Madan Lal Puri.

Asymptotic efficiency of a class of c-sample tests online

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IMM-NYU 314
NOVEMBER 1963



NEW YORK UNIVERSITY
COURANT INSTITUTE OF
MATHEMATICAL SCIENCES



Asymptotic Efficiency of a Class
of c-Sample Tests



MADAN LAL PURI



:3-



PREPARED UNDER
CONTRACT NONR-285(38)
WITH THE
OFFICE OF NAVAL RESEARCH






\



IM-NYU J>lk
Noveraber I965



NEW YORK UNIVERSITY
Courant Institute of Mathematical Sciences



ASYMPTOTIC EFFICIENCY OF A CLASS OF c-SAMPLE TESTS"'"



Madan Lai Purl



This paper was prepared with the partial support of the
Office of Naval Research, Contract Nonr"222-(43), while
the author was at the University of California, Berkeley.
It was revised at the Courant Institute of Mathematical
Sciences, Nev/ York University under the sponsorship of
the Office of Naval Research, Contract Nonr"285(58) .
Reproduction in whole or in part is permitted for any
purpose of the United States Government.



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7

Let Z^*^' = 1 If the 1th smallest of N = > m.

^"■^ (1)

observations Is from the jth set and otherwise let "^^j \ = 0»
Denote

(4.1) ^j'^J-Vj^^^N^i^fl



where the E,^ ^ are given numbers o Following Chernoff and
Savage [2], we shall use the representation

(4.2) T = r J,,[H„(x)ldS;^J)(x)

v;here E^ = Jj^d/N). V/hile Jjj need be defined only at
l/N, 2/n, ..., N/N, v;e shall find it convenient to extend its



domain of definition to (0,l] by letting J^ be constant



N



on



(1/N, d+D/N].
Let



Ij^ = [x: < Hj^(x) < 1].



Then I^ is a random Interval, given by



% =



X, X ... ■> ^"'•^X



f^'fi!!.-'.,.!



xioxfi'



3:



:■ bcis.



8



^ Joljat,^^a3;^-Dt_otic^ Before proving the asymp-

totic normality of the T,^ 's we state a fev; elementary



results.
(5.1)



H ^ X^ F^^^ ^Xq F^^^; l-l,.,.,c.



(5.2)
(5. 5)

(5.^)



, „(i) ^ 1-H ^ 1-H. ._,

1 - if S. —i~- < -=r-, 1-1,0.0,1



- A^ ^ Aq



F(/)(l-F(i))^£%lL_^H(l-Hl. ^^^/_^^^
dH ^ X^ dF^^^ > Xq dF^^^; 1=1, ...,c.



Lemma 5.I. If



(1) J(H) = lim J„(H) exists for < H < 1 and is not



il -) 00

cons tan t ,



K



(2)



'■N'-



j^m^,) - j(H^j



dS^J^x) = Op(N'-^l^'2)j^



(5) Jj,j(l) = o( v/!7)



(M



J^^^H(x))



d^j(H}



dH^



< K[H(l-H)]-i-(l/2)+6



for 1=0,1,2, and for some 5 > 0,
aTid almost all x (a^a.x).



then, for fixed F^^^ , , . . ,F^°^ and A ,...,X ,

J- v^






v'i.Q;;:fie"r3r3 we'i s> ejscfj



'i y-f*- -Ji 1 J iv,. ;v .'..V. -.'-.; v.^ I ■..■ i. -J \,-J









, -, r ... f^






.'■!■



t : ■{



:± > \J '-5 J j;



t J r i #«■'■:: *J-



. .^a^J



?i > .iH.;,lLH . {^^^q.l)-^^:i



f - -5 \



(i):






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-'■ I'iij'i



U)









J[r



\ (S)

T v./



(TTv )o - (£}j,^L (C)



t (J:), 1



i - A ^ • » • i J. (



(5.5) lim f( ■^''' "'^'^^" S t



1 -x^/2 ^
/27r



v;he re
(5.6)



^N,J =/'^J[H(x)]dF(J^x)



and



(5.7) NT



N,j



m' t.(l)r,.>ri 1.(1)



•U) rv^^T7( j)



- 2 ^X, // F^^Mx)[l-F^^'(y)]J'[H(x)]J'[H(y)]dP^^'(x)dF^"J'(y)

^y. -« .;•)



(\.)^'^'ufo(, }'-^^-[(^)H]s-:f(,:)via[(A:)-'^'5-j:](x>-^^






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: A'






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(y)^'-'^)(x}^^^:J>



■ • -l J . , kl

Ou> Tr> yJ- crj_ J • ^ -■■ ij



(viT'lTJfyilTl'X '"'■•-) ^'^^'^r-f' , 'q \\ ,/,/



Ji J.



"7 ™i-,_ X









9cr



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SVJii






•■YA' .



15

We shall now extend the proof of the above lemma to the
case v;here F ,o.,F and A,, }—- y B. ^(X. ) k, the proof of the lemma follows by apply-

l^T ri a^T ^•'- ^ J

ing the Central Limit Theorem to each of the c independent





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Online LibraryMadan Lal PuriAsymptotic efficiency of a class of c-sample tests → online text (page 1 of 3)