distribution has been obtained by using conditional expectation and recurrence relation of lower generalized order statistics. Special cases are also deduced. Acknowment The author are grateful to Dr. R.U. Khan, Aligarh Muslim University, Aligarh for his help and suggestions thought the preparation of this paper. The author also acknowledge with thanks to referees and Editor Tamsui Oxford Journal of Information and Mathematical Sciences for carefully reading the paper and for helpful suggestions which greatly improved the paper. References [1] A. K. Olapade, On extended type I generalized logistic distribution, Int. J. Math. Math. Sci., , [2] A. K. Olapade, Some properties and application of extended type I generalized logistic distribution, ICASTOR J. Math. Sci., , [3] D. Kumar, Recurrence relations for marginal and joint moment generating functions of generalized logistic distribution based on lower record values and its characterization, ProbStat Forum, , [4] D. Kumar, R. U. Khan, and A. Kulshrestha, Relations for marginal and joint moment generating functions of Erlang-truncated exponential distribution generalized order statistics and characterization, Open journal of Statistics to appear, [5] E. K. Al-Hussaini, A. A. Ahmad, and M. A. Al-Kashif, Recurrence relations for moment and conditional moment generating functions of generalized order statistics, Metrika, ,

19 Relations for Marginal and Joint Moment Generating Functions 237 [6] E. K. Al-Hussaini, A. A. Ahmad, and M. A. Al-Kashif, Recurrence relations for joint moment generating functions of generalized order statistics based on mixed population, J. Stat. Theory Appl., , [7] J. Saran and A. Pandey, Recurrence relations for marginal and joint moment generating function of lower generalized order statistics from power function distribution, Pak. J. Statist., , [8] J. Saran and A. Singh, Recurrence relations for marginal and joint moment generating functions of generalized order statistics from power function distribution, Metron, LI 2003, [9] J. S. Hwang and G. D. Lin, On a generalized moments problem II Proc. Amer. Math. Soc., , [10] M. Ahsanullah and M. Z. Raqab, Recurrence relations for the moment generating functions of record values from Pareto and Gumble distributions, Stoch. Model. Appl., , [11] M. Burkschat, E. Cramer, and U. Kamps, Dual generalized order statistics, Metron, LI 2003, [12] M. Z. Raqab and M. Ahsanullah, On moment generating function of records from extreme value distribution, Pakistan J. Statist., , [13] M. Z. Raqab and M. Ahsanullah, Relations for marginal and joint moment generating functions of record values from power function distribution, J. Appl. Statist. Sci., , [14] P. Pawlas and D. Szynal, Recurrence relations for single and product moments of lower generalized order statistics from the inverse Weibull distribution, Demonstratio Math., IV 2001, [15] R. U. Khan, B. Zia, and H. Athar, On moment generating function of generalized order statistics from Gompertz distribution and its characterization, J. Stat. Theory Appl., , [16] R. U. Khan, D. Kumar, and B. Zia, Relations for marginal and joint moment generating functions of generalized exponential distribution based

20 238 Devendra Kumar on lower generalized order statistics and characterization, To appear Canadian Journal of statistics,, [17] S. M. Ruiz, An algebraic identity leading to Wilson s theorem, Math. Gaz., , [18] U. Kamps, A concept of Generalized Order Statistics, J. Statist. Plann. Inference, , [19] U. Kamps, Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: Balakrishnan, N. And C.Rao, C.R. eds., Hand book of statistics, 16, Order Statistics, Theory and Methods, Amsterdam: North Holand, ,