ON THE HYPOTHESIS TESTING CONCERNING THE TYPE
OF THE SURVIVAL FUNCTION
12Evgeniy Chepurin, Innesa Chepurina
Moscow State University, Moscow, Russia 12echepurin@mail.ru, uchsov@cs.msu.ru
The aim of this paper is to discuss the problems of the testing hypothesis concerning a
survival distribution function form.
Let the random variable be a lifetime of a person which exists at the moment . Tt！0In other words, the random variable is the future lifetime of this entity measured from . Tt！0The probability that this person’s lifetime is greater than , i.e. t
StTt！，；,;；??
is called survival distribution function. It is supposed that
a) T，0;
b) S(0)1;！
c) St() is a non-increasing function;
d) S()0.)！
Suppose also that
e) St() is a differentiable function.
In this paper it is assumed that any other information about survival distribution function except the characteristics a) – e) is unknown.
Let us denote by
Sx(1)？pTxTx！，？，！；1| (1) ??xSx()
and
SxSx()(1)？？qp！？！1. (2) xxSx()
Consider the sequence of the embedded events
(3) TkTkTxTx，?，？?，？?，11,????????
xwhere and are integers. From the properties of conditional distributions, the embedding (3) k
and the assumption that , follows that xk，
(4) SxSxTxTxpSx()1|11,！？，，？！？；;；??;；x？1
x？1
(5) SxSkp()(),！?jjk！
and
x？1
(6) SxSkq()()1.！？;；?jjk！
St()Let be a veritable survival distribution function which generates our random life 0
data concerning the life durations of some people group, for example, the participants of some
St()pension fund. In reality an actual functional form of is unknown. 0
1
(：:()()StStAt the same time often it is necessary to test hypothesizes for 101
(?:()()StStttt((ttt((, under alternatives for on the basis of special censored 2011212
St()ttsample of size . Here will be well known survival distribution function, and will N121
St()be known constants. It is supposed that is either parametric function with known 1
ktkt:((parameters or is defined by the life table, for integers . To obtain survival data for 12
iN！1,the members of pension plan we use the following sample scheme: participant , , is i
[,]HHHHobserved only in the time interval , where and are calendar dates for the 1212
beginning and the termination of the observation period in which life status information of
statistical data members can be obtained.
YAAA！(,,,)Let be a sample data, where 12N
AWJJBDyz！(,,,,,,), iiiiiiii
W is calendar dates of withdrawal from observation (by reason other than by death), i
J is calendar dates of joining to pension plan, i
J is observable calendar dates of joining to group under observation, i
B is calendar dates of birth, i
DDW， is calendar dates of death (if ), iii
()yJB！？y is calendar dates of age under joining to observation group , iiii
thz is scheduled age at termination for sample member. ii
In our case
?JifJH，,?ii1J！ ?iHifJH，,?11i?
and
HBifWH？，,?22ii z！?iWBifWH？(.iii2?
Then let be
yy！min,(1)i
zz！max,ni
xintegerxy！，min, ??(1)txt((12
xintegerxz！(max,??()n((txt12
AxAforxintegeryxzxxyx(){:(,1)(1)！(，？?，，？?iiii
?，，？?(，，？(1)(1)},xzxxyzxiii
AAx?()n is the number of . ix
AAx?()For let us define i
zxwhenxzx？，，？1,?ii ??！！()x?ii11,whenzx，？i?
and
2
yxwhenxyx？，，？1,?ii rrx！！()?ii0,whenyx(i?
is the number died in interval ?(,1].xx？AAx?;；xi
The sample design is such that are non random (deterministic) NnJJBHHyz,,,,,,,,xiiiii12variables known to . Hx
At the same time events connected with are random events. Di
It is known [1] that
?xˆq！ (7) xxrx？;；;；?;；?ii:?iAAx;；i
and
x？1ˆˆSxStp！, (8) ;；;；?01jjt！1
ˆˆpq！？1,Sxwhere is unbiased moment estimator of and . q;；jj0x
xIf is exposure
；！？?)?)?xrxN,, (9) ;；;；;；?xii:?iAAx;；i
ˆˆSThen it can be shown in the usual way that q and are consistent and asymptotically normal xx
Stestimators. If under the value is known then it is possible to construct a test statistic (;；011
by using the estimator (2). In the opposite case the estimator (1) can be used.
From the conditions (3) and
2?xrx？;；;；;；?ii?iAAx:;；i (10) 1,,??)Nxrx？;；;；;；??ii?iAAx:;；i
it follows that
ˆˆpqxx！V (11) x?？xrx;；;；;；?ii:?iAAx;；i
ˆˆare consistent estimators for Dq. If under ( the conditions (3), (4) are fulfilled and qq： xx1x1
then we have
dˆqq？1xxSNo！！？0,11. (12) ;；;；xdVx
ˆ~！xqqUsing graphic methods [2], we find a set such that the difference between and ??1xx
，~mwill be obviously considerable. Let be the number of integer points x?~. Denote by ;；
((a test statistic for testing the hypothesis against . 12Lets assume that 2，~：= (13) ;；?xx?~
But it is impossible to find the correct value of its observed significance level. The problem is
：that are complicated dependent random variables. However, observed significance level can x
be estimated.
We have
3
??，~;；;；. (14) lim1;pKHIm，(？??~;；;；obsN?)??m??
2KHIum;Here is the distribution function of ， random variable. The criterion with test ;；m
statistic (13) is consistent.
References
1. London D. Survival models and their estimation. AXTEX Publications. Winsted and Avon,
Connecticut, 1986.
2. Чепурин Е.В. Об аналитико-компьютерных методах разведочного анализа данных.
Колмогоров и современная математика. Тезисы докладов. Москва, МГУ, 2003.
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