By M.M. Rao

Provides formerly unpublished fabric at the basic rules and houses of Orlicz series and serve as areas. Examines the pattern course habit of stochastic methods.

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Example text

10) given by n! (n- s)! 2 : /[ ti waxbfr,s:n(W,x) d w d x , 1 1 . b) Then, the double moments p~,s:, satisfy the recurrence relations presented in the following two theorems. THEOREM 3. , = [2(a+b) . ( a b - 1 ) H + O#r,;+l:,/kn - r) ; x(a,b) r,~+1. . . and for 1 <_ r < s <_ n, s - r > 2 and a , b = 1 , 2 , . . , ~,asb2 = ]/~a,b_)l:n + bl~,i':. 4) , (a) where /2r,s:n ~ i~r:n . PROOF. From Eqs. 2), for 1 _< r < s _< n and a , b _> 1 let us consider ]2(a,b_ ..... 1) = ( F - 1 ) [ ( s - r n!

6) Then by proceeding on lines very similar to those used in proving T h e o r e m s 1 and 2, we m a y prove the following two theorems. THEOREM 12. F o r n _> 2 and a = 1 , 2 , . . an 1) l:n = # -- [l~_J . 7) /~l:n- 1 with /Xl: • (0)n = 1 THEOREM 13. 0) _= 1. It should be mentioned here that T h e o r e m s 12 and 13 have been proved by Joshi (1978) and are presented here for the sake o f completeness. The recurrence relations presented in T h e o r e m s 12 and 13 will enable one to c o m p u t e the single m o m e n t s (of all order) o f order statistics for all sample sizes in a simple recursive way.

PROOV. F r o m Eqs. 2), for 1 < r < s _< n and a,b _> 1 let us consider #(a-l,b) r,s:n = ( r - 1 ) ! ( s - rn! - 1 ) ! ( n - s ) ! 10) where J(x) = fox[F(w)] r 'IF(x) - F(w)] s-r-] f ( w ) d(w a) = fX[F(w)]~-] [F(x) - F(w)] s-~-I d(w a) - foX[F(w)l [F(x) - F ( w ) ] s-~-I d(w a) 30 N. Balakrishnan and S. S. 3). Integrating by parts, we obtain for r = 1 and s = 2 that foxw a f ( w ) J ( x ) = x ~ - x~F(x) + forr_>2ands=r+l that J ( x ) = x~[F(x)] r-1 - (r - 1) -xa[F(x)]r +r forr=l dw , fo Xwa[F(w)]r-2 f ( w ) ~0xwa[F(w)]r-l f ( w ) d w dw , and s _> 3 that J ( x ) = (s - 2) + foxwa[F(x) fo Xw a IF(x) - - (s - 2) - F(w)]S-3 f ( w ) d w F(w)] ~-2 f ( w ) d w fo Xw a F ( w ) [ F ( x ) - F(w)] s J f ( w ) d w , and for r_> 2 and s - r >_ 2 that J ( x ) = (s - r - 1) fo Xw~[F(w)] r-1 IF(x) - - (r- l) foXwaIF(w);-2[F(x) + r foXwa[F(w)] r - (s - r - 1) F(w)]S-r-Z f ( w ) d w - F(w)]~-r-lf(w) dw l[F(x) - F(w)] s-r-1 f ( w ) dw ~0Xwa[F(w)]r[F(x) _ F(w)]S r 2 f ( w ) dw .

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