A.A.Abdushukurov R.S.Muradov N.A.Bushueva K.V.Kuzvesov A.K.Tsikh V.V.Denisenko A.A. Imomov CONTENTS On Estimation of Conditional Distribution Function under Dependent Random Right Censored Data 409-416 On the Asymptotics of Homological Solutions to Linear Multidimensional Difference Equations 417-430 Energy Method for Mathematical Modeling of Heat Transfer in 2-D Flow 431-442 On Long-Term Behavior of ContinuousTime Markov Branching Processes Allowing Immigration 443-454 O.V.Khodos G.Kh. <...> Khudayberganov Z.K.Matyakubov L.N.Krivonosov V.A.Lukyanov On Some Systems of Non-algebraic Equations in Cn 455-465 Boundary Version of the Morera Theorem for a Matrix Ball of the Second Type 466-471 Solving Yang-Mills Equations for 4-metrics of Petrov Types II, N, III 472-488 Редакторы: В. <...> Yury V. Zakharov (SibSTU, Krasnoyarsk, Russia) V.I.Kuzovatov A.A.Kytmanov N.A.Lukyanova D.V. Semenova E.K.Myshkina M.G. Sadovsky K.A. Nikitina V.I.Shmoylov G.A.Kirichenko On the Zeta-Function of Zeros of Some Class of Entire Functions 489-499 The Study of Discrete Probabilistic Distributions of Random Sets of Events Using Associative Function 500-514 Some Examples of Finding the Sums of Multiple Series 515-529 Very Low Ergodicity of Real Genomes 530-532 The Solution of Algebraic Equations of Continuous Fractions of Nikiports 533-547 Journal of Siberian Federal University. <...> Mathematics & Physics 2014, 7(4), 409–416 УДК 519.24 On Estimation of Conditional Distribution Function under Dependent Random Right Censored Data Abdurahim A.Abdushukurov∗ Dpt. <...> All rights reserved c – 409 – Abdurahim A.Abdushukurov, Rustamjon S.Muradov On Estimation of Conditional Distribution . or failure times) X1, . . . ,Xn and censoring times Y1, . . . ,Yn of identical objects, which are under study. <...> Then according to the Theorem of Sclar (see, [1]), the joint survival function Sx(t1, t2) can be expressed as x (t2)), t1, t2 0, where Cx(u, v) is a known copula function depending on x, SX x and SY (1) x in a general way. <...> The weak convergence of the empirical process (nhn)1/2{Fxh(·)−Fx(·)} in the space ℓ∞[0,T] of uniformly bounded functions on [0,T <...>
Журнал_Сибирского_федерального_университета._Сер._Математика_и_физика_№4_2014.pdf
Журнал Сибирского федерального университета
Journal of Siberian Federal University
Математика и Физика
Mathematics & Physics
Редакционный совет:
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Е.А. Ваганов
академик РАН
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чл.-корр. РАН,
д-р физ.-мат. наук
В.В. Зуев
чл.-корр. РАН,
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чл.-корр. РАН,
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чл.-корр. РАН,
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Editorial Advisory Board
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Свидетельство
о регистрации СМИ
ПИ №ФС77 - 28724
от 29.06.2007 г.
A.A.Abdushukurov
R.S.Muradov
N.A.Bushueva
K.V.Kuzvesov
A.K.Tsikh
V.V.Denisenko
A.A. Imomov
CONTENTS
On Estimation of Conditional Distribution
Function under Dependent Random
Right Censored Data
409-416
On the Asymptotics of Homological Solutions
to Linear Multidimensional Difference
Equations
417-430
Energy Method for Mathematical Modeling
of Heat Transfer in 2-D Flow
431-442
On Long-Term Behavior of ContinuousTime
Markov Branching Processes Allowing
Immigration
443-454
O.V.Khodos
G.Kh.Khudayberganov
Z.K.Matyakubov
L.N.Krivonosov
V.A.Lukyanov
On Some Systems of Non-algebraic Equations
in Cn
455-465
Boundary Version of the Morera Theorem
for a Matrix Ball of the Second
Type
466-471
Solving Yang-Mills Equations for 4-metrics
of Petrov Types II, N, III
472-488
Редакторы: В.Е.Зализняк, А.В.Щуплев
Компьютерная верстка: Г.В.Хрусталева
Подписано в печать 10.11.14 г. Формат 84×108/16. Усл.печ. л. 12,1.
Уч.-изд. л. 11,7. Бумага тип. Печать офсетная.
Тираж 1000 экз. Заказ 2999
Отпечатано ПЦ БИК СФУ. 660041 Красноярск, пр. Свободный, 82.
2014 7(4)
Стр.1
Editorial Board:
Editor-in-Chief :
Michail I. Gladyshev
Founding Editor:
Vladimir I. Kolmakov
Managing Editor:
Olga F. Aleksandrova
Subject Editor for
Mathematics & Physics:
Prof. Alexander M. Kytmanov
(SibFU, Krasnoyarsk, Russia)
Consulting Editors
Mathematics & Physics:
Prof. Lev Aizenberg
(Bar-Ilan Univ., Tel-Aviv, Israel)
Prof. Viktor K. Andreev
(ICM, Krasnoyarsk, Russia)
Prof. Alexander M. Baranov
(KSP Univ., Krasnoyarsk, Russia)
Prof. Yury Ya. Belov
(SibFU, Krasnoyarsk, Russia)
Prof. Sergey S. Goncharov
(IM, Novosibirsk, Russia)
Prof. Ari Laptev
(KTH, Stockholm, Sweden)
Prof. Vladimir M. Levchuk
(SibFU, Krasnoyarsk, Russia)
Prof. Yury Yu. Loginov
(SibSASU, Krasnoyarsk, Russia)
Prof. Mikhail V. Noskov
(SibFU, Krasnoyarsk, Russia)
Prof. Sergey G. Ovchinnikov
(IPh., Krasnoyarsk, Russia)
Prof. Gennady S. Patrin
(SibFU, Krasnoyarsk)
Prof. Azimbay Sadullaev
(NUUz., Tashkent, Uzbekistan)
Prof. Nikolai Tarkhanov
(Potsdam Univ., Germany)
Prof. Avgust K. Tsikh
(SibFU, Krasnoyarsk, Russia)
Prof. Valery V. Val’kov
(IPh., Krasnoyarsk, Russia)
Prof. Yury V. Zakharov
(SibSTU, Krasnoyarsk, Russia)
V.I.Kuzovatov
A.A.Kytmanov
N.A.Lukyanova
D.V. Semenova
E.K.Myshkina
M.G. Sadovsky
K.A. Nikitina
V.I.Shmoylov
G.A.Kirichenko
On the Zeta-Function of Zeros of
Some Class of Entire Functions
489-499
The Study of Discrete Probabilistic
Distributions of Random Sets of
Events Using Associative Function
500-514
Some Examples of Finding the Sums
of Multiple Series
515-529
Very Low Ergodicity of Real Genomes
530-532
The Solution of Algebraic Equations
of Continuous Fractions of Nikiports
533-547
Стр.2
Journal of Siberian Federal University. Mathematics & Physics 2014, 7(4), 409–416
УДК 519.24
On Estimation of Conditional Distribution Function
under Dependent Random Right Censored Data
Abdurahim A.Abdushukurov∗
Dpt. Probability Theory and Mathematical Statistics
National University of Uzbekistan
VUZ Gorodok, Tashkent, 100174
Uzbekistan
Rustamjon S.Muradov†
Institute of Mathematics
National University of Uzbekistan
VUZ Gorodok, Tashkent, 100174
Uzbekistan
Received 10.06.2014, received in revised form 20.08.2014, accepted 01.10.2014
In this article we study simple integral-type estimator of distribution function under random right censored
observations at fixed covariate values, where the dependence between a life time and a censoring variable
may expressed by a given Archimedean copula. We prove an almost sure asymptotic representation which
provides a key tool for obtaining weak convergence result for estimator.
Keywords: fixed design, right censoring, copulas, asymptotic representation, weak convergence, Gaussian
process.
Introduction
In such research areas as bio-medicine, engineering, insurance, social sciences, . . ., researchers
are interested in positive variables, which are expressed as a time until a certain event. For
example, in medicine the survival time of individual, while in industrial trials, time until breakdown
of a machine are non-negative random variables (r.v.-s) of interest. But in such practical
situations, the observed data may be incomplete, that is censored. This is the case, for example,
in medicine when the event of interest-death due to a given cause and the censoring event is
death due to other cause. In industrial study, it may occur that some piece of equipment is taken
away (that is censored) because it shows some sign of future failure. Moreover, the r.v.-s of
interest (lifetimes, failure times) and censoring r.v.-s usually can be influenced by other variable,
often called prognostic factor or covariate. In medicine, dose of a drug and in engineering some
environmental conditions (temperature, pressure, . . .) are influenced to the observed variables.
The basic problem consist in estimation of distribution of lifetime by such censored dependent
data. The aim of paper is considering this problem in the case of right random censoring model
in the presence of covariable.
Let’s consider the case when the support of covariate C is the interval [0, 1] and we describe our
results on fixed design points 0 x1 x2 . . . xn 1 at which we consider responses (survival
∗a abdushukurov@rambler.ru
†r muradov@myrambler.ru
- Siberian Federal University. All rights reserved
c
– 409 –
Стр.3