Open Science Journal of Statistics and Application
2015; 3(1): 1-4
Published online January 20, 2015 (http://www.openscienceonline.com/journal/osjsa)
On some estimates of mean residual life function
under random censoring from the right
A. A. Abdushukurov1, R. S. Muradov2, K. S. Sagidullayev1
1
2
Dpt. of Probability Theory and Mathematical Statistics, National University of Uzbekistan, Tashkent, Uzbekistan
Dpt. of Probability Theory and Mathematical Statistics, Institute of Mathematics & National University of Uzbekistan, Tashkent, Uzbekistan
Email address
[email protected] (A. A. Abdushukurov), [email protected] (R. S. Muradov), [email protected] (K. S. Sagidullayev)
To cite this article
A. A. Abdushukurov, R. S. Muradov, K. S. Sagidullayev. On Some Estimates of Mean Residual Life Function under Random Censoring
from the Right. Open Science Journal of Statistics and Application. Vol. 3, No. 1, 2015, pp. 1-4.
Abstract
The problem of analyzing time to event data arises in a number of applied fields, such as medicine, biology, public health,
epidemiology, engineering, economics, and demography. Let X be the time until some specified event. More precisely, in this
work, X is a nonnegative random variable (r. v.) from a homogeneous population. Four functions characterize the distribution
of X, namely, the survival function, which is the probability of an individual surviving to time x; the hazard rate (function),
sometimes termed risk function, which is the chance an individual of age x experiences the event in the next instant in time; the
probability density (or probability mass) function, which is the unconditional probability of the event’s occurring at time x;
and the mean residual life at time x, which is the mean time to the event of interest, given the event has not occurred at x. If we
know any one of these four functions, then the other three can be uniquely determined. We prove asymptotic results for
estimators of mean residual life function both in independent and dependent random right censoring models. Dependence is
described by Archimedean copula function.
Keywords
Survival Function, Mean Residual Life Function, Random Censoring, Copulas
1. Introduction
In survival analysis our interest focuses on a nonnegative
r.v.-s denoting death times of biological organisms or failure
times of mechanical systems. A difficulty in the analysis of
survival data is the possibility that the survival times can be
subjected to random censoring by other nonnegative r.v.-s and
therefore we observe incomplete data. There are various types
of censoring mechanisms. The estimation of distribution
function (d.f.) of lifetime and its functionals from incomplete
data is one of the main goals of statisticians in survival
analysis. In this article we consider only right censoring model
and problem of estimation of mean residual life function
(MRLF) both in independent and dependent censoring cases
assuming that the dependence structure is described by known
copula function. For survival function we use relative-risk
power estimator and its extension to dependent censoring
case.
2. Estimates of MRLF and Its
Properties
2.1. Independent Censoring Case
Let
, , ≥ 1 be a sequence of independent and
identically distributed pairs of positive r.v.-s, where
and
are supposed to be independent with common continuous
d.f.-s
and
=
≤
=
≤ ,
0 =
0 = 0, ∈
= 0, ∞ . Here ’s denotes lifetimes and
’s are censoring times from the right. The observed data
consist a sample of pairs
=
,
, 1≤ ≤
,
with
and
=
, ,
=
≤
! is the
is the
indicator of the event ! . Let " #
= 1−
survival function and the problem is consist in estimating of
2
A. A. Abdushukurov et al.:
On Some Estimates of Mean Residual Life Function under Random Censoring from the Right
the main functional of " # , i.e. MRLF of the object tested,
assuming % = & ' < ∞ :
%
= )" #
*
+'
0
,1 " # - .- ,
∈ 0, 23 .
/
(2.1)
Here % 0 = % and 23 = 4 ∈
∶ "#
= 0 ≤ ∞.
It should be noted that the problem of estimating of functional
(2.1) have been considered by many authors (see, references
review in [1],[4]-[5]). The bootstrap kernel estimation of
MRLF in lack of censoring is considered by recent article of
authors [5]. In the censoring situation many authors used the
product-limit estimator of Kaplan-Meier to estimate " # in
(2.1). Here we use the relative-risk power estimator of
survival function proposed in [1]-[3] and which have some
peculiarities with respect to the product-limit estimator (for
details, refer to [1],[3]).
have common continuous d.f. 6
Note that r.v.-s
=
1 − " # " 7 , ∈ , where " 7
= 1−
. Author
[1]-[3] proposed following estimator of " #
of power type:
1−
where
AAAAA
1, ,
= "#
= 89
'
=
+: <= 1
;
≤⋯≤
,
:
0,
≤ <
1,
<
≥
'
: '
,
, 1≤>≤
,
are the order statistics of
=
∑:D'
∑:D'
:
) : ≤ *
−>+1
,
) : ≤ *
−>+1
− 1 ,? (2.2)
,
=
-relative-risk function estimator and : corresponds to : .
In [1]-[3] for the estimator (2.2) have proved some asymptotic
results (as → ∞). Let
FG
=
,
H/J 3 1 +3 1
=
'+3 1
∈ K, LM,
≥ 1N
is a normed empirical processes sequence, where
K > PQ = R-S
L < PQ =
4
∈
∈
: 6
: 6
=0 ,
=1 .
is clear that PQ = UV P3 , PW ≥ 0 and 2Q =
23 , 2W ≤ ∞. Let X K, LM is Skorokhod’s space of
cádlág functions.
Theorem 2.1[3]. Suppose following conditions are hold:
(C1) 0 <
' ≤ ' < 1;
(C2)
)6 K , 1 − 6 L * ≥ Z for some Z ∈ 0,1 ;
It
(C3) Z
Then as
= ,]
1
→∞
G
[3 \
J
)'+3 \ * )'+W \ *
^
⇒ `
< ∞, for
in X K, LM,
< 2Q .
and Gaussian approximation results on weak and strong forms
up to some large order statistics in the sample with the rates
depending on the order of these statistics. In order to choose
the order statistics we choose a sequence a of integers
such that 1 ≤ a < and require the condition:
(C4) a > bcd for all
large enough and sequence
is asymptotically nonincreasing. For example,
a /
a = e M and a = K M for some K ∈ 0,1 are satisfies
(C4).
Theorem 2.2 [3]. Assume conditions (C3) and (C4) are hold.
Then
|3= 1 +3 1 |
fgh
1ij =kl=
'+3 1
(A)
=n
op 9a
+'/q
o9 aq+' bcd
(B) There exists a sequence r ∙ ,
processes such that
sup wG
1ij =kl=
=x
op )a +'
o9
'/q
aq+' bcd
*w
+a
; , a.s.
+q y/q
y/q
*,
+ a +q ;, a.s.
≥1
− r )Z
bcd + a +q
+ a +q ;,
?
(2.4)
of Wiener
?
(2.5)
` .
Note that for each : r )Z * ^
D
Using theorems 2.1 and 2.2 we investigate corresponding
properties of following estimator of MRLF:
%
= )" #
where as → ∞
(C5) L → ∞,
'/q
*
+'
,z " #
{
%
z
. → 0.
=
Let’s introduce function |
estimator |
=
,1 " # - .- ,
= ,1 " # - .- and its
{
= ,1 " # - .-. Then
{
= )" #
*
+'
9− ,1 = .| - ;.
z
(2.6)
The weak convergence results for MRLF estimator is
contents of next theorem.
Theorem 2.3. Let conditions (C1)-(C3) and (C5) are hold.
Assume also condition
{
(C6) ,] | q .Z
< ∞.
Then as → ∞
}
=
&~
~ R = )" #
'/q
)%
−%
^
*⇒ ~
in X K, LM,
(2.7)
where ~
is mean zero Gaussian process with covariance
function for , R ∈ K, LM:
"# R *
+'
,•€•
{
1,‚
| q - .Z - .
Proof of theorem 2.3. It is not difficult to get a
representation
(2.3)
where `
is mean zero Gaussian process with covariance
function &` ` R = Z)
, R * for , R ∈ K, LM.
In [3] have proved also more strong results on consistency
}
=G
+
'/q
%
)" #
+ )" #
*
+'
* !
+'
,z " # - .-,
{
=
+
(2.8)
Open Science Journal of Statistics and Application 2015; 3(1): 1-4
where !
= ,1 = G - .| - . Then the asymptotic
distribution of sequence of process (2.8) is equivalent to
asymptotic distribution of sequence
z
}
=G
∗
%
+ )"
* !
+'
#
!
= ,1 ` - .| - .
{
Hence by Slutsky’s theorem we see that process } ∗
weakly converges in X K, LM to the process
~
=`
%
+ )" #
*
,1 ` - .| - .
+'
{
The covariance function of ~
evaluated in [7]. The
proof is completed.
Now we prove consistency result for MRLF estimator
.
%
Theorem 2.4. Assume conditions (C4) and (C5) are hold.
Then
fgh
1ij =kl=
|%
| = op …
−%
z=
‡/J
†=
+
{
+ ,z " #
†= =
Jz
=
†=‡
. ˆ. (2.9)
Proof of theorem 2.4. We use representation (2.8) as
%
−%
+
+
=
|
'
'+3 1
'
'+3 1
1−
,1
'+3 1
,z )1 −
{
=
1−
≥1−6
*.-
z
z= )3= \ +3 \ *
It is easy to see that for all
1−
=
| ,1 )1 −
∙
1−
−
.| -
:
≥1−6
=
+
*. .
∈
'
∑ D'
>
+†=
)1 − 6
.
≤
≤
+†=
≤
)1 −
sup
+†=
*
+'
1−6
‰
1−6
≤
= op 1 ∙
Š∙
†=
≤
sup
sup )1 − 6
1ij
=kl=
= op 9 ;.
†=
*
*
+'
+'
(2.11)
On the other hand, with probability 1
0 ≤ ,1 = )1 −
z
By (2.4),
fgh
1ij =kl=
,1
- *.- ≤ 2L .
z= )3= \ +3 \ *
'+3 \
.| -
(2.12)
=kl=
H
J
+
)3= 1 +3 1 *
'+3 1
+ L a +q Š .
(2.13)
Now (2.9) follows from (2.10)-(2.13). The proof is
completed.
Estimation of MRFL under dependent right censored data
consider in section 2.2.
2.2. Dependent Censoring Case
Now we not require independence of sequences
, ≥1
Π,R =
and
, ≥ 1 . Let 6
> , > R , ,R ∈
q
=
× - a joint survival function of the pairs
, .
Πsubmitted the
Then by Sklar’s theorem (see [8]), 6
representation through the survival copula
-, • , -, • ∈
0,1M, as
Π, R = )" #
6
, " 7 R *,
,R ∈
q
.
where " #
and " 7
are marginal survival functions of
and . Supposing that -, • is Archimedean copula, i.e.
-, • = Ž +' Ž - + Ž • M, -, • ∈ 0,1Mq ,
where Ž: 0,1M →
is strong generator function Ž 0 =
∞ and Ž +' is its inverse, the authors [6] proposed following
generalization of estimator (2.2) for " #
by dependent
censoring data
:
= Ž +' •Ž)" j
where
Then by inequalities (1.9.12) and (2.3.25) in [3] we have
sup
= op ‰L a
"• #
(2.10)
,
≤ 2L ∙ 1ijfgh
,
under condition (C5). By the theorem 2.1 and Cramer-Wold
device the sequence of two dimensional processes
weakly converges as → ∞ in Skorokhod
G
,!
space X K, LM × K, LM to the process ` , !
, where
3
1
Ž)" j
* = −•
Œ
¡
= ∑ D'
"j
]
'
*
–
–
…+ ,— ‘
ž R > 0 ŸŽ ‰
= ∑ D'
'
≤ ,
˜= ™
=
˜ ™
”• 9 =
=
…+ ,— ‘ ’= \ “] ”• 9
ž R
>
’= \ “]
Š−Ž‰
, ž
Œj
=1 , ¡
ž R
Œ= \ ˆ
;[š
Œ=› \ ˆ
;[š
œ, (2.14)
1
Œj R ,
− Š .¡
= "j
− ,
= ∑ D'
'
≤
.
Authors [6] showed that estimator (2.14) is generalization
of (2.2), which follows from (2.14) under
-, • =
-•, -, • ∈ 0,1M, i.e. Ž - = −bcd-, - ∈ 0,1M. They
proved the consistency result of estimator (2.14) and proposed
the following estimator of MRLF % :
%¢
=x
9"¢
;
+'
0,
{
,1 "¢
- .-,
≥
∈ £0,
*.
,
?
Consider following conditions:
(C7) Function Ž ∙ is strictly decreasing on (0,1] and first
two derivatives of Ž
and Ψ
are bounded
= − Ž¥
for ∈ ¦, 1M, where ¦ > 0 is arbitrary. Moreover, the first
derivative Ž ¥ is bounded away from zero on 0,1M;
∗
ª
(C8) 0 < ,/ £Ψ)S ¨ *© .Λ∗
= 1,2,
< ∞, for
]
where
4
A. A. Abdushukurov et al.:
2 ∗ = R-S
Λ∗
On Some Estimates of Mean Residual Life Function under Random Censoring from the Right
∈
: S¨
denote both of Λ¬ t = −logS ¨
Λ
= ,]
1 [± j² i‚,³² D'
´µ ‚
>0 ,
and
;
(C9) ,/ wΨ ¥ )S ¨ *w.Λ∗
< ∞;
]
In order to state result on consistency of %¢
with weight
function ¶ ∙ : 0,1M → , we assume also conditions:
(C10) Function ¶ ∶ 0,1M → 0, ∞M is measurable and for
every · > 0,
∗
sup
\∈ ],'+¸M
¶ -
(C11) Function ¶ - 1 − neighborhood of - = 1;
(C12) ,] 0 F)" #
*
< ∞;
0
,1 ¶)
+'
/
+'
is nondecreasing in a
R *.R N .
/
< ∞.
Note that the conditions (C7)-(C9) satisfies, for example,
for copula generators of Clayton and Frank.
Theorem 2.5[6]. Let % = & ' < ∞, conditions (C7)-(C12)
are hold. Then for → ∞,
ε• F = sup ¶)
*|%¢
1»/ ∗
p
| → 0.
−%
(for detailed proof, see [6]).
3. Conclusion
In the case of dependent censoring Zeng & Klein (1995),
Rivest & Wells (2001) investigated copula-graphic estimators:
"¼ #
"• #
=Ž
+'
1
½,]
ž - > 0 …Ž 9
= Ž +' ½− ,]
'
1
’= \ +'
;−Ž9
ž - > 0 Ž¥ 9
’= \
’= \
Π- ,
;ˆ¾ .¡
= ¿VS À− •
"• #
0
)ž - > 0 *
ž -
= Â À1 −
Œ - ¾, (3.2)
; .¡
'iÃ
Œ - Á,
.¡
Œ .¡
Á.
ž -
Figure 2. Plots of estimates " #
(thin-solid), "¼ #
(medium one) and
"• #
(thick-solid) for copula generator Ž - = −bcd- q , - ∈ 0,1M.
Acknowledgements
The authors expressed their sincerely thanks to two
anonymous reviewers and Associate Editor for theirs valuable
comments and suggestions on a previous variant of
manuscript, which drive to the present version of this article.
This work was supported by the grant F4-01 Republic of
Uzbekistan.
References
[1]
Abdushukurov A.A. Nonparametric estimation of the
distribution function based on relative risk function. Commun.
Statist.: Theory & Methods. 1998. v. 27. № 8. pp. 1991-2012.
[2]
Abdushukurov A.A. On nonparametric estimation of reliability
indices by censored samples. Theory. Probab. Appl. 1999. v. 43.
№ 1. pp. 3-11.
[3]
Abdushukurov A.A. Statistics of Incomplete Observations.
Tashkent. University Press. 2009. (In Russian)
[4]
Abdushukurov A.A., Sagidullayev K.S. Asymptotical analysis
of kernel estimates, connected with mean value. LAMBERT
Academic Publishing. 2011. (In Russian)
[5]
Abdushukurov A.A., Sagidullayev K.S. Bootstrap kernel
estimator of mean residual life function. J. Math. Sci. 2013. v.
189. № 6. pp. 884-888.
[6]
Abdushukurov A.A., Muradov R.S. Estimation of survival and
mean residual life functions from dependent random censored
data. New Trends in Math. Sci. 2014. v. 2. № 1. pp. 35-48.
[7]
Gill R.D. Large sample behaviour of the product limit estimator
on the whole line. Ann. Statist. 1983. v. 11. № 3. pp. 49-58.
[8]
Nelsen R.B. An introduction to copulas.-Springer, New York.
2006.
(3.1)
and these estimators are shown to be uniformly consistent and
asymptotical normal. In the case of independent censoring
model these copula estimators are equivalent, consequently, to
the exponential-hazard estimator of Altschuler-Breslow
(1972), and product-limit estimator of Kaplan-Meier (1958):
"¼ #
Figure 1. Plots of estimates " #
(thin-solid), "¼ #
(medium one) and
(thick-solid) for copula generator Ž - = −bcd-, - ∈ 0,1M.
"• #
It is easy to verify that from (2.14) we obtain relative-risk
power estimator of Abdushukurov (1998) (see, (2.2)).
In figures 1 and 2 below we demonstrate plots of estimators
(3.2), (2.14) and (3.1) of " #
using well-known Channing
House data of size n=97 (see [3]). Here, thin-solid line stands
for " # , medium-one for "¼ #
and thick-solid line stands
for estimator "• # . Note that estimate "• #
is defined in
whole line.