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1 A05le 258 FOR 16N TECHNOLO6Y DV WRGHT PATTERSON AFB OHO F/S 12/1 LASS F led FT R rc,1

2 t H ! 25 Qfl 4 llll V N \\ \, L \ \ h

3 FTD D(RS)T FOREGN TEC HNOLOGY DVSON c2 ON THE THEOREM OF N V SMRNOV WTH RESPECT TO A COMPARSON OF TWO SAMPLNGS by tj DDC Approved for public release ; distribution unlimited

4 r rr,crr c FTD D(RS)T EDTED TRANSLAT ON FTD D(RS)T November 1977 MCROFCHE NR : ON THE THEOREM OF N V SNRNOV WTH RESPECT TO A COMPARSON OF TWO SAMPLN GS By : D Kvit English pages: 14 Source: Dokiady Akademli Nauk SSSR, zd vo Moscow Leningrad, Vol 71, No 1, 1950, pp Country of origin: USSR Trans lated by: Robert D Hill Requester : AFFDL/FBRD Approved for public release; distribution unlimited ic; a t C fl THS TRANSLATON S A RENDTON OF THE ORG HAL FOREGN TEXT WTHOUT ANY ANA LYTCAL OR EDTORAL COMMENT STATEMENTS OR THEORES PREPARED BY: ADVOCA TED OR MPL ED A RE THOSE OF TH E SO UR CE AND DO NOT NECESSARLY REFLECT THE POSTON TRANSLATO N DVSON OR OPNON OF THE FOREGN TECHNOLOGY D FOREGN TECHNOLOGY DVSON VSON WPAFB OHO FTD D(RS)T Date22 Nov 1977 _

5 U S 3OARD ON GEOGRAPHC NAMES TRANSLTERATON SYSTEM Blo ck talic Transliteration Block talic Transliteration A a A d A, a P p p p R, r L 5 6 B, b C c C c B B B a V, V T T T m i, t rr r a G,g Yy y y U, 0 V, : E e 5 S Y, ye; E, x x x Kt, h ) c Zli, i [4 i Li a T, t 3 3, Z, H Ch, L [ H v H u, W w LU w h, :[ H R a, y L4 LU :tich, si h R H X x K, k b J1 h A L, 1 Nb Y,y i M M i, m b b b H H H H il, n 3 3 a E, e Do 0 0 0, 0 kj o Yu,yu fl n 17 P,p Ha * initially, aft er vowels, and after elsewhere When wr itten as ë in Russian, transl iterate as y e or ë RU 3AN AND NJLSH TRGONOMETRC FUNCTONS Russ ian English Russian English Russian English sin sin sh sinh arc sh sinh cos cos ch cosh arc ch cosh 1 tan th tanh arc th tanh 1 ct g cot cth coth arc cth coth 1 sec sec sch sech arc sch sech cosec csc csch csc h arc csch csch Russ ian rot lg Engl ish curl log U,

6 Th THE THEOREM OF 1 V WTH RESPECT TO A COMPARSON OP TWO S[\rV PT TNOS D Kvlt (Present ed by f c demic an A N Ko) mocerev on Januai v lq O fn werk [1] H Sm Pfl V st Udi ed the foil owlnc mrnr nnt n r ohlem of stat st cs: there are two series of i esu li s of m dc nendent ehs rvations on the random auantitles and and Yi y,,, y t is asked under what conditions is t possible to consider that the distribu t ion functions P1(x) Pg1<x) and fl (x)=p{ 1<x ) are eq ual, and when is the divergence of the exp erment data 50 cons derab le that the hyrothes s P,(x) P, (x) shoul (1 be i e, oct 1 n the rrcsent memorandum we show that the enoral 17at on of the theorems of AN Kelmogoorov and Ny nim oy, ob t : 1ned by GM Maniva L] for one samnllnr, is t ransferred w thout dt ft l cull y to the nrohlem of N V Smirnov about we sarnnlin s Let us examine the empirl cal dist i ihut ion k1( ) S (x), funet ons 1 where k3 (x) is the num b: i 01 observed values of les s than x anc+ T (x) r h,(! & *

7 r where k,(x) is the number of observed values of 2 less t han x At t he discont inul t v noints we sinehlement the emnimical di st ri but on funct ens with t he vert t cal segment n he indicated work N V : mt rnov nroved that if F1(x) F,(x) func t on F1 (x) is cent nueus and increases everywhere, const then when nn N = n n P ( D (m ii) 1) e 2 z, () :hero D(m, n) sti r, Sm (x) T,,(x) cz Let us assum t hat and 2 o<e <e <1 and Let us assume tliat by means of eouaiities are arbitrary numbers, F1( x) F (x) (x) and dot e1 rn ne F( )=O L F(f ) =02 Let us denote further mm ts (x) = 7,, (x) = 0 J, = max [S5, (x) = 0 ; 7, (x), Then when m, fl n virt ue of t he theorem of 1 venko there should be let us introduce the notations p m,, : D4,, (0i, 02) = SUp (Sn,, mr) (Sm (x) T (x)}, (0 k, 02) = SUP Sm (x) Ti, (x) ( m, em,) The obtained results can be formulated n the form of the roliowing two theorems r

8 Theorem 1 f when n oo e& ) =O1+o( L) 0<01<01<1 then where Os ) _ D (01, 0,; z) l+(0 0,; z) = r =rz 5 e sq ( z a) dz1dz, 2 /? 1 1 e Z )dz i dz, C i= o,) E o t 0,) b T QT 0 1 R = 1 O,(i 0,) 0 ( 0,) :)= (z1, z) = ç [z 2Rz iz2+z 1 Henc e, in particular, P D ( O, 1) < j }N 1_ e t We see thus that the theorem of N V Smirnov about the one sided deviations of the emnirical function from the theoretical is carried over to the case of the maximum of one sided deviat ons of two enmlrical functions Theorem 2 n the suppositions of theorem 1 where P { Dm1a (0 >, 0 > ) 1)(0, 0,; z) 3

9 (D (Ok, 0,; = 2i e Q( dz 1d;!_, i 1)*_ e_ k /, cj r,, 7) dz1d Z 2k g ak= 1 Ye, (1 0 ) Henc e when 0 =o, 0,= we get (1) b k j/ Similar to the theorems of G Man iy a, the results discussed make it nossible to use that nterval of the observed values in which the results of the observations are more reliable n the work we used the method of Laplace transforms apolied by W Feller 113] for proofs of the theorems of A N Ko lmogorov and Ny Smirnov n conclusion wish to express my deep thanks to Prof B V Gnedenko for the statement of the problem and his guidance in the solving of t L vov State Unversty im Franko Submitted 12 January 1q50 Bibliograrhy H 0 CMRpH0 6ioia HMHM Mry 2, 2 (1939) r M MaHHR aah 69, M 4 (1949) W Fell r, Ann of Math Staitstics, 19, (1948) 14,,