Asymptotic properties of modified empirical Kac processes under general random censorship model.pdf 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 1 let A(1),..., A(k) be pairwise disjoint random events, and define the subdistribution functions H(x;i) = P(Z< x,A(i)), i 6 3 = {1,...,k}. Suppose that when observing Z we are interested in the joint behaviour of the pairs (Z, A( °), i 6 3. Let {(Zj, Aj1,..., Ajk)), j > 1} be a sequence of independent replicas of the (Z,A(1),...,A(k)) defined on some probability space {fi,A,Pj. We assume throughout that the functions H(x),H(x;l),...,H(x;k) 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 6 3 1 j=1 where R = [-»;да], §(') = /(i^) isan indicator of event A(l) and 1 n _ Я (x;l) +... + Я (x;L) = - £/(ZJ < x) = Я (x), x e Ж, n ,=1 n 1 n j=1 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: (1) and (2) A crucial role is played the vector-valued empirical process where a 27 Обработка информации / Data processing 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 12 logn. 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°^(x ),B^k (x),...,B(k) (x) such that for an (t) and Bn (0 = (ВПО)(хо), В Ч В),..., Bik)(xk )), t = {to,..., В ), wehave (3) p\\ sup\\\\an(t)-Bn(t)fk+r> >n ^(M(log/?) + z)i l} by distributional equality {k(y;n), 0 l|={w(y;n)-yW(l;n), 0 < y < 1, n > l}. (7) Consequently, in view of (6) and (7) the Wiener process {w,i e з} also admits representations for all (x;/)eKx3: W (1)( H ( x;l); n )=W (H ( x;l); n ), D W(2) (H(x;2);n) = W(H(x;2)+ H (+;l);n)- W(H (+;l);n), W(i) (H(x;i);n) = W(H(x;i) + H (+;i -1);n)- W(H(+;1) + ... + H(+;i -1);n). Now by directly calculations of covariance of processes {w^, i e з} 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 {Z}., j > l} on a probability space {Q, A,P} consider also a sequence {vn,n > 1} of r.v.-s having Poisson distribution with parameter Evn= n, n = 1,2,.... Assume throughout that the two sequences {Z, j > l} and {vn,n > 1} are independent. Kac’s empirical d.f. is 4 I - Ё1 (Zj < x) , if Vn > 1 ^ Hn (x)=i nl=1 0, if vn = 0 a.s., 29 Обработка информации / Data processing while the empirical sub-d.f. one is 1 V'n / (Л\\ „ - VI1Z < x, Av),i e3, if v > 1 a.s., H(x;i) = jn^ v J j ' 0, i e3 if vn = 0 a.s., with H* (x;l) +... + H* (x;k) = H* (x) for all x e R . Here we suppose that sequence {vn,n > 1} is independent of random vectors {(Zj,5(1-1,...,5(kl), J > l}, where 5.”)= I(A^). Note that statistics H* (x;i) (consequently also H* (x)) are unbiased estimators of H (x; i), i e 3 (consequently also of H (x)): E (HП ( x; i )) = - E E = - E 'll E ^).I(Zk < x) ІУі>. I (Zk < r)/vn = .k =1 m ,v = m > = P (v n = m )i = 1 ^ 1 “ nme-n = - V H ( x-i)mP (v = m ) = -H ( x; i)V m-n ( ; ) ( n ) n ( ; )V=1 m! “ nm - = і/(х;/)е_я^- = i/(x;/), (x;/)e 1x3. m=о m! Consequently, £[Я» (x)] = llE[H*n (x;0] = Y,H(x;i) = H(x), xeR. Let’s define a(l) ( x') = sfn (H* ( x;i)- H ( x;i)), i e3 and a(0) ( x) = yfn (H* ( x)- H ( x)) the empirical Kac processes. Theorem 1. If the underlying probability space {Q, A, P} is rich enough, then one can define k + 1 sequences of Gaussian processes Wn0)(x),W(1)(x),...,Wk)(x) such that for a*(t) = (a(0)*(t0),a(1)*(t),...,a(k)*(tk)) and W* (f) = (wi°)(f0),W(\\tl),...,W(k)(tk)), t = (^t,...tk), we have pjsup ||a*(i) -fFj(i)|(i+1) >CV^log/?j 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, (i,;)elx3 and for any i,je3, іф j, rjel: EWf] (x)W® (y) = mm {tf (x),tf (j)}, (9) EW„} (x )Wn0) (y) = min {H (x;*'), H (y; j )} EW(i) (x) W(0) (y) = min {H (x;i),H(y)}. The basic relation between an (t) and a* (t) is the following easily checked identity .,x(vn - n) (10) a* (x) = J-< (x)+H (x;i) г- , V n fn Hence the approximating sequence have respectively the form I - r( (x) = BV‘> (x) + H(x;i) W * (n ) -Jn 30 Abdushukurov, A.A., Sayfulloyeva, G.S. Asymptotic properties of modified empirical Kac processes where B^ (x) is a Poisson indexed Brownian bridge type process of Teorem A and {w1-*-^x), x > o| is a Wiener process. Easy to verify that {^'-’(x), (x;i) e M. x (Я(х;7')),(х,7')е Rx 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* (+да) = -, then by Stirlings formula P ( - n = n)= P (Hi (+») = i): ?! sf2 (i + o(1)), n Inn v and “ nke-n P (H* (+“)> i) = P (Vn > n)= Z = 0 (i) , n k =n+1 k ! Thus H* (x) with positive probability t0 be greater than 1. In order to avoid these undesirable properties, we propose following modifications of Kac statistics: (11) Hn (x) = 1 - (l - Hi (x))l (tf; (x) < l), X eR, I e IK x .д H„ (x;z) = 1 -(l-tf * (x;z ))/(#* (x;z) 1} be a sequence of Poisson r.v.-s with Evn = n. Then for any s> 0 such that n > s log n 8(1 + e/ 3)2’ we have P v - n > - 1 n 1 2 1 f S -n log n V.2 5 1/2 Л P P sup|H*(x;z)-H(x;z)| > 2* Sl0g** Hn (x;i)-H(x;i) sup \\x\\ 2 2n ^s log n V 2n < 2n 12 Л < 4n , 4 3, J 12 У < 4n-4sw, i £3, (12) (13) (14) (15) where w = [16 (1 + e/3)J . Proof. Let y1,y2,... be a sequence of Poisson r.v.-s with Eyk = 1 for all k = 1,2,.... Then t Лк Sn =vn -v = Z(yк -1) = Z^k and Eexp(^) = e^exp(11) = exp(-(t + 1))Z= exp {et(f +1)}. k=1 k=1 k=0 Using Taylor expansion for et , we get E exp (t^k) = exp j1+t+f-+v(t) -(t+1)| = exp j у+v(t )|, 13 where y(t) = - exp(0t), 0

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