Журнал Сибирского федерального университета
Journal of Siberian Federal University
Математика и Физика
Mathematics & Physics
Редакционный совет:
академик РАН
Е.А. Ваганов
академик РАН
И.И. Гительзон
академик РАН
А.Г. Дегерменджи
академик РАН
В.Ф.Шабанов
чл.-корр. РАН,
д-р физ.-мат. наук
В.В. Зуев
чл.-корр. РАН,
д-р физ.-мат. наук
В.Л. Миронов
чл.-корр. РАН,
д-р техн. наук
Г.Л. Пашков
чл.-корр. РАН,
д-р физ.-мат. наук
В.В.Шайдуров
Editorial Advisory Board
Chairman:
Eugene A. Vaganov
Members:
Josef J. Gitelzon
Vasily F. Shabanov
Andrey G. Degermendzhy
Vladimir V. Zuev
Valery L. Mironov
Gennady L. Pashkov
Vladimir V. Shaidurov
Свидетельство
о регистрации СМИ
ПИ №ФС77 - 28724
от 29.06.2007 г.
CONTENTS
A.A.Abdushukurov,
N.S.Nurmuhamedova
Yu.Ya.Belov,
K.V.Korshun,
O.A.Chuesheva
I.V.Frolenkov,
M.A.Darzhaa
A.A.Kuznetsov,
K.V. Safonov
D.V.Lytkina,
V.D.Mazurov
A.A.Makhnev,
M.S. Nirova
Local Asymptotic Normality of Family
of Distributions from Incomplete Observations
141–154
On
Solvability of the Cauchy Problem
for a Loaded System
155-161
The Spaces of Meromorphic Prym
Differentials on Finite Tori
162-172
On the Existence of Solution of Some
Problems for Nonlinear Loaded Parabolic
Equations with Cauchy Data
173-185
Hall’s Polynomials of Finite Two -
Generator Groups of Exponent Seven
186-190
Groups with Given Element Orders
191-203
OnDistance-RegularGraphs with λ = 2
204-210
2014 7(2)
Редакторы: В.Е.Зализняк, А.В.Щуплев
Компьютерная верстка: Г.В.Хрусталева
Подписано в печать 10.04.14 г. Формат 84×108/16. Усл.печ. л. 12,1.
Уч.-изд. л. 11,7. Бумага тип. Печать офсетная.
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Отпечатано ПЦ БИК СФУ. 660041 Красноярск, пр. Свободный, 82.
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. 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)
T.K.Yuldashev
S.A. Solov’eva
S. Sharakhmetov
F.A. Shamoyan,
E.G.Rodikova
S.P.Moshchenko,
A.A.Lyamkina
Modelling the Localized Surface
Plasmon Resonan of Nanoclusters of
Group III Metals and Semimetallic
Antimony
211-217
S.I. Senashov,
O.N.Cherepanova,
A.V.Kondrin
A.S. Serdyukov,
A.V.Patutin,
T.V. Shilova
Elasto-Plastic Bending of a Beam
218-223
Numerical Evaluation of the Truncated
Singular Value Decomposition
Within the Seismic Traveltimes Tomography
Framework
224-234
On Interpolation in the Class of
Analytic Functions in the Unit Disk
with Power Growth of the Nevanlinna
Characteristic
235-243
Construction of Measure by Given
Projections
244-253
Special Version of the CollocationMethod
for a Class of Integral Equations
of the Third Kind Based on HermiteFejer
Interpolation Polynomials
254-259
On Differentiability of the Solution of
the Mixed Boundary Value Problem
for a Nonlinear Pseudohyperbolic
Equation with Respect to Small
Parameters
260-271
Стр.2
Journal of Siberian Federal University. Mathematics & Physics 2014, 7(2), 141–154
УДК 519.24
Local Asymptotic Normality of Family
of Distributions from Incomplete Observations
Abdurahim A.Abdushukurov∗
Nargiza S.Nurmuhamedova
National University of Uzbekistan,
VUZgorodok, Tashkent, 100174,
Uzbekistan
Received 27.12.2013, received in revised form 10.01.2014, accepted 20.02.2014
In this paper we prove the property of local asymptotic normality of the likelihood ratio statistics in the
competing risks model under random censoring by non-observation intervals.
Keywords: competing risks, random censoring, likelihood ratio, local asymptotic normality.
Introduction
The likelihood ratio statistics (LRS) plays an important role in decision theory. For example,
while testing a simple hypothesis H0 against a complicated alternative H1 with an undefined law
of distribution the criterions based on the LRS, according to the Neyman-Pearson lemma, are
uniformly more powerful for any size n of observations (see [1,2]). Here appear some interesting
examples when the alternative H1 depends on n and is close to H0, i.e. H1 = H1n → H0 as
n→∞. In such cases asymptotic properties of the LRS become transparent, which are useful for
estimation theory and hypothesis testing. Among them there is the local asymptotic normality
(LAN) of LRS. There is a number of papers devoted to investigations of the LAN for LRS
and its applications in statistics. The most remarkable works are [2–5], which show that the
LAN allows the development of asymptotic theory for most maximum likelihood and Bayesian
type estimators and prove the contiguality properties of the family of probability distributions.
In the papers [6–11] the properties of the LAN for LRS in the competing risks model (CRM)
under random censoring of observations on the right and both sides were established. This
paper includes investigations of the LAN for LRS in the CRM under random censoring by nonobservation
intervals.
1. Competing risks model under random censoring by
non-observation intervals
a measurable space (X,B) and events (A(1), . . . ,A(k)) forming a complete group, where k is
fixed. In practice, a r.v. X means, obviously, the survival or reliability time of some object
(individual, physical system) exposed to k competing risks and failing in case one of the events
{A(i), i = 1, ..., k}. The pairs {(X,A(i)), i = 1, ..., k} denote the time and reason the object fails
(see more about the CRM in [6,12,13]). During the experiment under homogenous conditions an
In the CRM it is interesting to investigate a random variable (r.v.) X with values from
ensemble (X,A(1), ...,A(k)) is observed, and we obtain a sequence {(Xj,A(1)
Let δ(i)
j = I(A(i)
- Siberian Federal University. All rights reserved
c
– 141 –
j ) be the indicator of the event A(i)
∗a_abdushukurov@rambler.ru
j . Every vector ζj = (Xj, δ(1)
j , ...,A(k)
j , ..., δ(k)
j ), j 1}.
j ) induces
Стр.3