Научная статья на тему 'ASYMPTOTIC PROPERTIES OF MODIFIED EMPIRICAL KAC PROCESSES UNDER GENERAL RANDOM CENSORSHIP MODEL'

ASYMPTOTIC PROPERTIES OF MODIFIED EMPIRICAL KAC PROCESSES UNDER GENERAL RANDOM CENSORSHIP MODEL Текст научной статьи по специальности «Математика»

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Ключевые слова
CENSORED DATA / EMPIRICAL ESTIMATES / KAC ESTIMATE / STRONG APPROXIMATION / GAUSSIAN PROCESS

Аннотация научной статьи по математике, автор научной работы — Abdushukurov Abdurakhim A., Sayfulloyeva Gulnoz S.

In this paper, we consider a general random censorship model and prove asymptotic properties of modified empirical Kac processes. This model includes such important special cases as random censorship on the right and competing risks model. Our results use strong approximation method. Cumulative hazard processes also investigated in a similar manner in the general setting. Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests.

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Текст научной работы на тему «ASYMPTOTIC PROPERTIES OF MODIFIED EMPIRICAL KAC PROCESSES UNDER GENERAL RANDOM CENSORSHIP MODEL»

ВЕСТНИК ТОМСКОГО ГОСУДАРСТВЕННОГО УНИВЕРСИТЕТА

2022 Управление, вычислительная техника и информатика № 61

Tomsk: State UniversityJournal of Control and Computer Science

ОБРАБОТКА ИНФОРМАЦИИ DATA PROCESSING

Original article

doi: 10.17223/19988605/61/3

Asymptotic properties of modified empirical Kac processes under general random censorship model

Abdurakhim A. Abdushukurov1, Gulnoz S. Sayfulloyeva2

1 Moscow State University named after M.V.Lomonosov, Tashkent Branch, Tashkent, Uzbekistan, [email protected] 2 Navoi State Pedagogical Institute, Navoi, Uzbekistan, [email protected]

Abstract. In this paper, we consider a general random censorship model and prove asymptotic properties of modified empirical Kac processes. This model includes such important special cases as random censorship on the right and competing risks model. Our results use strong approximation method. Cumulative hazard processes also investigated in a similar manner in the general setting.

Keywords: censored data; empirical estimates; Kac estimate; strong approximation; Gaussian process

For citation: Abdushukurov, A.A., Sayfulloyeva, G.S. (2022) Asymptotic properties of modified empirical Kac processes under general random censorship model. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitelnaja tehnika i informatika - Tomsk State University Journal of Control and Computer Science. 61. pp. 26-36. doi: 10.17223/19988605/61/3

Научная статья

УДК 519.2

doi: 10.17223/19988605/61/3

Асимптотические свойства модифицированных эмпирических процессов Каца в общей модели случайного цензурирования

Абдурахим Ахмедович Абдушукуров1, Гулноз Сайфуллоевна Сайфуллоева2

1 Ташкентский филиал Московского государственного университета им. М.В. Ломоносова, Ташкент, Узбекистан,

a_abdushukurov@rambler. ru 2Навоийский государственный педагогический институт, Навои, Узбекистан, [email protected]

Аннотация. Рассматривается общая модель случайного цензурирования, включающая в себя модели случайного цензурирования справа и конкурирующих рисков. В ней определены эмпирические процессы Каца и определены их модификации. Также исследованы кумулятивные процессы риска с доказательством свойств сильной аппроксимации последовательностями гауссовских процессов.

Ключевые слова: цензурированные данные; эмпирические оценки; оценка Каца; сильная аппроксимация; гауссовские процессы

© A.A. Abdushukurov, G.S. Sayfulloyeva, 2022

Для цитирования: Абдушукуров А.А., Сайфуллоева Г.С. Асимптотические свойства модифицированных эмпирических процессов Каца в общей модели случайного цензурирования // Вестник Томского государственного университета. Управление, вычислительная техника и информатика. 2022. № 61. С. 26-36. doi: 10.17223/19988605/61/3

The empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample. It is not, however, appropriate when the observations are incomplete. Developing the corresponding theory of convergence of considered empirical and concentrated processes to a Gaussian process has been obtained by many scientists. A generalization of these results for the case of competing risks or when present various types of censorship considered by authors [see, for example, [1-3]]. These results have numerous statistical application in areas such as medical follow-up studies, life testing, actuarial sciences and demography (see, also, [4-6]). A general scheme of random censorship was considered by authors includes an competing risks model and random censoring from both sides.

1. Mathematical model

Let Z be a real random variable (r.v.) with distribution function (d.f.) H(x) = P(Z<x), x e R. For a fixed integer к> 1 let A(1),...,A(k) be pairwise disjoint random events, and define the subdistribution functions H (x; i) = P(Z< x, i e3 = {1,...,к}. Suppose that when observing Z we are interested in the joint behaviour of the pairs (Z, A(i)), i e3. Let {(Zj, A^,..., Ajk)), j > 1} be a sequence of independent replicas

of the (Z, A(1),..., A(k)) defined on some probability space {Q, A, P}. We assume throughout that the functions H(x),H(x;l),...,H(x;к) are continuous. Let Hn (x) denote the ordinary empirical d.f. of Z1,...,Zn and introduce the empirical sub d.f. Hn (x;i), i ёЗ

1 n _

Hn (x;i) = ~Y^PiZj < x)> fo) G № x 3>

n J=1

where R = [-да;да], 5^= I(A^ isan indicator of event A() and

1 n _

Я (x;l) +... + Я {х-к) = -YJl(ZJ<x) = Я (x), x e Ж,

n j=l

is the ordinary empirical d.f. Properties of many biometrical estimates depends on limit behaviours of proposed empirical statistics.

The following results are a straightforward consequences of exponential inequality of Dvoretzky-Kiefer-Wolfowits with exactly constant D = 2 from [7, 8]: For all n = 1,2,... and s> 0:

P

and

P

( //,4 \1/2^

n

sup | Hn ( x )-H ( x )|>(il^. iS£

|x|<® ^ 2 n

J

< 2n"(1+s), (1)

sup |Hn (x;i)-H (x;i) > 2

x <да

where

"(1 + s) logn

v ^ 2 n ' ,

V J

A crucial role is played the vector-valued empirical process

a(0)( x) = 4n (Hn (x)-H (x)), dyi ( x) =4n (Hn (x; i)-H (x; i)):

< 4n"(1+s). (2)

i ёз.

The results of our approximation theorems presented here is, quite naturally, the approximation theorems of Komlos-Major-Tusnady's, for the ordinary empirical process with the approximation with the rate of order n 1/2 log n. We will construct the approximation Gaussian processes in terms of Wiener sequences. The following theorem of Burke-Csorgo-Horvath [9, 10] is an extended analogue of Komlos-Major-Tusnady's result [11, 12].

Theorem A [9, 10]. If the underlying probability space {Q, A,P} is rich enough, then one can define k +1 sequences of Gaussian processes B(0)(x),B(1)(x),...,B(k)(x) such that for aB(t) and Bn (t) = (Bno)(x0), ^(x),..., B^(xk)), t = (t0.....^), we have

P | sup |\an (t)-Bn (t |(k+1) > nl/2 (M (log n) + z )|< K exp (-Xz), (3)

for all real z, where M = ( 2k +1) A, K = ( 2k +1) A2 and X = A3 /(2k +1) with A1, A2 and A3 are absolute constants. Moreover, Bn itself is a (k +1) dimensional vector-valued Gaussian process having the same covariance structure as the vector an(t), namely EE>f,(x) = 0, (x,/)eRx3 = 3u{0j and for any

/', j e 3, /' ^ j, ijgI:

EB^\x)B^\y) = min{H(x),H(y)}-H(x).H(y),

EBn\ x) Bn\ y) = min {H ( x;i), H (y;i)}-H ( x; i )• H (y;i),

EBn)( x) Bnj)( y ) = -H ( x;i )• H ( y; j ),

EBno)( x) B^i y ) = min {H ( x; i), H ( y; j )}-H ( x )• H ( y;i). Note that in proving of theorem A (theorem 3.1 in [10]) authors constructed sequence of two -parametrical Gaussian processes Q^(x;«),Q^(x;«),...,Q^(x;«) such that for a„(t) and

Q(t;n) = ^Q^ (x;«),...(x;n)j, t e K^1 the following its Borel-Cantelly consequence of approximation have used

sup

-1

(t )- n/ 2 n )

(¿+i) (

= O In 72 log2 n

(5)

where Q(t;n) is the (k +1) dimensional vector-valued Gaussian process that Q(/;/?)=/?'1 aH (t). Hence

(x,n) = Q, (jc,i)eRx3

and for any i, j e 3, / ^ /', jjeM:

EQ(0) (x-n)<Q(0) (j;m) = min(«,m){min {H{x),H(y)} -H(x)H (j)}, £Q(0)(x;«)Q(0)(j;«) = mm(«,«){mm{i7(x;/),i7(j;7)}-i7(x)i7(j;/)}, E^i\x-n)4\y,m) = mm(n,m){mm{H(xj),H(y^

E<Q{l) (x;n)QU) (y;m) = -mm(n,m)H(x;i)-H(y; j). Observe that jQ''1, / e 3j are Kiefer processes and they satisfying the distributional equality

Q(i) (x;n)=W{'] (H(x;i);n)-H(x;i)Wi) (l;n), (6)

where {W') (y; n), 0 < y < 1, n > 1, i gs} itself are two-parametrical Wiener processes with EW() (y; n) = 0 and

an

l>

EW° (y; n)W° (u; m) = min(n,m)min(y,u), i e 3. It is important to note that though Kiefer processes j'^"1,/ e 3j are dependent processes, but corresponding Wiener processes are independent. Indeed, from proof of theorem A are follows that

Q(r)(x-,n)=K(H(x;\);«),

D

Q{2](x;n) = K(H(x;2)-H(+oD;l);n)-K(H(+oD;l);n),

f>(i)(x;n) = K(H (x;/) + H (+oo;l) +... + H (+00;/-\);n)~ Kin (+oo;l) +... + H (+<x>;i-\);n), i e 3, where H (+«;z ) = lim H ( x; i), H (+<;1) +... + H (+<; k ) = 1.

The Kiefer processes j^(y; n), 0 < y < 1, n > 1| are represented through two-parametrical Wiener processes jW(y;n), 0 < y < 1, n > 1| by distributional equality

j* (y;n), 0 <y < 1, n > 1}=jw(y;n)-yW(1;n), 0 < y < 1, n > 1}. (7)

Consequently, in view of (6) and (7) the Wiener process jw^, i e 3} also admits representations for all (x;z)eKx3:

W (1)( H ( x;1); n )=W (H ( x;1); n ),

D

W( 2) (H (x;2); n ) = W (H (x;2) + H (+<;1); n)- W (H (+<;1); n),

W(i) (H(x;i);n)DW(H(x;i) + H(+<;i -1);n)- W(H(+<;1) +... + H(+<;i -1);n). Now by directly calculations of covariance of processes jW, i e 3} it is easy to believing on its independency.

This paper further structured as follows. In section 1 we introduce the classical Kac processes analogues and their modifications. For its we prove approximation results. Then in section 2 we propose corresponding estimators of hazard functions. For them we also prove approximation results.

2. Kac processes under general censoring

Authors [9] proved the general theorems to obtain approximation for the usual empirical and corresponding cumulative hazard estimates by Gaussian processes for the competing risk generalizations. We prove these results for a corresponding Kac-type processes.

Following of [12] we introduce the modified empirical d.f. of Kac by the following way. Along with sequence jZj, j > 1} on a probability space jQ, A,P} consider also a sequence {vK,n > 1} of r.v.-s having

Poisson distribution with parameter Evn= n, n = 1,2,.... Assume throughout that the two sequences

jZj, j > 1} and {vK,n > 1} are independent. Kac's empirical d.f. is

' 1 V" / \ . -yI(Z < x), if v > 1 as.,

Hn (x) = Jny ( j ), f n ,

0, if Vn = 0 a.s.,

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H* ( x; i )-

while the empirical sub-d.f. one is

1 ¿I(Zj < x, Aj'0),i e 3, f V > 1 a.s.,

n j=1

0, i e 3 if vn = 0 as.,

with H* (x;1) +... + H* (x;k) = H* (x) for all x e R . Here we suppose that sequence jvB,n > 1} is independent of random vectors j(Zj,S'1,...,5(k)), j > 1}, where 5(i)= I(A()). Note that statistics H* (x;i) (consequently also H* (x)) are unbiased estimators of H (x; i), i e3 (consequently also of H (x)):

£ ( H* ( x; i )) =1 £ j£ £

X Si")-1 (< x )

,v = m :

-1 £ H £

>• I ( Zt < x ) / v * - m

k -1

P (v * - m )U

1 да 1 да

- - X H ( x; i )mP (v* - m )--H ( x; i )X

* m -1 * m -1

m •-

m!

^ yt-" _

= #(jc;i)e_,,£— = H(x;i), (х;г)еЖхЗ.

Consequently,

E[H'n (*)] = iE[K fcO] = = H(x),xeR.

1=1 1=1

Let's define a(i) (x) = V« (H* (x;i)-H(x;i)), i e3 and a(0) (x) = Jn (H* (x)-H(x)) the empirical Kac processes.

Theorem 1. If the underlying probability space jQ, A, P} is rich enough, then one can define k + 1 sequences of Gaussian processes W0)(x),W(1)(x),...,W^k)(x) such that for a*(/) = (a(0)*(/„),a(1)*(t),...,a(k'*(tk)) and W* (t) = (W^>(t0),W?)(t,),...,Wik)(tk)), t = (t0,t1,...,tk), we have

P j sup lla*(t)-W„

|(k+- -К , К -r

> C * /2log* K * r,

(8)

Ließ

where r > 2 is an arbitrary integer, C* = C* (r) -depends only on r and K* is an absolute constant. Moreover, W* (t) itself is a (k +1) -dimensional vector-valued Gaussian process with expectation eW'\x) = 0, (x,/)eKx3 and for any i,je 3, / ^ j, rjel:

EW{f ] (x)r„(0) (j) = mm{H(x),H(y)}, EW(i) (x)Wnj (y) = min jH(x;i),H(y; j)}, (9)

EW(i) (x) W(0) (y) = min jH (x;i),H(y)}. The basic relation between aK (t) and a* (t) is the following easily checked identity

(i)

/V )

a* (x) = *рЧ?(x) + H (x; i )

.A(v* - *)

n......4n '

Hence the approximating sequence have respectively the form

i e ^

W°( x )- BVi)( x ) + H ( x; i )

rt

W * ( * )

i e ^,

m-0

where B(l)(x) is a Poisson indexed Brownian bridge type process of Teorem A and {w(n)(x), x > o| is

a Wiener process. Easy to verify that (x;z) e Mx3|={jF* (H (x;z')),(x,z) e Rx^j. The proof of

Teorem 1 is coincides with the proof of theorem 1 of Stute in [13] hence it is omitted. In so far as lim H* (x) = H* (+<») = —, then by Stirlings formula

P (v„ = n)= P (H* (+x) = 1):

n!

~t= (1+° (1)) '

n ^ œ,

and

œ k —n

n e

n ^ œ.

P ( H* (+«)> 1) = P (vB > n )=£ ^ = o (1),

k =n+1 k !

Thus H* (x) with positive probability i0 be greater than 1. In order to avoid these undesirable properties, we propose following modifications of Kac statistics:

tfB(x) = l-(l-#;(x))/(tf;(x)<l),

H„ (x;i) = 1 - (l -H*n (.x;i))l{H*n (x;i) < l), (x;z) e M x 3. The following inequalities are useful in investigating of Kac processes.

Theorem 2. Let {vB,n > 1} be a sequence of Poisson r.v.-s with Evn = n. Then for any s > 0 such that

(11)

>-

log n 8(1 + e/ 3)

we have

P

v — n > — 1 n 1 2

—n log n 2

1/2 \

< 2n

P

P

sup|H*(x;i) — H(x;i)| > 2

Hn (x;i)— H(x;i)

sup

> 2

J

e log n y/2 ^

2n J

e log n Y/2 ^

2n J

< 4n

i e 3,

< 4n-4sw, i e3,

(12)

(13)

(14)

(15)

where w = [16(1 + e/3)] \

Proof. Let y1,Y2,.- be a sequence of Poisson r.v.-s with Ejk = 1 for all k = 1,2,.... Then

i Ak

n n № (e )

=—n-v = Z(Yk - and Eexp(^) = e" exp(ty) = exp(-(t +1))X^p = exp{e(-(t +1)}.

k=1

k=1

Using Taylor expansion for e', we get

E exp (t%k) = exp |1 +1 + + y(t) - (t +1)| = exp | *— + y(t )|,

13

where y(t) = — exp(9t), 0<9< 1. Estimate y(t) taking into account that t3 < t2 under 0<t< 1:

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y(t )< — e < e —. Thus, E exp (t&) = exp J—{1 + e U, 0 < t < 1.

6 6 "k/ * [ 2 ^ 3^

The following result (theorem from [14]) is necessary for our further investigations. Lemma 1 [14]. Let ,n > 1} be a sequence of independent r.v.-s with = 0, n = 1,2,... . Suppose that U,\,...,Xn positive real numbers such that

e

n

2

k=0

£exp)< exp\t2kJ for k = 1,2,...,* |t| <U. Let Л -^j +... + . Then

( v2\

V if o < z <ли,

(16)

P(1^1 +... + > z)<

2 exp

V 2ЛУ Uz

1 iE

2 V 2

V 1/2

2exp —— j, if z >Ли.

1 is

Let in lemma 1 \= 1 + e/ 3, U = 1, z = -\ — n log n I , then we obtain (13). Here 0 < z = ~ I - n log n I <

2 V 2

< (1 + e 3) n = AU. Consider probability in (14). By total probability formula

P

sup j H* ( x; i )- H ( x; i )| > 2

- P

sup

Ixl <да

Hn (x;i)-H(x;i) +1 X Щ)I(Zk < x)

* t=„

>2

s log * 2*

s log * 2*

1/2 \

v„ > *

P (v* > * )+

+P

sup

H(x;i)-H(x;i)-1 X ^IZ <x)

- k -v„+1

>2

slog* 2*

j

\

v „ < *

P(v* <*)<

<P

sup IHn (x; i)- H (x; i)|

>

s log * 2*

1/2 Л

+ P

sup

-t max(*,v* )

1 X ^I ( Zk < x)

* k-min(*,v* )+1

>

slog« 2

12

< 2*-4e+ P

v„ - *

>

s log n 2n

V 1/2 Л

<

< 2*-4e+ 2n4 ws< 4n4ws, i e3,

where we applied (2) and (13) that proves (14). Let's define T{n,) = inf jx: H* (x;i) = 1},i e3. If x > Ti? and vn > n, then Hn (x;i) = 1 and H* (x;i) - H (x;i) > H* (x;i) - HB (x;i) > 0. Then assuming vn > n , we obtain

sup

Ixl <да

H„ (x;i)- H (x;i)

- < max

sup |H* (x;i)- H (x;i)|, sup H*(x;i)-H (x;i)

. <

<i max

sup |H* (x;i) - H (x;i)|, sup |H* (x;i)- H (x;i)|

(17)

sup| H* ( x;i)-H(x;i)|, i

e3.

Under vn < n, it is obvious that Hn (x;z) = H*n (x;z), for all (x;z) e R x 3.

Now taking into account last two relations, total probability formula and (14) we obtain (15). Theorem 2 is proved.

Let an (t) = (a!/1 (t0), a^1 (tx),..., a? (tk )), where al0)(x) = V* (h* (x)-H (x)), ai')(x)^^/* (H«(x;i )--//(x;/)j, (x;z) e Mx 3. We shall prove an approximation theorem of the vector-valued modified empirical

Kac process a„ (/) by the appropriate Gaussian vector-valued process W* (t),t e K from theorem 2.

Theorem 3. Let jT, n > 1} be a numerical sequence satisfying, for each n, the condition T < TH = inf jx: H (x) = 1} < < such that

min |p ( A«)-H (Tn, i )}> 1 - H (Tn )> 2

r log * 2w*

1/2

(18)

If for any s > 0 condition (12) hold, then on a probability space of theorem 2 one can define k +1 sequences of mean zero Gaussian processes WW(x),W^ (x),..., Wk) (x) with the covariance structure (9) such that for

an (t) and W* (t)-(w„(0)(t0 ) ,W„(1)(t1 ) ,...,W„(k)(tk )) we have

P < sup

[ e(-œ;Tn f

an (t)- W* (t)

(k+1)

> Cn 12 log n > < Kn

(19)

where K is an absolute constant, C = C(s) and p = min (r, sw) for any s > 0 . Proof. It is easy to seen that probability in (19) can be estimated by sum

P \ sup

in] ( x ) - W*0) ( x ) > Cn-12 log n 1 + £ PI sup ^ ( x ) - Wn(,) ( x )

J i=1 V x<T„

> Cn-2 logn I- q* + q2n. (20)

Taking into account that for any x < Tn, H*n (x) < H*n (Tn) and if H* (T ) < 1, then an ^ (x) = an' (x) and by formula of total probability

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(0)*

qrn < PI sup

a® ( x ) - W*0 ( x ) > Cn'12 log (Tn ) < 1 | + P ( H* (T* ) > 1)<

< P f sup a^ (x) - W*0 (x) > Cn-% log n 1 + P (H* (Tn ) > 1) <

< Kn r + P ( Hn (T* )-H (T* )> 1 - H (T* ))<

(21)

< Kn-r + P

sup|H* ( x )-H ( x )|>( ^

< Ln

where we have used theorem 1 and analogue of (14) for H* - H, L = K* + 4. Analogously,

q2n < ¿Pf sup ^ (x) - W*,} (x) > Cn'12 logn 1 + ¿P (h* (Tn) > P (A«))

i=1

x<T„

i=1 V x<Tn J i=1

UVPI c,r % I /7 (i)* ^ _ MM".

PI sup a(,)* ( x)-WB(,)( x) > Cn1 log n | + £ PI sup a(,)* (x )-W(i) ( x) > Cn ^log n

i=1 V lxl<œ \V2A

A

(22)

J

f

+kP

v* -v| 11 4r log *

n

2 V 2wn

< kLnr + 2kn

-4r

where we also have used inequalities (13), (15) and theorem 1. Now from (21) and (22) follows (19). Theorem 3 is proved.

3. Estimation of hazard function

In many practical situations, when we are interested in the joint behaviors of the pairs {( Z, A1-1-1), i es}, a crucial role is played by the so-called cumulative hazard functions

{,S,|-,)(x) = exp(-A(,)(x)), i e^}, where A(l)(x) is the i-th hazard function

x = J

-œ (-œ;x]

A (X)-_J 1 - h (u), "

with A(1)(x) +... + A(k)(x)-A(x)= J-is the corresponding hazard function of d.f. H(x).

-œ 1 H (U )

Consider two important special cases of considered generalized censorship model:

a) Let jX,X2,...} be a sequence of independent r.v.-s with common continuous d.f. F. These are

censored on the right by jY , Y2,...} a sequence of independent r.v.-s, independent of the X - sequence, with common continuous d.f. G. One can only observe the sequence of pairs j(Z, §*), k = 1, n}, where Z = min(Xj,Yj) and 8y =8(1 is the indicator of event A = A(1)=jZ = Xj}. In this case

x

k = 2, 1 - H (x) = (1 - F (x))(1 - G (x)), H (x;1)= J(1 - G (u)) dF (u), thus 5(1)(x) = 5 (x) = 1 - F (x).

The useful special case when 1 - G (x ) = (1 - F (x ))P, P > 0, which corresponds to independence of r.v.-s

Z, and 8,., j > 1.

J J 7 ^

b) For k > 1 consider independent sequences jl^^Y^,...} (i = 1,...,k) of independent r.v.-s with common continuous d.f. F and let Z = min ,...,Y(k)). One observes the sequences j(Zy, , i = 1, k} , where 8(i) is the indicator of the event aA\'=jZj = y'. This is the competing risks model with

e3.

S W( x) = 1 - F «( x), i

Define the natural Kac-type estimator

A<ni)(x) = XdHtlK, i

-œ 1 - Hn (u)

of A(i)( x ), i e3. Let w() ( x ) = Jn (a!"1 ( x )-A(i)( x )), i e3 is an Kac-type hazard process and wn (t ) = ( = ),..., w^^ )), t = ,..., tk ), Yn (0 = ^% ),...,)) corresponding vector process with

Y(i)(x) rWn(0)( u ) dH ( u; i ) + W( )( x ) x )(u ) dH (u ) i e3 n ( ) i (1 - H (u ))2 1 - H{x) (1 _ H{u ))2 ,

and x),W()(x),...,W(k)(x)| are Wiener processes with the covariance structure (9). Then for

i e3, EY[J]( x ) = 0 and

EYn i)( x)Y i)( y) = C ( x, y ),

where x, y < ^ = inf |x : H(x) = 1| < œ.

Theorem 4. Let |TK, n > l| be a numerical sequence satisfying for each n, the condition Tn < such

that

n ^ Lo 2 2rbn 2sb2n1

> max < 32sw ,--,-- ^, (23)

log n [ w w

where bn = (l - H (Tn )) 1, s > 0, r > 2. Then on a probability space of theorem 2

P f sup ||w* (t)-7* (i)|(k)> r(*)^

^ te(-ry,Tn ](k) ^

< kOj *-», (24)

where r (n) = 00b2n 1/2 log n, 00 = 00 (s, r), 01 - (absolute) constants. Proof. It is enough to prove that for each i e3

P ^sup (w*) (x) - 7*(,) (x)) > r(*) j < Ф1*-13. (25)

-да

For difference we have representation for each i e3:

x

W )( x )- YÏ )( x )=J

(^ (u) - ^ (u))dH (u;i) Ji) (x) - W> (x)

-r--1--;—---

(1 - H (u ))2

1 - H ( x )

I ( J: )( u )-W( )(u )) dH (u ) ^ 2| ( a* ( u )) dH ( u; i)

(1 - H (u))2

(1 - H (u))2 (1 - H„ (u))

+n

x (0)/ W/ \ 4

12 x a* (u ) da* (u ) = f R(i) (x)

Jœ(1 - H ( u ))(1 - Hn (u )) £ "" ( ).

For sum R® (x) + R® (x) + Ri (x) using (15) and (19) we have

P

Rewrite R{'} as

sup

x<T

£ R*( x)

> 3Cn"V2 log n + sn-/2b3 log n < 3Knp + 2LnW < (3K + 2L ) n~p, i e 3. (26)

al^u)) d (H (u; i)-H (u; i ))

R(,>( x) = n 1/2 I"

4 i (1 - H (u))2 (1 - H* (u))

(0^ / \ 2 v / \

12 fa*( u ) da* (2u ) - r4*( x )+r ;*( x ).

i (1 - H (u ))2 W W

Then by (15) for i e3

P | sup

x<T

rH ( x )

> 2sn-1/2b3 log n | < 2L*-Ws < 2Ln~p.

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There exists an absolute constant A such that

PI sup

x<T

R* ( x ) > 3An-12b2log n |< P (h* (T* )> 1)

(

+P

sup n

x<T„

-V2

x

J

a*** (u ) dO? (u )

(i)* ,

(1 - H (u))2

> 3 An~1/2b2 log n

< Ln- + pn,

(27)

(28)

(29)

so that for any x < T*, H* (x)< H* (T*) and if H* (T* )< 1, then H* (x; i)< H* (T*) and hence

a*^ (x) = a(i)*(x) for i e3. It is enough to estimate probability pn. According to proof of theorem 1 in [13], supposing a^(x) =-y/—^(h* (x)-H(x)), x) = y—(H^ (x;i)-H(x;i)), i e3 and using representation (10), we have proved the theorem 4.

Conclusion

We consider Kac processes in a general censorship scheme, including competing risks model and random censoring from both sides. Our results uses strong approximation method. Cumulative hazard processes also investigated in a similar manner in the general setting. In paper we obtain corresponding approximation results for ordinary empirical processes, for a Kac processes and their modifications and for hazard processes. All results are new and have approximation rates of order n-/2 log n.

References

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œ

œ

2

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Information about the authors:

Abdushukurov Abdurakhim A. (Doctor of Physical and Mathematical Sciences, Professor, Branch of Moscow State University in Tashkent, Tashkent, Uzbekistan). E-mail: [email protected]

Sayfulloyeva Gulnoz S. (Post-graduate Student of the Department of Probability Theory and Mathematical Statistics of the Navoi State Pedagogical Institute, Navoi, Uzbekistan). E-mail: [email protected]

Contribution of the authors: the authors contributed equally to this article. The authors declare no conflicts of interests. Информация об авторах:

Абдушукуров Абдурахим Ахмедович - профессор, доктор физико-математических наук, профессор Ташкентского филиала Московского государственного университета им. М.В. Ломоносова (Ташкент, Республика Узбекистан). E-mail: [email protected]

Сайфуллоева Гулноз Сайфуллоевна - докторант кафедры теории вероятностей и математической статистики Навоийского государственного педагогического института (Навои, Республика Узбекистан). E-mail: [email protected]

Вклад авторов: все авторы сделали эквивалентный вклад в подготовку публикации. Авторы заявляют об отсутствии конфликта интересов.

Поступила в редакцию 04.03.2022; принята к публикации 29.11.2022 Received 04.03.2022; accepted for publication 29.11.2022

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