Riyaziyyat ensiklopediyası
1185
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Transcript of Riyaziyyat ensiklopediyası
-
- ; Abel integral equation
)(xf = x
sxdss
0
)(, x 0 (1)
, )( xf , )(s - . )( xf , - ( )(xf - ) . . .- . , )0(f = 0 ,
)( s =
s
zsdzzf
0
)( (2)
. )0(f = 0 -, . . .- :
)( s =
+ s
zsdzzf
sf
0
)()0(1
.
. . .- . X = = )( 21 , XX , )( 21, ,21 xxp XX
r = 2221 xx + - ( X ) ; 1X )( 11 xp X - . X ( ;
). R = ,2221 XX + X - , )( rpR ;
)( 21, ,21xxp
XX= .)()2( 1 rpr R
1X - - ; - R - , e = )( 21 , ee 1e - ; e - ( , [2], 1, 10 ). - 1x > 0
)( 11 xp X = =
dzzpz
xpz eR
)(11
11
0
= .1
02
1
1
1221 dz
zzxp
R
21
1x
= ,x
2zx = ,s
xp
x X11
1= ,)( xf
sp
s R11
= )( x
, )( xf )( s - (1) . f (2) . , . . .- . - 2 , . . . . . . . ( , [1] ) 1823- - ( ) . )( , xu . - x )00( , t )( xt . .
. . . (1), )(xf = )(2 xtg , g
, )( s =sin
1, s -
.
u = x
dss0
2 1)(
. )(xf = const -; )( xt = const , [ . ( Ch. Huygens, 1673 ) ]. .: [1] A b e l N. H., Oeuvers compltes, t. 1, Christiania, 1839, p. 2730; [2] ., , . ., . 2, ., 1984; [3] . ., , 2 ., . ., 1937; [4] ., ., , . ., . 1, 3 ., ., 1951.
7
-
; Abelian group , ( ) . . .- n - ( ) - . . .- - ( , , ). . .- - . ( 2mod ). 0 1 ,
00 + = 0 , 10 + = 01 + = ,1 11 + = 0
. . G . ,...,,...,, 21 n 0 1 , n{P = }0 = nq ,
n{P = }1 = np = nq1
nS = n +++ ...21
, (
,nS n ,...,, 21 -
2 - - ). : ) 0n 0n G - ,
0np = 0nq = 21 , n > 0n n nS ; )
= 1
)(min ,n
nn qp =
, n nS . ) ) , ) nS - , ; G -. . .- , ( , - ). .: [1] ., , . ., ., 1965; [2] . ., - . ., . ., 1966, . 11, . 1, . 335; [3] ., - , . ., ., 1981; [4] D v o r e t z k y A., W o l f o w i t z J., Duke Math. J., 1951, v. 18, p. 50107; [5] Proba-bility measures on metric spaces, L. N. Y., 1967. ; Abel theo-rems , - . 8
1.
0
1 dxxe xs =
s)(
, 0> , s > 0
, x )( xf ,
)(xf ~ 1x
, 0s
0
)( dxxfe xs ~
s)(
. )( xf , , . 2. ,
= 1nna
A - . || z < 1 z -
)(zf =
= 1n
nn za
1z
Azf )(
( - ). 1- . 0 < z < 1 z = se
)( xG = xn
na
.
)( zf =
0
)( xGde xs = .0
)(
dxxGes xs
x AxG )( , 0s
)( zf ~
0
dxAes xs = A
. , . . -. .: [1] ., , . ., . 2, ., 1984. -; Abel theorems , .
-
; abstract ergodic theorem , . ; abstract decision rule , . ; open queueing system , . - ; open random set G . G gT
: S , G ; ,GG ,K S - .
GG = GMMM I,:{ G = ,}
KG = KMMM I,:{ G = ,}
( ) gT ,
G . , . ( ) ( ) - ; sample range , niiX 1}{ =
niiX 1)( }{ =
nW = )1()( XX n -. iX , 1 i n , F , nW -
nW{P < }x =
+ )())()(( 1 tFdtFtxFn n , x 0
. . - . ( - , ) . - ( . ) srW , = ,)()( rs XX 1 r < s n . - . - . .: [1] ., , . ., ., 1979; [2] . ., . ., - , 3 ., ., 1983. ; Hada-mard matrix - 1+ 1 N > 1
NH ,
NN HHT = NEN
, NE . . .- - . - ( , [1] ). , N = 2 . ( , [2] ).
. .- 1+ , . . .- : N )4mod(0 N = 2
. -, , -. , . .- p -
p( )N . I , II I . , I . .- 1N 1p II . .- p , ( - ). . .- . .: [1] H a d a m a r d J., Bull. sci. math., sr. 2, 1893, t. 17, p. 24046; [2] S y l v e s t e r J. J., Phil. Mag. 1867, v. 34, p. 46172; [3] , ., 1969; [4] H e d a y a t A., W a l l i s W. D., Ann. Statist., 1978, v. 6, p. 1184238. -; adaptation algorithm - , - , - . . . - ( ) . - . . .- ( ) . .: [1] . ., , ., 1984; [2] B i c k e l P. J., Ann. Statist., 1982, v. 10, 3, p. 64771. ( ) - ; adapted random process , .
F F - ; F adapted function , . ; adaptive controlled random process i n d i s c r e t e t i m e , . - ; adaptive procedure , -.
; adaptive estimator - ; - . . . . . .: [1] B i c k e l P. J., Ann. Statis., 1982, v. 10, 3, p. 64771. ( ) - ; spain o f a d i s t r i b u t i o n , .
9
-
; step fac-tor , . , ; step of arithme-tic distribution , , .
- ; additive functional o f i n t e g r a l e t y p e
st = s
t
uduuf ,, ))(( 0 t < s
, )(
, 11]0[: , RR +f , . .: [1] . ., - , 2 ., ., 1986; [2] . ., . ., - , ., 1970. ( ) - ; additive functional o f M a r - k o v p r o c e s s , ; . ( ) ; additiv functional o f t h e W i e n e r p r o c e s s , ( st = suut + , t < u < s )
)(tw , t 0 ( ,st ,)(uw ,, ][ stu t < s stA ) -
,st 0 t < s . .: [1] . ., - , 2 ., ., 1986; [2] . ., . ., , ., 1970. ; additive function , ( ) , - . - ; additive set func- tion ( ) . K , G + ( G ). KA , KB , BAI = , KBAU A B
)( BAU = )()( BA +
10
, GK: . K , , G )( = 0 . G ,1R C , n ,nR , . - - . , K - ,
KA KB KBAU ( K U - ). . . . - . , K , , K ,
)( ...1 mAA UU = )()( ...1 mAA ++
K - miiA 1)( . K U I ,
GK:
)()( BABA IU + = )()( BA +
, ,, BA K . , , - .
.: [1] ., , . . ., ., 1953. ; additive noise , . ( ) - - ; additive problems o f n u m b e r t h e o r y ( ) . . )( xf - , p . N
N = )()()( ...21 nxfxfxf +++
)( , nNI , 0 nxxx ...,,, 21 1p n - . - . n ...,,, 21
1{ P = })(kf = ,1 p k = 110 ...,,, p
, a = ,1M 1D = 2 > 0 . n = n ++ ...1 .
-
n{P = }N = dzepen
p
k
xfziNzi
=
1
0
1
0
)(22 1 . (*)
,
n{P = }N = )( , nNIp n .
(*) , )( , nNI : 1) n , p ; 2) n , p ; 3) n p = )(np . - . )( , nNI - . )( , nNI - . , - - ,
)( , zfS =
=
1
0
)(21p
x
xfziep
z - p , )( , zfS - , . )( xf . . .- . ( G. Castelnuovo, 1933 ) , , )(xf = x - , )( , nNI . . . ( 1956, , [1] ) . ,
)(xf = x )(xf = 2x )( , nNI . . .- : , ( , [3] ). . . - , ., [4].
.: [1] . ., , ., 1971; [2] . ., - . ., . ., - , ., 1975; [3] . ., . . . , 8, 1983, . 121, . 6282; [4] . ., - . ., ., ., 1984. ; additive model , , . ; additive measure , .
* ; additive random ( stochastic ) process , .
; affine shape , .
( ) ; f a u l t tree , .
; r a n d o m tree , .
- ; linear tree code , .
; tree code , ( ) . ( ) -. - ; . , . . .- , - ( ) . . .- .
.: [1] ., , - , . ., ., 1986. ; flow )( ,, AX - }{ , RtT t :
R21, tt xTtt 21 + = xTT tt 21
Xx . tT - t - .
, ; B e r - n o u l l i f l o w , . ; c o n t i -n u o u s flow , . ; m e a s u r - a b l e flow , . K K ; K flow , K .
* ; parameter of flow , . , ; filtra-tion / family of algebras )( ,, PA . T ; , Tt )( ,, PA tA . Ttt )( A
Tt AA t ,, Tts s t ts AA , Ttt )( A
. )( ,BM )(tX ,B ,Ts s T
11
-
})(:{ , sX - tN - Ttt )(N . T = +R -, 0)( ttA : ) , t 0 tA = ;I
tss
>
A ) A P
0A A - . .: [1] ., , . ., ., 1975. ; intensity of flow , . ( ) - ( ); last income first outcome ( LIFO ) - . , , . - . . , n - . 1 - - )( tB . - ; n - , , n - .
.: [1] . ., - , . ., ., 1965. ; Akaike information criterion, AIC , , , .
; Akaike criterion , ; .
; axio-matic quantum field theory - ; . . . -
( , [2] ). ., dM , H - , )( dMSf
dMSff d ,, )()({ }2 ; )( ff f - , 12
)( 1f )( 2f 1f 2f - , - . , ) H ( dM - ) gUg , g )(
dMSf
1)( gg UfU = ,)( fg
)()( xfg = ,)( 1xgf dMx ;
) dM - gU
,{ P = }...,, dq dM - ; ) - H gUg -
)}()({ , dMSff .
)( ...,,1 nn ffw = Hnff ))()(( ,...,,1 - ( , [1] ). - .
.: [1] ., , . ., ., 1967; [2] W i g h t m a n A., G a r d i n g L., Ark. Phys., 1964, v. 28, p. 12984; [3] . ., [ .], , ., 1987.
; active variable , .
; active experiment - -. ; Aldous Rebolled condition , ; .
-; Alexandrov space , . -; Alexandrow theorem -. ,M )( , GX MP . 21 \ GBG X )( 1GP = = )\( 2GXP G21,GG -, )(G B P - . M - )( P - MP : 1) P , )(G B
)(lim
BP = )( BP
;
-
2) GG
)(inflim
GP )(GP )(lim XP = )(XP
; 3) F
)(suplim
FP )( FP )(lim XP = )(XP
. . , . . . [1]- .
.: [1] . ., . ., 1940, . 8, . 30748; 1941, . 9, . 563628; 1943, . 13, . 169238; [2] - . ., . , 1976, . 31, . 2, . 368. - ; alpha excessive function , . ; alpha faktor analysis , ; .
, ; potential alpha kernel , .
; alpha potential , - . ; Kolmogorov complexity entropy , - , , . . .
( ) - - ( , ) . G - x - )( xKG , p - , )( pG = x . - F , G
)( xKF )1()( OxKG +
( , [1] ). - ( - ) , )( xK .
. . ; , . .- . .- . . .- ( )1(O+ ); - ( - ). . .- ( , [2] ). . .- , . . - ( , [2] [4] ); ., y x -
. . ;)|( yxK
):(I xy = )|()( yxKxK y - x . .. I , :
):(I):(I xyyx = ,)))()((log( ylxlO +
,)( zl z . . .- - . . )(xMK - - ( [3] [5] ). - : n . .-, - , )1(On + - , ]10[ , ( , - ) ( , [3], [5] ). )(xM -, x )(log2 xM . M .
(1) )( xM < + ; (2) r )( xM - xr , ; - . . .- ( , [3] ). . . . ( C. Shannon ) , . 1) x , p ,q x - , H = + pp
2log(
,)log2
qq+ 0 1 , p q . )( xl , )()( xlxK )1(OH + . , . ., ( ) . x , ; , x - . 2) 0 1 - p q ( ) n . .- K - n ,
)loglog( 22 qqpp + - . .: [1] . ., -, 1965, . 1, . 1, . 311; ( , . ., , ., 1987, . 21323 ); [2] - . ., . ., . , 1970, . 25, . 6, . 85127; [3] . ., , ., 1981, . 16, . 1443; [4] . ., . , 1984, . 276, 3, . 56366; [5] . ., . , 1973, . 212, 3, . 54850; [6] H a r t m a n i s J., Bull. European Ass. Theor. Comp. Sci., 1984, 24 ( Oct. ) p. 7378 [7] ., ., . , 1988, . 43, . 6, . 12966.
13
-
; Allais para-dox . }{F - f }{F -
F f )(FUG > )(GU , (1)
]10[ ,
))1(( GFU + = )()1()( GUFU + (2)
U ( , ). . [1]- { 0 ., 5 . ., 25 . . } - , ,, 21 FF
43 , FF . - .
0 .
5 . .
25 . .
1F
2F
3F
4F
0
0,01
0,9
0,89
1
0,89
0
0,11
0
0,1
0,1
0
-, 5 . , , 5 .- 25 .- . , ( ) . , , - , -, . , f , 1F f 2F 3F f 4F . (1) (2)- U ,
21)( 31 FF + f ,21)( 42 FF +
,
21)( 31 FF + = .21)( 42 FF +
( , ). ( , [2] ). .: [1] A l l a i s M., Econometrika, 1953, v. 21, p. 50346; [2] . ., . ., . ., -- , ., 1980. * - ; lower ladder time , . 14
* - ; lower ladder height , . * -; lower ladder index , . * - ; lower ladder random variables , .
; alternative , - , - ( ) , . , -, , . . - . ., . - .
-; alternative hypothesis , . - ; forecast of meteorological binary variables of events / alterna-ting meteorolgical forecast , - - - ( ., - . ). . . . - - . . . .- ( , , Y - Y - ), , . . . , , . . . . - ( , , ; ) . .: [1] . ., . . . ., 1937, . 14, . 4957; [2] . ., . . . ., 1955, 4, . 33949; [3] . ., - , ., 1981. * - ; alternating rene-wal process . }{ ,..., 21 XX
}{ ,..., 21 YY . )(1 xf )(2 xf . , -. , ( 1 ) .
o
; ;
- - - - - ; o - . 1. . . ,
-
. , . . .-.
2. - . t = 0
1X ; 1X - )(1 xf -. . 1X 1X . 1X
1X - xe - . , 2X
1X 2X - - )(2 xf -. )1( i i - ii XX + - . ( ) , ; , )(1 xf -
xe . , , xe - . . ( - ) , . , . . . .
. ., - k . , k
ijp ; ,jip i - j - . - .
.: [1] . ., . ., , . ., ., 1967. ; subprocess .
)( ,BX
))(( ,,, xss
tt PF )~~)(
~( ,,, xs
stt PF -
. , :
) )(~ < )( ;
) 0 t < )(~ , )(~ t = )(t ;
) stF = ]~[ t
st F , t
~ = )(~:{ > }t , ]~[ tst F , stF st
~ , stA F
tA ~
I .
)~~
)(~
( ,,, xss
tt PF ))(( ,,, xss
tt PF - , .
, ~ . )( , xs {,xsP ~ } = 1 t < ~
)( t = )(~ t , )(~ t )(t - , .
)~~~
)(~
( ,,, xss
tt PF ))(( ,,, xss
tt PF - , )~~
~)(
~( ,,, xs
stt PF ,,)(( t
)~, xss
t PF .
)( ,,, txsP ))(( ,,, xss
tt PF -
, )(~ ,,, txsP - , ,
)(~ ,,, txsP )( ,,, txsP .
.: [1] . ., . ., - , . 2, ., 1973; [2] , ., 1985; [3] - . ., , ., 1959.
, - ; sub- class o f a M a r k o v c h a i n , ; .
; subnet , ( ) .
-; amplitude modulation )(t ,
)(tZ = ,)())(1( 0 ttmc + < t <
, )(tM = ,0 )( tD = 1 , c , ,)(0 t < t < , )(0 tM = ,0
)(0 tD = 1 , m , 0 < m < 1 , , m - .
)(t )(0 t , )(B )(0 B ,
)( tZ
)(ZB = ))(1()( 022 BmBc +
.
)(t )(0 t - , )(f , )(0 f
)(Zf :
)(Zf = ,))()()(( 022 ffmfc +
15
-
. )(t )(0 t -
1 2
, 0 < 2
-
, )(t -, ,)(dK = 0 - . - ; amplitude frequency response , . - ; analytic characteristic function X - P - z = 0 . z = ,ist + || z < r f |Im| z < z
)( zf =
)(dxe xzi P
. f , , r > 0
||{ XP > }A = )( AreO , A
. . .-.
a = t{sup > XteM:0 < } ,
b = t{sup > XteM:0 < }
, ,ai ,bi f az :{ < zIm < ,}b f
. . . .- , - .
.: [1] . ., . ., , ., 1972; [2] ., - , . ., ., 1979. ; analo-gous method : , . - , - , . .- , - - . , . .- , - , ( - , , - ). - - - - . . . ( - ) X - ( )
. )1(X )2(X )( )2()1( , XXD - :
)( )2()1( , XXD = )( )1()2( , XXD , )( )2()1( , XXD )()( )2()3()3()1( ,, XXDXXD + ,
)( )2()1( , XXD ,0 )( )1()1( , XXD = 0 .
:
)( )2()1( , XXD =21
1
2)2()1(2 )(
=
n
iiii xx
. :
)( )2()1( , XXD = ,1
)2()1(=
n
iiii xx
, ,)( jix i = n,...,1 )( jX , i - i - . -, ., ( , , ; - ). jX jY - ( j ) -, . .- (
) )( 0XY = jY , 0X -
, Y -, j ,
)( ,0 jXXD = )(min ,0 jj XXD .
. . ( , [3] ). . - .
.: [1] . ., . ., - , ., 1983; [2] . ., - . ., , ., 1982; [3] , ., 1985. -; Andersons inequality -, - . F - , FC ,, ]10[t Fy )( ytC + )( yC + , , )(C )( yC + , - , .
17
-
nR - . . , ., , f : 1) )( xf = )( xf ,
;nx R 2) > 0 )({ ; xfx nR } - . . . . . , - . .: [1] A n d e r s o n T., Proc. Amer. Math. Soc., 1955, v. 6, 2, p. 17076; [2] B o r e l l K., Ark. mat., 1974, v. 12, 2, p. 23952; [3] . ., . ., , ., 1985. ; Anderson Jensen theorem , ( ) . -; Andres reflection principle , .
( ) ; s t o p p i n g time , . - ; instantaneous spectral density , ( ) - . ; instanta-neous state x , :
)(lim ,,0
xxtpt
= ,1
xq = txxtp
t
)(1lim ,,0
= + ,
)( ,, xxtp t x - x - . . .- 1951- . . - . ( P. Lvy ) , - . . . . . . .- ( ) , - . . . . .: [1] - , , . ., ., 1964. - ; Anosov dynamical system
)( ;t }210{ ...,;; t
D - DD :tS , D )(x )()(
*xS t =
= )( xSdS tt tS*
-
.
18
. . . ( , [1], [2] ), t K . D -
)( xf -. . . . - ( , ).
.: [1] . ., . . - , 1967, . 90, . 1210; [2] . ., . . . ., 1966, . 30, 1, . 1568. ; Ansari Bradley test - . , , . : ; 1 - ; 2 . . . . . .- . , . . .- ; . . ( ) - , . , - . , ( ) ( , [2], [3] ). - .
.: [1] A n s a r i A. R., B r a d l e y R. A., nn. th. Statist, 1960, v. 31, 4, p. 117489; [2] ., ., , . ., ., 1983; [3] M o s e s L. E., Ann. th. Statist., 1963, v. 34, 3, p. 97383. - ; antiferromagnetic model . , . .- }{ , dt tx Z
, dt Z etx + = tx , de Z . . . U :
)( AA xU =
===
,,
,,,,
,,
01}{
}{
lardig AtstsAxx
tAxh
ts
t
tx = ,1 dt Z , Rh . - || h < d2 , ,1x 2x -:
1tx = d
tt ++ ...1)1( , 2tx = 1tx .
h = 0 . . . t
ttt xx d
++ ...1)1(
-
. h 0 d 2 ( ) .
( , [1] ), h - , - h - . , -. .: [1] . ., . ., 1968, . 2, . 4, . 4457. - ; antisimmetric Fock space , .
- ; antisymmetric variate method , .
- ; antithetic variables method , f s
2)]11()([ ..,.,...,, 11 ss ff +
, ,i ]10[ , .
.: [1] H a m m e r s l e y J., M o r t o n K., Proc. Cambr. Phil. Soc., 1956, v. 52, p. 44975. ; drift coef- ficients , , .
/ -; drift coefficients . . . . ( - )
)(txd = )())(())(( ,, tdtxtdttxta w+
( , - ), . . ( ) )( , xta - ( ).
)( , xta : t - tt + -
,)()( txttx + 0 t )( tx = x , )( to
txta )( , ( , - ). , )( , xta x t ( ) .
; drift vector , .
; leading function , , .
* , -; leading measure of flow , .
-; posteriori probability , -, , . - . , .
, , - .
. . .
; posterior mean .
-; posteriori distribution - , .
, )(p -
, ~ - )|(xp { = } ~ , - ~ - . .-
)|( xp = +
)|()(
)|()(
dxpp
xpp
. )( xK )|( xp , . . x - , )( xK - . ix - -
)|( 0xp ~
, )|( ,...,1 nxxp . .- n - .
. .- .
.: [1] . ., , 4 ., . ., 1946.
19
-
( ) ; a posterior risk o f a d e c i s i o n f u n c t i o n - X ( ) x
))(( , XL . )( xG . .-
. , . * ; approximation theorem A - A A - . )( ,, PF , ,U F - A .
UA
)(lim AAAA nnnUP
= 0 (1)
AnA ; (1)
)\(lim nnAAP
= )\(lim AAnn P = 0
.
)(AP = )()( AAAA nn PP + =
= )()()( AAAAA nnn PPP +
, )(AP = = )(lim nn AP UA
A . FA (1) , A - .
, - ; approximation of complex distributions b y s i m p l i e r o n e s . . . y = )( xf
dxdy
y1 = 2
210
1
xCxCCCx++
+
, 10 ,CC 2C 2 , 1 2
20
. , ,2 1 2 - .
2 , 1 2 - . , . .: [1] . ., ., -, . ., 1966; [2] E l d e r t o n W. P., Frequency curves and correlation, Camb., 1953; [3] . ., . ., , 3 ., ., 1983.
)( , K
)( , K ; )( , K approximating functional ,
. * - ; approximable event , - .
; a priori probability , - , , , - . , -. , , - .
-; a prior information - ; , - . , -
)( , A , , A - . ,
,}{ dP )( , A - )(Cap , A - P = }{ , P -, . - P - )( , B - }{dQ . - . . , - ( , [3] ).
.: [1] ., , . ., ., 1960 ( ); [2] ., -, . ., ., 1975; [3] . ., . ., 1981, . 26, 1, . 1531. - ; a prior information usage -
. )( 2,, W -
-
. . .- .
pR A = r , A q -
)( pq , 0 < q < ,p r q . R = r = B+0 ; 0 , )( qp , B
qp )]([ qpp . -
)( 20 ,, WB . -
= B 0 + ,
. r = +R , r q , R )( pq , q M = 0 , cov = V
)( qq .
; )( 02
00 ,, W ,
0 = rY
, 0 = R
, 0W = VW
200
.
)()( 1211121 rVRYWRVRW ++ (*)
. )( dP
- ( pR ) .
r =
,)( dP
V =
,)()()( drr P R = ,pI
pI )( pp . V 0 (*) ;
~
L -
)()~
()~
( dPM
.
,
= )()(:{ 1 rUrp R kk ,} > 0
, R = ,pI V = Uk (*) ;
-
aa )~()~(max
M
pa R . r = 0 - RVR 1 , (*) . .: [1] . ., . ., - , ., 1987; [2] - , ., 1983. ; priori distribution , , , . )( , ( ) . , . - - ( - . .- ) - P - . - ( - ) . . . , , . . .- .
( ) ; a prior risk o f a d e s i g n f u n c t i o n .
; sequential analysis , ( ) , . . ( A. Wald ) . , ( ) , , ( , ). . .- . ,..., 21 ,
)( xF = 1{ P }x - . , . ( ) d D
21
-
( - ) . , ; ( , ) . , nA = = ,,...,; )( 1 n n ,...,1 - , = )( +...,, 10 - ; n 0 { nn A} , ( 0A = = }{ , ). A n 0 {IA nn A} A . nA n ( n ) - , A ( ) . ( ) d = )(d ( ) D A . = )( , d ( ) . )( ,, dW
)( ,, dW M . * = )( ** , d
. ; , = )(d .
)(R =
)()( ,, ddW M
)(*R )(R
( ) , * = = )( ** , d ( ) . )( ,, dW - )( ,1 dWc + , c 0 , )( ,1 dW ( ) .
*d , - , * - . . .- . .
X = )( ,, xnnx PA , n 0 Ex , )( , BE , nx , n -; nA n ( n ) , xP ( ) Ex . , n , )( nxg . )( xgxM - , x . )( xs = )(sup xgxM
22
, sup . Ex x -
)( xs +)( xgxM
; . : )( xs , , , ? .
)( xg : )( xg c < . )( xs )( xg -, )( xg )( xf , )( xfT )( xf - )( xf , )( xfT = = )( 1xgxM .
= n{inf )(:0 nxs })( +nxg
> 0 . )( xs )( xs = )}()({max , xsTxg -
, )( xs = )(lim xgQnn
,
)( xgQ = )}()({max , xgTxg . E
0 = n{inf )(:0 nxs = )}( nxg
. 0{xP < } = ,1 Ex , 0 . = )(:{ xsx > ,)}( xg = )(:{ xsx = )}( xg .
0 = n{inf }:0 nx .
, . , - , - . . . - . 1 1 0 . ( ) D - : d = 1 ( :1H = 1 ) d = 0 ( :0H = 0 ). )( ,1 dW
)( ,1 dW =
====
,
,,,
,,,
01001
dbda
)( ,, dW = )( ,1 dWc + , )(R )(R = )()( dac ++M , )( = d{P = |0 = ,}1 )( = d{P = |1 = }0
, , P
-
. n = {P = }|1 nA :1H = 1 nA = ):( ,...,1 n - ,
)(R = )]([ gc +M
, )(g = ))1((min , ba . nx = = )( , nn - , )( = )(inf
R
)( = )}()({min , Tcg +
. ,)( ,)(g )(T - , , A B , 0 A < B 1 , = = A:{ < < }B , = )(\]10[ ,, BA . 0 = n{inf }:0 n 0( = ) . )(0 xp )(1 xp )(0 xF )(1 xF - d = = 2)( 10 dFdF + -,
n = )()()()( 010111 ...... nn pppp
, - ( 1 )
1.
=
1
1:
AA
n < n <
1
1 BB
0 = n{inf }:0 Cn .
0
11 B
B , d = 1 ,
:1H = 1 ; 0
1
1 AA
, d = 0 , :0H = 0 .
. = )( , d )( = d{1P = }0 , )( = d{0P = }1 > 0 , > 0 ; , ,, )( )( , )( 0M < ,
1M < . . . + < 1
= n{inf )}(:0 , ban ,
d =
,
,
01
ab,
, n d --
= )( , d a = a b = b , - , * = )( ** , d a = a b = b )( , , ,
)( , 0M 0M , 1M 1M - .
* = )( ** , d :0H = 0 :1H = 1 - t . , * = )( ** , d :
= t{inf })()(:0 , bat ,
*d =
,,
,,
01
ab
t = tln ( = 1 - = 0 - )
t = 2/tte , b a
b = /)1(ln , a = )1(/ln
( 2 ). = )( , d :
t~ = )( , t , d~ =
)(f ( c < )(f )
c
c
TT
MM
)()1(ln
)()1(ln
1
1
cB
cB
(1)
,
)( cB = )( ,inf cD, )( cB = )( ,inf cD
,
)( , = )](ln)(ln[ ,, nn xpxp M ,
cD = )(:{ f > }c , cD = )(:{ f < }c .
nf = ,,...,1min rnr f= n = ...,, 21 (2)
)(f ,
nf = )(:{inf cc n })1(ln1 nA ,
)(cn = =
n
rr
DXp
c 1
)(lnsup ,
,
nA = =
n
rrr Xp
11 )(ln ,
n = )( ,...,1 nn XX , n = ...,, 21 n .
*n
f =rnr
f...,,1
max=
)(f ,
nf = nc :{sup })1(ln
1 nA ,
)(cn = =
n
rr
DXp
c 1
)(lnsup ,
.
,, )( xp ,)(f n - (2), (3) (1)- cTM ,
25
?
yox
( = 0)
2
-
cT M 1 - . )(f a - )( 21,, zzN , ,N nA , n = ...,, 21 ,
N{P < } = 1 , 1z 2z NA ,
12{ zz P }a = 1 , 1{zP )(f }2z .
, 1 ,
NM )]1(1[)()1(ln
1o
aW+
,
)( aW =
= }])([])([{maxinf 0 , tfBtafBat + . (4)
(2), (3) )( *2*1
* ,, zzN )(f 1 , , (4)- a - )12( ,
*N = **
:{minnn
ffn ,}a
*1z =
*N
f , *2z =*Nf .
)( , xp = )]([exp bx , R
[6]- . nI = In
rrI
1
~
=
-
,
nI~ = )(:{ nM })1(
1 , n = ...,, 21 ,
)(nM = ==
n
rr
n
rr xpdFxp
11
)()()( ,,
, ,F - . I )( IF > 0
,nI n = ...,, 21 ( , [6] ). ,...,,...,1 nXX a
2 )( 2, aN , - . ( , [1] ) a . 2 - , a
26
N = nn :{min }4 222 Kc (5)
, K
2)1( + ( , [2] ). 2 ,
N = nn :{min )()}(max 2221, kSKcn nn +
1z = ,)2( caN 2z = )2(caN +
a (5)- 0 , 1n 2 , KKn , na ,
2nS n -
, )(K
. 4nxM < ( , [3] [5] , [7] ). .: [1] S t e i n C., Ann. Math. Statis., 1945, v. 16, p. 24358; [2] S t e i n C., W a l d A., Ann. Math. Statis., 1947, v. 18, p. 42733; [3] S t a r r N., Ann. Math. Statist., 1966, v. 37, p. 3650; [4] C h o w Y., R o b b i n s H., Ann. Math. Statist., 1965, v. 36, p. 45762; [5] S i m o n s G., Ann. Math. Statist. 1968, v. 39, p. 194652; [6] L a i T., Ann. Statist., 1976, v. 4, p. 26580; [7] ., , . ., ., 1975. , -; sequential probability ratio test
)(:0 xpH = )(0 xp )(:1 xpH = )(1 xp
)( , d , )( xp ,..., 21 XX . . . . .-
nL = =
n
kkk xpxp
101 )()(
)( , BA , 0 < A < 1 < B ,
= n{inf nL:1 )}( , BA ;
d = 0 ( ) ( 0H )
L A , d = 1 L B . [1]- ( , [2]- ). , . . . .- ( - ) - , A B : A ,)1( B )1( . [3] [4]- ,
dq {[ 0P = dqc {[)1(]}1 100 PM ++ = ]}0 11 Mc+
. . . . ( A B - ) ,
iP iM i iH
-
, i = 10, , ,q 0H - , 0c ,
1c . . . . . ,
d{0P = }1 d{0P = }1 ,
d{1P = }0 d{1P = }0
)( , d 0M 1M . . . . .- [5]- ( , [2] ), [6]- . . . . .- ( A B ) [7] [8]- . .: [1] ., , . ., ., 1960; [2] . ., -, ., 1976; [3] W a l d A., W o l f o w i t z J., Ann. Math. Statist., 1948, v. 19, 3, p. 32639; [4] W a l d A., W o l f o - w i t z J., Ann. Math. Statist., 1950, v. 21, 1, p. 5299; [5] - . ., ii i , 1958, . 1, 1, . 10104; [6] I r l e A., S c h m i t z N., Math. Oper. und Statist., 1984, Bd 15, 1, S. 91104; [7] . ., . ., 1987, . 32, . 1, . 6272; [7] . ., . ., 1988, . 33, . 2, . 295304. ; sequential estimation . t ,}{ AtA t 0 )( ,, PA .
}{ tA ( ), ~
tA , )~
( , - . < -, )
~( , ,
~M =
, )~
( , ; M . - - - ,
)~
( ,
2 )
~( M 1 ))((
MI (*)
( , [1] ), )(I ( ). - )( xp - , [ ., )( xp ] , ( , [3] ); )( xp ,
)( M n n ( ). , ( , [2] ) (*) .
.: [1] W o l f w i t z J., Ann. Math. Statist., 1946, v. 17, 4, p. 48993; [2] . ., . ., - , ., 1974; [3] . ., - . ., . ., 1974, . 19, . 4, . 70013. - ; sequential design of estimation , . , ; sequential design of experiment )( ,...,1 Nyy = Ny1 , ny - - ( N , n - ) 1+ny -
Xyx nn + )( 11 . - . . . - . . . . , . . . .- . . . .- )( ,YY
xP - . )( ,XX -
x . XYx nn 1: ,
n > 1 U = )( ,...,...,1 nxx
)( ,XX - 1x )(q ny1
nA N - . = )( , NU . YB , n = ...,, 21
}|{ 11 nn
U yByP = )( BnxP ( UP ) (1)
, YX - UP . (1)- , ny nx
11ny - . :
n - , - , . )(
P UP - NY = }{ 1Ny -
. N(inf P < ) = 1 , P ,
)( , = )()(
PP dd = )(1
, i
N
i
x yi=
, )(, x = )()(
xx dd PP .
27
-
, Nx1 Ny1
. d ( ) NY - ; )( ,w . , )( ,, NU s .
))(( ,1nyw M . ,
,ln , )( fS =
=
N
ii
x yf i1
)( .
N = NM < x
f = )()( yfdyxx P .
)( x
fSD = 0 , :
)( fSM = )( dxfN
x
=
fN ,
)( fSD = )|)(( 11
1
= nnx
N
n
yyf n DM ,
XB -
)( , )( B
= nNn
{1
1
=
P }, BxN n
.
, - - - :
1) ,K = , lnM ;
2) pR -
I = T)ln(ln ,, M = )( ,, T SM ( ,)( )( - );
3) )( 1
TfvfS M .
IK , : Ny ; -
)( 1Ny )(
, )( , K ,, K )( I I [ )( II - ].
.
1) ,iH i = 10, i - iiH : , i = 10, , 10 U = ,
10 I = :
0N ,, ,11*
infsup)( K
Xii
i
28
10 + 1 , )( 1, ii = ]1[ln 1 iii )(min ,
1
Ki
,
*X , )( , XX - .
2) {P = }j = )( j MPP ...,,1 - M )( ,
- , 1 M : 0N ,,
/supinf
*
R
X ,R = ,, )( KK -
.
3) max N
,
supinf
*R
X .
4) , pR
,
T)()( M )()()( 1 bJNbbb +++ II
T ,
b = M , I , b = )( ...,,1 pbb . 5) . . . ., nN
)|( 11n
ny yM = )( ,nx , )|( 11 nny yD = )( ,nx , (2)
. )( ,x = )( xfT , )( ,x ,)( x N = const )(mS const , - :
= )()(1 yfSmS T (3)
- , xm = )( xf )()(1 xfx T . (2) n -
nN
= na , n
- 0 - . ,
0
ny~ = )( 0,nn xy (3) , )(0 xf = )( 0,x , y y~ . - , )( 0 xf
T , -
))()((diag ...,,1 Nxx - - ( , - ).
1) 3) ; M , 1)( q ( , - ); - - ( , [2] ). 4) 5)
-
)()()( 1 dxxBBSan - n , 4)- B = I , 5)- B = m . , . ., 4) : n , }{ mA
nT1y nL ,
0 nLnI ,
=
n
n
xn
na
1
11 0I n
P
. ph R -:
+ )()(ln 2/1
nn
ndd
ah PP =
= , )21( nn hhh n + ITT (4)
0)( , hn nP ,
)( , nn I )( , I , I ,
( ~ )0( 1, IN ). (4)- - , xP - Xx - , . .: [1] , ., 1983; [2] . ., . . . ., 1979, . 43, 6, . 120326. - ; sequential simplex me-thod , . - ; sequential estimator , - . . [1]- , ( ) - . . ., , , . . - [2]- , . . . . . . . , , . .: [1] ., , . ., ., 1960; [2] . ., . ., . ., 1974, . 19, . 2, . 24556.
; sequential estimators P , - . . . . P - D - ( . )
, ( . ) D - . , . . . , . - 1950- . . ( , [1] ) . . N P n c - . - ,d d c , P , d > c , P . D . . .- . d c , D = ,dD d > c , D = 0 . = NnD dn DNh ,, )(dp = ,!ded , . , = )(d > 0
=
0
)()(d
dpd = =
c
d
dpdnN0
)()(
. . . .
)(d =
++
>=
101
)(
,
,
,,
cdcdndNcddnN
. .: [1] . ., - , ., 1986, . 34063; [2] . ., , ., 1975; [3] - . ., , ., 1979. -; sequential structure - . . .
)(x = i
ix = )(min ...,,1 nxx
, n - n , x = ,,..., )( 1 nxx )(x - , - . i -
ix :
ix =
,,
,,
01
-
-
i
i
i = ,,...,1 n n . :
29
-
=
.
,
,,
01
. ., , ( , ).
.: [1] ., ., , ., 1984.
, - ; sequential hy- potheses testing - , -. , }{ ,, PA - }{ tA - tA( )A ( ), ,d A , k...,,0 ( ) ( ) , )( , d ,iH i = k...,,0 . . .- ( - ) .
, , . ( ) }{ P - , ( , [1] [3] ). }{ P
:0H 0 :1H 1 - M - - ( ) ; )( 10 ,n
, )( 10 ,n , :0H = 0 :1H = 1 - - [ )( 10 , ].
30
)( 21, )( 10 ,n -
( , [4] ): )( , d ,
sup M ( -
). ,
sup M
)( 21, M ( - ); - ( , [5], [6] ). - ( ) , ( , [7], [6] ). :1H 0 :0H = 0 - , 0H - , , 1H , ( , [8] ). ., iX
)1( ,N ,
=
+++> =
21
1
]))1(ln()1([:inf annXmnh
ii
< 1H . m a - {
1P < } , , 10
-
[13] . ., . ., 1987, . 32, . 1, . 14953. ( ) ; arithmetic simulation o f r a n d o m p r o c e s s e s ; , .
k , m . )(mf )( mkf + = = )()( mfkf + , , )( mkf = = )()( mfkf k m , . p ,
)(mp = }:{max mp , ,,, ]1[]10[: nyn 1 -
, )0(ny = 1 , )1(ny = n . nf
)( , mtHn =
)(
)( )()(typ
nm
nn
p tapf , 10 t
. )( , mtH n - }{ ,, nnn PA ,
n = }1{ ...,, n , ,nA - , )}{( mnP = n1 . )(tan - . )( , tH ]10[ ,D . ]10[ ,C - - . )( , mtH n . . ( [1], , [2], [3] ) , n )( , tH - nP -
)(tw . )( , mtH n
. )( , mtH n )( tw - ( , [9], [10] ); ( , [11] ); ( , [6] ) ; ( , [5], [6] )
)(tw - ( , [6] ). ( , [7] ).
ng ,
)( , mtUn =
+nhtk
nn kmgd )(1 , 10 t ;
n nh , nd - . ,, )( mtU n }{ ,, nnn PA - . ng . . . . - ( , [4] ) , )( , mtU n )(tw - . . . ( 1967 ) :
)(mgn = )(m , n )()( , twmtU n ? )(m . [8]- .
.: [1] . ., , 2 ., , 1962; [2] . ., . , 1955, . 103, . 36163; [3] . ., Lect. Notes Math., 1976, v. 550, p. 33550; [4] . ., - . ., . . ., 1959, 6 (13), . 8895; [5] . ., . ., . . , 1982, . 25, . 20711; [6] ., . . ., 1984, . 24, 3, . 14861; [7] ., ., . . ., 1984, . 24, 2, . 7281; [8] . ., . , 1986, . 290, . 786 88; [9] P h i l i p p W., Proc. Symp. Pure Math., 1973, v. 24, p. 23346; [10] B a b u G. J., Probabilistic methods in the theory of arithme-ticcal functions, Calcutta, 1973 ( Diss. ); [11] B i l l i n g s l e y P., Ann. Probab., 1974, v. 2, p. 74991. -; arithmetic distribution ,hnx =
,...,,, 210 =n 0>h - . h - . .- . . . - ( , ). - . .-. . .- 1=h , . . . . .- h2 . ,
)( z 00 z 1)( 0 =z , . .-. 1=h . .-.
.: [1] ., , . ., . 2, ., 1984. , ; arithmetics of probability distributions , .
; ARIMA process , .
; arcsine law - , - . . 1939-
tt ,{ ;0 }00 = . t ]0[ , t , , ,t uu :{ > }00, tu . tt . . :
{P tt < }x = )(21 xF = xarcsin21 ,
10 x , .0>t - ( , [2] ):
31
-
...,,...,1 n ,
kS = ,...1 k ++ 01 0, = Snk ;
nv = kSk :{min = }max0 mnmS
,
nK n,...,, 10 0>kS k - ( , ).
nKnn {lim P < }x = nvnn {lim P < }x = )( xF ,
n
SS nn
}0{}0{lim ...1
-
1) , ts s t ; 2) tA A = 0)( ttA ; 3) 0>t P t <
tt
lim = .
M < , . . . ,
nM < 1n nn ,( )1 nn
lim = P
, . . . . - -
~ = tt ,
~( )0
, ~
.
M = ~
M
0
)( sdsH M =
0
~)( sdsH M
)( , sHH = - . . . .- . t
~ t . .: [1] ., , . ., ., 1975; [2] . ., . ., , ., 1986; [3] ., . ., , . ., ., 1994. ; increasing point , . ( ) ; de-pendent events )( ,, PF AA , AB
)( BAIP )()( BA PP
. , . ; indepen-dence , . . . , , , - , , - .- . )( ,, PF , F F - P . , . .- . A B . . . A B ,, )( FBA
)( AP )( BP . 0)( >BP - A B
)|( BAP =)(
)(B
BAP
P I
, ,)( BAIP A B - .
)( BAIP = )()( BA PP (1)
, A B . 0)( >BP (1)
)|( BAP = )(AP (2)
. : 1) FA , A -; 2) ,0)( =SP FA , S A ; 3) A iB , 21,=i 21 BB , A 21 \ BB ;
4) A B , A B , A B . n )2( >n nAA ,...,1 . .- . ., nm 2 - m - nkkk m ...,,, 21
mkk AA ...,,1 ,
)( ...1 mkk AA IIP = )()( ...1 mkk AA PP (3)
, - ( - ). nAAA ,...,, 21 . .- ( . .- ) - . .- , ,ji
iA ,jA ni ,1= -. - , , , . - . . . . . , - ; - , , , ( , [1], . 24 ). . . - . )( ,, PA , ,A , ,P A - . . .- ( A B ).
nnAA BB ,...,11 (3)
33
-
( ) ,
nBBB ,...,, 21 .
T , 2n - Ttt n ,...,1
mtt BB ,...,1 , tB ( Tt )
. nk 1 kA . .-
kB = }{ ,,, kk AA
. .- . . .- ( , [1] ). ,tX Tt . . )( tXB - . .- , )( tXB tX .
nAA ,...,1 . .- -
kAI - . . ,
)(kAI =
k
k
AA
,
,,
01
. . -. nXX ,...,1 . .- . 1) naa ,...,1
)( ,...,1,...,1 nXX aaF n = )(:{ 1 XP < )(,...,1 nXa < }na
.
)( ...,,1,...,1 nXX aaF n = )()( ...11 nXX aFaF n .
2) )()( ...,,11 nXX apap n nR -
)( ...,,1 naa )( ...,,1,...,1 nXX aap n )()( ...,,11 nXX apap n -
.
3) )( ...,,1...,,1 nXX uuf n =nn XuiXuie ++ ...11M -
nuuu ...,,, 21 - :
)( ...,,1...,,1 nXX uuf n = ,... )()( 11 nXX ufuf n
)( kX uf k =kk XuieM .
- . .- : ( , ., , , ), ( , ., , ) . , .
. 1) . nXX ...,,1 . .- 34
( . . ) - ; ., kXX ...,,1 ,...,,1 nk XX + nk
-
3) .
...,...,,,, 210 nYYYY - )( , yxh ( ) .
1X = ,, )( 10 YYh 2X = ...,,, )( 21 YXh nX = ...,, )( 1 nn YXh - . , ., - . - . 4) . - , , ,
...,...,,, 21 nXXX - - ( m - , kX ,lX lk > m ). . . . 5) - . 2p 2q , . N 1 - N - - ( - N1 ). pA ( qA ) p - ( q - ) .
)( pAP = ,1
pN
N )( qAP = ,
1
qN
N
)( qp AA IP =
qp
NN1
.
N pA qA
. , N S = NS ,
spAAA ,...,, 32 ( ,jp j - ) , - . - , . . . 6) . . - .- . .: [1] . ., , 4 ., ., 1924; [2] . ., , 2 ., ., 1974; [3] - . ., , .: , - , ., 1956; [4] ., - , , . ., ., 1963; [5] ., - , ., . 12, ., 1984; [6] . ., , 2 ., , 1962. , -; property independence of class events , .
- ; test of independence - - . p - n - q - ; qpp ...,,1 .
2q -,
qqqq
q
q
...
............
...
...
21
22221
11211
, , kj jk = 0 .
Hl = =
q
j
njj
n
1
22
, jk
2nHl =
=
q
jjj
1
.
, l - , jj kk - ,
)( rlM =
= ==
= ==
+
+
q
k
p
j
np
j
q
k
p
j
np
j
k
k
rjnjn
jnrjn
1 11
1 11
222
222.
, lln2 2 -
f =
++
=
q
ijj pppp
1
)1()1(21
.
=
jj
jj
jj
ppn
pppp
22
2233
6
92
1
. j - jp = 1 -, p
35
-
, f = 2/)1( pp . = 1 np 6/)112( + .
- , -
2n - , , 2n - . ]
.: [1] ., ., - , . ., ., 1976. ( ) ( - ) ; independent ( statistically inde-pendent ) events )( ,, PF ,
,FA FB
)( BAIP = )()( BA PP (*)
; P , P ( , [2] ). 0)( >AP )()|( BAB PP = , , A B
)|( ABP =)(
)(A
BAP
P I ,
)( BAIP = )()( AB PP . , 0)( >BP )()|( ABA PP = , )( BAIP = )()( BA PP
. 1 . A B , A B , A B , A B ( , [1] ). 2 . 1B 2B A , A 21 BB U . . .- ( , [1] ). nAAA ,...,, 21 ( FiA , =i n,1 ) . . .-, iA - . . . .
3 . iA ( =i n,1 ) -
)( ji AA IP = ,)()( ji AA PP i ;j nji ,, 1=
( 2nC ) . nBBB ,...,, 21 ( FiB )
=I
k
rirB
1
P = =
k
rirB
1
)(P
nk 2 k -
niii k
-
/ - - ; effective number of indepen- dent trials , ( ) , ( - ) ( ) . . . . . .- ( Tt 0 Tt ,...,, 21= ) )(t T m = )( tM . m , *Tm - , t
2)( mmM * T =2
Tm = T
dbTT0
2 )()()2(
t
2)( mmM * T = 2 Tm =
=
1
0
2 )0()()()2(T
TbbTT
. )(b = = ,])([])([ mmM + tt )(t - [ , ,)0(b )(t ]. N - N ...,,1 m = M -
*Nm = =
N
iiN
1
1
N2 - (
,2 - ), . . . . .,
efN =2)0(
Tb m . ., t
( t ) )(t
1T =
0
)())0(1( dbb
, 1TT >> efN
12TT . 2m = )(2 tM 2 = )([ tM
2])( tM km = )( tkM , . . . . .
T )(t . )(t
)(b = eC cos ,
m , 2m 2 , )()1( ,..., t ,
. . . . . T - [1]- .
.: [1] B a y l e y G. V., H a m m e r s l e y J. M., J. Roy. Sta-tist. Soc. Suppl., 1946, v. 8, 2, p. 18497.
, ; indepen-dent spectral types of measures , , .
; independent random events , . - - ; pairwise independent random events , . ( ) ; independent shift o f a p o i n t p r o c e s s , . ; de- pendent trials ( ) - ( - ) . F . f F - - , .
)( = )()( ; dxxf P ...; )([ 11 + fN )]( ;Nf+ )(
*
= k ,...,1 )( dxP i .
)(* )( - C - , )(nC - . P = )( ;dxP - .
.: [1] . ., , 2 ., ., 1975; [2] . ., .: , ., 1964, . 563; [3] . ., . ., . VI . . . . , , 1962, . 42537. * ; dependence - , - , . - , - - ; - - . .
37
-
. ( , , ):
)(cov ; = ,))()(( MMM (1)
)( ; = DD )(cov ; . (2)
, = ba + ( a b , 0a ) , )( , 1
)( ; =+
0101
,
,,
aa
( , [2], ., 249 ). . - - ( ) -. )( , > 0 , - , )( , < 0 , ( , ). ( 1906- ) . - ( , [1] ). )( ,, PF , }{ nF , nFFF ...10 . n m kk FB , nF ,
,...,,...,, 111111{:{ ++ nnnn BBBPP
=++ }| nmnmn FB }/{ 1111 ...,, nnn FBBP
}|{ ,...,11 nmnmnnn F++++ BBP (3)
, nF P , . (3) . . .: [1] ., , . ., ., 1962; [2] . ., , ., , 1980; [3] - . ., , ., 1969. ; inte-raction radius , . ; asymmetric channel , . -; coefficient of skewness ,
As = 33
38
, 3 -
, 2 = D , , . . .,
0=As . , - : As > 0 ; -, , As < 0 . 0M . : , . . , , . . .
a) b)
. . -. ,10=n 51=p ( ) ,10=n 54=p ( )
As =)1(
21ppn
p
;
As =
3312
;
= ;2
)( ;; prk
As =)1(
2pr
p
. . . .
. . , . . . , - . . , - . . . , , -.
As =
=
n
ii xxns 1
33 )(
1
3
, nxx ,...,1 , x s -.
-
=n 0 ),;( pnkP -
: ) =p ,5/1 , ) =p 5/4 .
, - ; skewness of a distribution -. - . . . ( , [1] ). ( ) , . . ( ). . . ( ) , ( ). .: [1] ., , . ., 2 ., ., 1975. * - ; asymptotic independence - . nk
n = =
n
knk
1
-
)( nL - . - , ; - n )( nL - )( nL - . , - . n )( nL )( nL ( ) ,
)( nL )( nL ,
; , LL .)( zn ,
LL .)( zn ; )()(
.n
zn LL
. LL .)( zn ,
FF Ln . .
.
, , .
~)(tam
nL )( nL .
n = =
n
knk
1
n =*n =
=
n
knk
1
*
, *nk -
*nk nk
)( *nkL = )( nkL . , , )( *nL )( nL
. .
~)(tam
nL )(*nL ,
nk - . , [1], . VIII, 28.1, 28.2.
.: [1] ., , ., 1962. * ; asymptotically independent random variables , . * - ; asymptotic expansion - . )( xF ( -, ) - , 1c > 0 ,
)}()(1{lim xFxFxx
+
= 1c
. - - p (
-
)( ,; xf =
=
+
0 !
)/)1(()1(1
k
kk xk
k
++
2111
2cos
kk
. (2)
1x , N
)1ln2( ,, xxf + = )(
!1 2
0
=
+ NN
k
kk xOxkb
x,
kb = tdtiitek
kt
+
0
ln2Im .
- .
kk ,{ }1 , . k - ,an =M
nD =2 3r -
)(lim zgz
< 1
, ,)( zg }{ xn ,0 kX1M < 3k k
1suplim||
Xti
teM
< 1
, n
-
.: [1] . ., . . ., . 2, . ., 1947; [2] E d g e w o r t h F. V., Trans. Comb. Philos. Soc., 1905, v. 20, p. 3665; [3] C r a m e r H., Scand. Aktuarietidscrift, 1928, v. 11, p. 1374, 14180; [4] E s s e e n C.-G., Acta math., 1945, t. 77, p. 125; [5] . ., ., - -, . ., . 1972; [6] . ., , ., 1972; [7] H l l P., Rates of convergence in the central limit theorem, Boston, 1982.
( ) ; asymptotic expansion o f t h e r i s k o f a n e s t i m a t o r ( , ) . ., . . - n - )( ,xf
2 || M = )(
2)1(1
22
231
1 ... + ++++ kk
k nOgngngn , ig . . -.
.: [1] . ., ., 1976, . 21, . 1, . 1633; [2] . ., . . . . 1981, . 45, 3, . 50939. ( ) - ; asymptotic expansion o f e s t i m a t o r s , - , - ( , ) . ., n
)( ,xf . .
i = nkk hnhn +++ 21
21 ...
, ih - , n n . . .-, , .
.: [1] . ., . ., . , 1963, . 149, 3, . 51820; [2] . ., . ., 1973, . 18, 2, . 30311; [3] . ., . ., 1977, . 104, 2, . 179206.
- ; asymptotically Bayes test , . : ,nP , n nX
11 : H 00 : H , 0 ,1 - . n - 0H 1H , , .
}{ n , )|( nnn X nX
n )}({ nn X ,
)|()(suplim ,,
nnnnnnnnXX MM
= 0 (*)
, )}({ nn X - , . . . .- . ., (*) ,
))|(()(lim nnnn nnrr
= 0
, ,)( nnr n n . , ., [1]- . . . .- , - }{ n . . . - . . . ( . . .- ). . . .- , - - . .: [1] . ., , ., 1984; [2] L i n d l e y D., Proc. 4-th Berkeley symp. math. statist. probab., v. 1, Berk., 1961, p. 45368. * ; asymptotically mutually effective sequence of estimators -. ,)( n
( n
, )()(nI
, ,)()(nD )( n(
. n
1)()( )(1)( nn DI
, )( n(
- . .
)( ,,axf =2
2
2)(
21
ax
e
41
-
a . - n , )( ,, axf
n ,...,1 .
2
1
)(
=
n
kk aM =
=
n
kk a
1
)(D = )( an k D = ,n
==
n
kk
n
kk naa
1
2
1
)()( M =
=
=
n
jkjk aa
1,
2)()( M = ,0
2
1
2 )(
=
n
kk naM =
=
n
kk a
1
)(D =
= 2][ an k D = ,22n
, - :
2ln
aM =
n
,
lnln
aM = 0 ,
2
ln
M = 22
n
( , [1] ). ,
)(I = 0
20
0
2
n
n
.
a
= ,1
1 =
n
kkn
2s = 21
)(1
1 =
n
kkn
. 2s - . ,
)()( 2 sM =
= )()()(1
1
1
22 aanan
n
kk
=M =
=
=n
kk anaan 1
32 )()()(1
1 MM = 0 ,
42
ak .
)(D = 0
120
0
2
n
n
)()( DI =
20
01
nn
. , a - 2s . . . . .-. .: [1] . ., . ., - . ., , ., 1979. , - ; asymptotic mergence of states o f a M a r k o v c h a i n , ; . - ; asymptotic stability i n p r o b a b i l i t y , ( ), - . -; asymptotic deficiency , ( ) - .
; asympto-tically efficient estimator . , , - , . . . . . , - , . - . . . .-. .: [1] . ., -, . ., ., 1968; [2] . ., . ., , ., 1979; [2] . ., , ., 1984. ( ) ; asymptotic efficiency o f a t e s t - . 30 40- , - , . . .- . , P
-
:1H 0 :0H = 0 . ,
, - 1N , 2N . 12e = 12 NN . 12e , , , , - , . . - , )(lim ,,12 0
e
(
) - ; - , )(lim ,,120
e
( )
; 1 , )(lim ,,121
e
(
) .
. ., , .
., , 1 ; :1H > 0
:0H = 0 . X t - . t -
X - . , . , X - t - > 0 1 - . - 1 0 . , , 0 , - .
12e .
.: [1] ., ., , . ., ., 1973; [2] B a h a d u r R., Ann. Math. Statist., 1967, v. 38, 2, p. 30324; [3] H o d g e s J., L e h m a n n E., Ann. Math. Statist., 1956, v. 27, 2, p. 32435; [4] . ., - , ., 1995; [5] L a i L., Ann. Statist., 1978, v. 6, 5, p. 102747; [6] B e r k R., B r o w n L., Ann. Statist., 1978, v. 6, 3, p. 56781; [7] K a l l e n b e r g W., Ann. Statist., 1982, v. 10, 2, p. 58394; [8] W i e a n d H., Ann. Statist., 1976, v. 4, 5, p. 110311; [9] K a l l e n b e r g W., Ann. Statist., 1983, v. 11, 1, p. 17082; [10] G r o e n e - b o o m P., O o s t e r h o f f J., Int. Statist. Rev., 1981, v. 49, 2, p. 12741.
, - - ; B a h a - d u r asymptotic efficiency of estimators - , . [1]- . - , . - . -. .: [1] B a h a d u r R., Sankhya, 1960, v. 22, 34, p. 22952; [2] ., , . ., ., 1976; [3] . ., - . ., , ., 1979. , - ; R a o asymptotic efficiency of estimators , - . . , 20- 60- . . . - , . . .: [1] R a o C. R., J. Roy. Statist. Soc., 1962, v. B 24, 1, p. 4672; [2] . ., , . ., ., 1968; [3] . ., . ., , ., 1979. , - - ; W o l f o w i t z asymptotic efficiency of estimators - , . [1] . , - . }{ n b > 0 b -
)}({lim 2121 ,
+ bnbnnn P
- , }{ n . - -.
43
-
.: [1] W o l f w i t z J., . ., 1965, . 10, 2, . 26781; [2] W e i s s L., W o l f o w i t z J., Maximum probability estimators and related topics, [ B. N. Y. ], 1974; [3] . ., . ., - , ., 1979. - - - ; sequence of asymptotic neglectabi- lity random variables nkX , nk ,1( = ; )21 ...,,=n ,
nkX , k - )21( ...,,, nk = - , 0>
nk ,...,, 21=
||{ ,nkXP > } <
. ( -
) )1( ,, nkX nk =
nS , ( , , ).
- ; asymptotic neglecta-bility , .
- ; asympto-tically most powerful test - ( - ) - . ,nX n . 11 : H = 0\ 00 : H . - K }{ n ( Kn }{ )
)(suplim1, nnn
n
M 0
, Kn }{ K
11 . K -
nnn
, 0suplim M
- . . . . . . , n - 44
n - . Kn }{ 11 . . . . , . - ; asymptotically most powerful unbiased test .
; asymptotic estimation theory - , , . . 1. . . .- n
)(nX = )( ,...,1 nXX (1)
-. . . .- ( n ) . , , . . . ( , ), .
(1) P T = )(PT - (1) )( ...,,1 nn XXT ; , . . ., , )( ...,,1 nn XXT
n . , TTn .
1. (1) = )(P =1XM
.
a = =
n
jjXn 1
1 -
( , |)(|{ aPPP > 0} ), 2( an )() 2/12
a .
2. )( xf (1) - p - p . p - -
pz . )( pfn
)())1(( 2/1 ppzpp
.
3. )( xF = 1{XP < }x , )( xFn (1) . )(nF )( ,C - )(F
, ))()(( xFxFn n )( , C -
-
)(xw 2)( xwM = = ))(1()( xFxF . 2. . . . . )(nX X - . X }{ , AX -
X - ,P
}{ , P ,
. - )( X . )( XT . T - . , 0>
0 T ; , ,
}{ . ( ), . . . - uP
P u
, 0 , . 4. 1 3 X =
nX , = 1n . 5. ,X k
X = + (2)
, kR , k . ( ) 0 . , , : X = )(tX
)(tXd = )()( tdtdtS w+ , 10 t (3)
, S , )10( ,2LS , w . , , . - . . ., , . . . . . - - . . .- - . 3. ( - ) )( -
}{ , P X -
. . . .- )( T - , 0 , , )( (
). T -
T - )( - P -
. 6. (1) . 1 3 ,d ,p F . 3- - . ,F jX )( Fg - , )( nFg . 7. 5- ,X
, )(tX t
uduS0
)( .
8. TX m )(tX , Tt 0
. TX = ,0
1 )(T
tdtXT m
, T ;
+
T
T tXXtXT0
1 )(())((
tdXT ) )(R , 0> . . . .- , , .
; P - - , ,
)( uP = ,)(
X
dd uP u
. kR , ,
.
9. TX = )( ,...,1 TXX jX - 1- jX = jjX +1
, ,i )0(2,
, )11( , .
T , T ,
1
2
21
== T
T
jj
T
jjj XXX
.
45
-
10. , (2) X
X . (3) S = )( ;tS
, kR .
1
0
1
0
2 )(21)()(
;; tdtStXdtS
dd
= 0
, ., )( ;tS = )( tS , R ,
= )()()(1
0
11
0
2 tdXtSdttS
= )1( o+ .
, T - T T - T . , . . .- 0 ||{
TT P > } -
. aTT || M ,
}||{expaTTb
M - - . 11. (1) jX 1
0 1 .
n = =
n
jjXn
1
1
-
nM = nD = )1(1 n ;1)4( n -
, ||{ nP > } }2{exp2 2n . 1 10 . 4. T T - ,
TT - T . )()( TTa - .
)()( TTa T - . X }{ ,,,
PAX ( , , - . ). , (1) - . 46
( 1 ), ( 2 ), ( 3 ) , TT . 1 3 , ,...,, )( aa
,...,, )( qp zz )( nF -
,..., a ppz , .
, )( XT TT . . (1) , jX -
}{ , AX )( ;xf . kR . , )}({ ;xf
)}({ ;f
)(I =
Xf
dffji
-
. )( nn - , )(1I
pp
nnn MM )( ,
))(( 1, IN , 0>p .
.
)(uln = =
n
jj uXf
1
)(ln ;
, - . ,
)( nn = =
+n
jj oXfd
dnI1
2/11 )1()(ln
)( ;
.
12. R , )( ;xf = )( xf .
)(I = I =
xdxfxf )()( 12
, )( nn ,
)0( 1, I -
. I , n -, , ,
-
n - . ., )( xf = 1))(( xex 1 , 0>x . 2>
-
; F , X
. 2/1z )( 2/1 zn - . F -
2
)( XnM = 1 , 2
2/1 )( znM = 2))1(1( o+
( 2 ).
14- , , ( ), - , M - . , ., nT - , .
7. . . .- , n - - . -
. , ., 2
)( nnM .
, . k , . , n k . k = )(nk ,
., 0)( nnk nnk /)( 1 , . . .- .
, . -. , , . - . , ,
:l , )(diam1
l 0)(
, . -
. , : - .
15. (1) R - jX )(xf - ; f , - .
dyyfyxbHa )())(( ;
,)(naa = ,)(nbb = H .
n -
)())(( ydFyxbHa n (
) .
.: [1] . ., , ., 1984; [2] . ., . ., - , ., 1979; [3] ., ., , . ., ., 1973; [4] ., ., 48
, . ., ., 1976; [5] ., , . ., 2 ., ., 1975; [6] . ., , ., 1972; [7] G r e n a n d e r U., Abstract inference, N. Y. [a. o.], 1981; [8] L e C a m L., Asymptotic Methods in Statistical Decision Theory, N. Y. [a. o.], 1986; [9] L e h - m a n n E., Theory of Point Estimation, N. Y. [a. o.], 1983. - ; asymptotically unbiased test . ,nX
n . nX 11 : H = 0\ 01 : H n .
]infsup[suplim ,, 10nnnn
n MM
0
, }{ n - . . . . .
.: [1] ., . , . ., ., 1975; [2] - . ., , ., 1984.
; asymp- totically unbiased estimator ; , - . ,}{ nT - )(nb = , nTM ,...,, 21=n nT - . n 0)( nb ,
,}{ nT - . - . . X = )( ,...,1 nXX a = iXM ,
2 = iXD - .
2ns =
=
n
ini Xxn 1
2)(1 , nX = =
n
iiXn 1
1
}{ 2ns 2 . .
. . , , 2 X n -
)( 2nb =22 nsM = 0
2 n
. .: [1] ., - , . ., 2 ., ., 1975; [2] . ., - . ., , ., 1989.
-
( ) - - ; asymptotic admissibi-lity o f s t a t i s t i c a l t e s t , ( ) .
- ; asymptotically minimax tst , . ,nX n . n - : nX 11 : H 00 : H ( , , n - ). ,)( nn X nX n - . n n
)()(suplim ,, 1
nnnnnnnXX MM
= 0 (*)
, )}({ nn X - , . (*)
)(infsup)(inflim ,}{, 11 nnnKnnnnXX
n
MM
= 0
. . . -,
K = )(supinflim:}{{ , 0
nnnn
n X M
} .
. . . .
.: [1] . ., , ., 1984; [2] . ., . ., .: - , -., 1982, . 7990. ; asymp- totically minimax estimator - , - . ,n n
, 1R , .
2
])([supsuplim
n
nnM
2
)]([supinflim *
nn
nM
*n ,
n - . - . - , ., -.
.: [1] . ., , ., 1984; [2] . ., . ., , ., 1979; [3] W a l d A., .: Proc. 2-th Berkeley symp. math. statist. probab., Berk., 1951, p. 111. - ; asymptotically uniformly most powerful test - . . . . . .
)(suplim1, nnn
n
M 0
)(supsuplim1
11,
nnn
n
M 0
.
; asymptotically uniform distribution . U = )( ...,, 21 kS = k ++ ...1 . - h
nnS{lim P
= }mod hj = ,1 h
j = 110 ,...,, h
, U - .
,A U , , . . . . AU -
ahL = mm :{ = }kha +
0:)({min ahkk
L P a }1h =
( , [1] ). nS - . . .-
n 0)2( hrfn -, h , r = 110 ,...,, h ( , [4] ). nS . . .- ( , [2], [3] ).
.: [1] . ., . ., , 3 ., ., 1987; [2] . ., . , 1954, . 98, 4, . 53538; [3] . ., - , ., 1987; [4] D v o r e t z k y A., W o l f o w i t z J., Duke Math. J., 1951, v. 18, p. 50107.
49
-
- ; B a h a d u r asymptotic relative efficiency , ( ) . - ; H o d g e s L e h m a n n asymptotic relative efficiency , ( ) . - ; P i t m n asymptotic rela-tive efficiency , ( ) - .
- ; asym-ptotically normal sequence a - , 2 ( 0 < 2 < ) , n nS
*nS
*{ nSP < }x - n x - ( < x < ) )( x -
}{ *nS ,
nS = n ++ ...1 .
*nS =
nnaSn
,
,)( x )10( , -
. *nS . }{ n , (
) . { }~
nS - -: > 0 n
)(nL = =
n
kkkkk
naa
B 12
2 ;)((1 M > 0) nB ,
kM = ka , kD =2k ,
2nB =
=
n
kk
1
2 , nS~ =
=n
n
kkn
B
aS =
1 .
.: [1] . ., , ., 1972. - ; asymptoti-cally normal transformation
50
-. n
nX{P < }x = )()( xxFn
, )( x . )( nn Xu - nX -
, - )( xun )( xFn -
)( xFn = =
+ ++r
k
rk nOnxPxx
1
2)1(21 )()()()(
,
)( x = )( x ,
,)( xPk x - ,
)( xy = ])([1 xFn =
=
+ ++r
k
rkk nOnxQx
1
2)1(2 )()(
( [1] [2]- ).
)( xun = =
+ ++r
k
rkk nOnxQx
1
2)1(2 )()(
, n
)({ nn XuP < }x = )()(2)1( ++ rnOx
. . . .- .
.: [1] . ., . ., 1959, . 4, 2, . 13649; [2] W a z o w W., Proceeding of sympo-sia in applied mathematics, 1956, v. 6, p. 25159. ; asymptoti-cally normal estimator - , .
., nXX ...,,1 iXM = , iXD = ,2
4iXM < .
X = =
n
iiXn 1
1 2s =
=
n
ii XXn 1
2)(1
2 . . . .-.
, })()({ 22, snXn ,
2243
32
, k = .)(k
iX M .: [1] . ., , ., 1984; [2] . ., . ., - , ., 1979.
-
* - - ; asymptotically normal random variables , . * ; asymptotic normality , - . - ; asymptotically op-timal test - . . . . . . . . : . . . .- : , , . - ; asymptotic Pearson transformation - , - . . . . . t -
X{P < }x = )()( , xtxF
; )( x - . X )( ,txF - :
)( , txF = =
+++r
k
rkk tOtxPxx
1
1)()()()( ,
)( x = ,)( x )( xPk . , :
)( xy = ])([ ,1 txF = ,)()( 1++ rr tOxy
)( xyr = =
+r
k
kk txSx
1
)( .
, )( xur = )()(1++ rr tOxy , 0t
)({ XurP < }x = )()(1++ rtOx .
., X )( , qp - , t = ,01 q p = const -,
)( , txF = )()( , tOpxI +
, ,, )( pxI p -.
)(1 xu = )]1(1[ + pxsx , s = )]1(2[ + ptt
, 0s
)({ 1 XuP < }x = )()(2, sOpxI + .
.: [1] . ., . - ., 1963, . 8, 2, . 12955; [2] . ., . ., , 3 ., ., 1983. ; asymptotic design , .
( ) - ; asymptotic density o f a s e t , . - ; asynchro- nous channel of multiple access , . ; asyn-chronous channel , . , ; assosiated spectrum of a process , .
- ; slowly changing function , . ( ), ; lower value of a game , , .
- ; lower confidence bound , . * ; lower
quartiler p =41
. ,
. ( ) - - ; lower limit o f s e q u e n c e o
