b - Database DPPM UII - Universitas Islam Indonesia

Prosiding Seminar Nasional Penelitian, Pendidikan dan Penerapan MIPA, Universitas Negeri
Yogyakarta, ISBN 978-979-9314-4-3, 15 Mei 2010, 105-111
CONFIDENCE BANDS FOR SURVIVOR FUNCTION
OF ONE PARAMETER EXPONENTIAL DISTRIBUTION
UNDER DOUBLE TYPE-II CENSORING
Akhmad Fauzy and R.B. Fajriya Hakim
Department of Statistics, Universitas Islam Indonesia, Jogjakarta 55584, Indonesia
Abstract
This paper describes existing methods and develops new methods for constructing confidence
bands for survivor function of one parameter exponential distribution under double type-II
censoring. Our results are built on extensions of previous work by Raqab (1995) and
Balakrishnan (1990). They use maximum likelihood estimator to construct interval estimation
under double type-II censoring. The confidence bands are developed for survivor function using
the confidence region about survivor function. We will use another method, known as the
bootstrap percentile from Efron (1979). This method gives shorter confidence bands compared
to the traditional method.
Keywords: bootstrap percentile, confidence bands, double type-II censoring
Introduction
The survivor function or reliability function is a property of any random variable that
maps a set of events, usually associated with mortality or failure of some system, onto time. It
captures the probability that the system will survive beyond a specified time. The term
reliability function is common in engineering while the term survivor function is used in a
broader range of applications, including human mortality.
The exponential distribution is often proposed as a model in life testing and reliability
because of its simplicity and the ease with which estimates can be calculated. [2] deals with
inference procedures for the one-parameter exponential model. Inference for the two-parameter
exponential model has been studied by [9], [10], [12] and many others, based on complete
samples and type-II censored data.
In reliability studies, due to time limitations and/or other restrictions on data collection,
several lifetimes of units put on test may not be observed. In addition, sometimes the lowest
and/or highest few observations in a sample could be due to some negligence or some other
extraordinary reasons. In such cases, it is convenient to remove those outlying observations.
Type-II censored samples are considered here, whereby, in an ordered sample of size n , a
known number of observations is missing at either end (single censoring) or both ends (double
censoring). Doubly censored samples have been considered, by authors, like [1] and [11]. They
used maximum likelihood estimator to construct interval estimation for survivor function of one
parameter exponential distribution under double type-II censoring. Using the intervals
estimation for survivor functions at every lifetime develops confidence bands for survivor
function. This band allows us to see the region in which the survivor function lies.
Bootstrap method is a computer-based method for assigning measures of accuracy to
statistical estimates, especially to calculate the confidence interval. The aim of using bootstrap
method is to gain the best estimation from minimal data [5].
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[6] used bootstrap method to construct interval estimation for one parameter exponential
distribution under double type-II censoring. In [7] bootstrap method was utilised to construct the
interval estimation for survivor function for one parameter exponential distribution under
double type-II censoring. In this paper the focus is to make comparison of the confidence bands
for survivor function obtained by the conventional method and bootstrap percentile method.
Methodology
An example of real data is analysed to illustrate the procedure. The data is an air quality
date extracted from the Malaysian Data Report 2000 obtained from the Department of
Environment, Ministry of Science, Technology and Environment. The confidence band for the
survivor function was first constructed by the traditional approach. From the bootstrap’s
repeated samples, the convergence condition is determined. This will be fallowed by developing
the confidence band for the survivor function. The S-Plus software was used in the development
of the programme.
Theory
The actual survival time of an individual, t , can be regarded as the value of a variable T ,
which can take any non-negative value. The survivor function, S t
, is defined to be the
probability that the survival time is greater than or equal to t , and so:
S t P T t 1 F t . (1)
The survivor function can therefore be used to represent the probability that an individual
survives from the time origin to some time beyond t [3].
The essential element in lifetime data analysis is the presence of a nonnegative response,
t , with appreciable dispersion and often with censoring. Due to sampling methods or the
occurrence of some competing risk of removal from the study, several lifetimes of individuals
may be censored. By censored data we mean that, in a potential sample of size n , a known
number of observations is missing at either end (single censoring) or both ends (double
censoring). The type of censoring just described is often called type-II censoring [8].
Suppose some initial observations are censored in addition to some final observations
being censored. Out of the n components put to test, suppose the experimenter fails to observe
the first r and the last s , observations are then said to be double type-II censoring.
t r 1: n tr
2:
n ... tns:
n.
(2)
One Parameter Exponential Distribution
One parameter exponential distribution has probability density function [10]:
1 t
f t
; =
exp
; 0
(3)
Here, is the expected lifetime. Then it is simple exercise to derive the maximum likelihood
estimation of the as [1]:
ns
i r
t i n s t n s n T
ˆ 1 : :
(4)
n s r n s r
T is sometimes referred to as the total observed lifetime or the total time on test, since it is the
total of the observed lifetimes for all n individuals. Confidence intervals on are obtained
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from the exact sampling distribution of the following quantity related to the total time on test T
:
ns
2n
s r
ir
t i : n s t n
s : n n
s rˆ
1
2
2
~ 2n-s-r
(5)
That is, the quantity n
s r/
2 n
s r
degree of freedom. A two-sided, equal-tail,
2 is exactly distributed as a chi-squared variable with
1 level confidence interval can
therefore be constructed from the probability statement:
2
2 n s r ˆ
2
Pr
1
(6)
2 n-s-r
; / 2
2 n-s-r
; 1
/ 2
2
Here 2n sr;
q
is the q-quantile of the
The 1
confidence intervals for is:
2T
ˆ
2T
min
2
2
2
;1
n-s-r -
/ 2
2n-s-r
;
2
distribution, at n
s r
/ 2
ˆ
max
Survivor function on one parameter exponential distribution is:
S
t
t
-1
t
Ft
f t
dt
1
exp
t/
dt
exp
t/
2 degree of freedom.
1 1
(8)
0 0
The 1
confidence for survivor function is:
exp t / ˆ
S t exp t ˆ
(9)
min /
max
(7)
Bootstrap Percentile Method
In setting up of the bootstrap method to find the confidence intervals and estimating
significance levels, the method consists of approximating the distribution of a function of the
observations and the underlying distribution, such as the pivot, denoted by Efron as the
bootstrap distribution of this quantity. This distribution is obtained by replacing the unknown
distribution by the empirical distribution of the data in the definition of the statistical function,
and then resampling the data to obtain a Monte Carlo distribution for the resulting random
variable [5].
Bootstrap method is a computer-based method for assigning measures of accuracy to statistical
estimates, especially to calculate the confidence interval. Bootstrap itself comes from the phrase
“pull oneself up by one’s Bootstrap” which means to stand up by one’s own feet and do with
minimal resources. The minimal resource is a minimum data, data which are free from certain
assumption or data with no assumption at all about the population distribution. The aim of using
bootstrap method is to gain the best estimation from minimal observation [4].
The Bootstrap’s percentile procedures for the interval estimation for survivor function of
exponential distribution under double type-II censoring are as follows:
1 n s r to every observation of type-II censoring,
1. give an equal opportunity
2. take n
s r
sample with replication,
3. do step 2 until B times in order to get an “independent bootstrap replications”,
ˆ 1 2
, ˆ B
, ..., ˆ , and search for convergence condition. Calculate:
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S
b
*b *b
t
t /
exp with
ns
r
4. define the confidence interval at the level
b
b
i
t s t
b
i n n s n
ˆ 1 : :
, (10)
n s r
1 of the bootstrap percentile for
survivor function of one and two parameters exponential distribution under double type
II censoring as:
S
b
/
2 b
1
/ 2
t , S t
Expressions (11) refer to the ideal bootstrap situation where the bootstrap replications
b
are infinite. So if B = 2000 and = 0.05,
/
2
b
is the 50th and
1 / 2
S t
1950th ordered value of the replications.
S t
(11)
is the
5. confidence bands for survivor function are developed using the intervals estimation for
survivor functions at every lifetime.
Results And Discussion
The data presented in air pollutant (special case: carbon monoxide) data report 2000
Malaysia from Department of Environment, Ministry of Science, Technology and Environment.
The first 22 observations in a random sample of 28 lifetimes from carbon monoxide (ppm/part
per million)) on 1 st December 2000 are as follow:
- , - , - , 0.1100, 0.4950, 0.5338, 0.6075,
0.6150, 0.7029, 0.7350, 0.7871, 0.8650, 0.8925, 0.8938,
0.9429, 0.9543, 0.9629, 1.0186, 1.0500, 1.0514, 1.0625,
1.2171, 2.3050, 2.8038, 2.9275, - , - , - .
These data are of double type-II censoring. Using Lilliefors test, the data are exponentially
distributed. As an illustration we will use these data to construct confidence bands for the
survivor function.
Based on the one parameter exponential distribution under double type-II, the intervals
estimation for survivor functions at every lifetime, are tabulated in Table 1.
Table 1.
The floor (F) and ceiling (C) for survivor functions to every lifetime at the level of
99% and 95 % with traditional method
lifetime
Traditional method
99% 95%
F C F C
0.1100 0.884833 0.960657 0.896491 0.954154
0.4950 0.576601 0.834752 0.611585 0.809624
0.5338 0.552245 0.823017 0.588462 0.796332
0.6075 0.508779 0.801179 0.546921 0.771682
0.6150 0.504552 0.798989 0.542861 0.769217
0.7029 0.457555 0.773769 0.497472 0.740904
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0.7350 0.441505 0.764759 0.481860 0.730827
0.7871 0.416646 0.750358 0.457557 0.714761
0.8650 0.382064 0.729329 0.423486 0.691396
0.8925 0.370554 0.722047 0.412074 0.683332
0.8938 0.370018 0.721705 0.411542 0.682953
0.9429 0.350351 0.708890 0.391952 0.668795
0.9543 0.345937 0.705947 0.387539 0.665550
0.9629 0.342643 0.703735 0.384242 0.663113
1.0186 0.322058 0.689576 0.363560 0.647541
1.0500 0.311004 0.681721 0.352395 0.638924
1.0514 0.310520 0.681373 0.351905 0.638542
1.0625 0.306710 0.678618 0.348047 0.635526
1.2171 0.258252 0.641396 0.298499 0.594960
2.3050 0.077002 0.431247 0.101303 0.374037
2.8038 0.044212 0.359486 0.061722 0.302338
2.9275 0.038529 0.343620 0.054585 0.286796
Table 2.
The floor (F) and ceiling (C) for survivor functions to every lifetime at the level of
99% and 95 % bootstrap percentile method
lifetime
Bootstrap percentile method
99% 95%
F C F C
0.1100 0.886015 0.943217 0.894880 0.939989
0.4950 0.580075 0.768692 0.606653 0.756924
0.5338 0.555834 0.753004 0.583346 0.740580
0.6075 0.512543 0.724081 0.541513 0.710500
0.6150 0.508331 0.721200 0.537428 0.707508
0.7029 0.461474 0.688285 0.491785 0.673371
0.7350 0.445461 0.676643 0.476101 0.661319
0.7871 0.420645 0.658165 0.451703 0.642216
0.8650 0.386095 0.631473 0.417536 0.614677
0.8925 0.374589 0.622312 0.406102 0.605240
0.8938 0.374053 0.621882 0.405569 0.604798
0.9429 0.354383 0.605864 0.385953 0.588319
0.9543 0.349966 0.602205 0.381536 0.584558
0.9629 0.346670 0.599459 0.378237 0.581736
1.0186 0.326064 0.581974 0.357552 0.563789
1.0500 0.314992 0.572343 0.346394 0.553916
1.0514 0.314507 0.571917 0.345905 0.553480
1.0625 0.310690 0.568554 0.342050 0.550035
1.2171 0.262094 0.523708 0.292615 0.504214
2.3050 0.079186 0.293764 0.097555 0.273400
2.8038 0.045742 0.225358 0.058955 0.206501
2.9275 0.039922 0.211020 0.052033 0.192619
Connecting the intervals estimation of every lifetime develops confidence bands for survivor
function with traditional method and bootstrap percentile method.
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Comparison of Confidence Bands
Figure 1 and 2 give the confidence bands for survivor function on one parameter exponential
distribution under double type-II censoring using the traditional method and the bootstrap
percentile method.
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
t
traditional t method
bootstrap method
0.110.4950.5340.6080.6150.7030.7350.7870.8650.8930.8940.9430.9540.9631.0191.0501.051.0631.2172.3052.8042.928
Figure 1.
99% Confidence bands for survivor function
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.11 0.4950.5340.6080.6150.7030.7350.7870.8650.8930.8940.9430.9540.9631.0191.0501.05 1.0631.2172.3052.8042.928
traditional method
bootstrap method
t
Figure 2.
95% Confidence bands for survivor function
From these figures 1 and 2 with 99% and 95% respectively, the width of the confidence regions
for the survivor function with bootstrap percentile are narrower compared to the traditional
method.
Conclusion
Using the intervals estimation for survivor functions at every lifetime develops the confidence
bands for survivor function. Bootstrap percentile method proved to be more potential in
constructing confidence bands for survivor function on one parameter exponential distribution
under double type-II censoring than the traditional method. Therefore, bootstrap method can be
used as an alternative method in constructing confidence bands.
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Acknowledgement
The author would like to acknowledged the financial support from Indonesian Directorate
General of Higher Education under Hibah Bersaing 2009-2010.
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