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Prosiding Seminar Nasional Penelitian, Pendidikan dan Penerapan MIPA, <strong>Universitas</strong> Negeri<br />
Yogyakarta, ISBN 978-979-9314-4-3, 15 Mei 2010, 105-111<br />
CONFIDENCE BANDS FOR SURVIVOR FUNCTION<br />
OF ONE PARAMETER EXPONENTIAL DISTRIBUTION<br />
UNDER DOUBLE TYPE-II CENSORING<br />
Akhmad Fauzy and R.B. Fajriya Hakim<br />
Department of Statistics, <strong>Universitas</strong> <strong>Islam</strong> <strong>Indonesia</strong>, Jogjakarta 55584, <strong>Indonesia</strong><br />
Abstract<br />
This paper describes existing methods and develops new methods for constructing confidence<br />
bands for survivor function of one parameter exponential distribution under double type-II<br />
censoring. Our results are built on extensions of previous work by Raqab (1995) and<br />
Balakrishnan (1990). They use maximum likelihood estimator to construct interval estimation<br />
under double type-II censoring. The confidence bands are developed for survivor function using<br />
the confidence region about survivor function. We will use another method, known as the<br />
bootstrap percentile from Efron (1979). This method gives shorter confidence bands compared<br />
to the traditional method.<br />
Keywords: bootstrap percentile, confidence bands, double type-II censoring<br />
Introduction<br />
The survivor function or reliability function is a property of any random variable that<br />
maps a set of events, usually associated with mortality or failure of some system, onto time. It<br />
captures the probability that the system will survive beyond a specified time. The term<br />
reliability function is common in engineering while the term survivor function is used in a<br />
broader range of applications, including human mortality.<br />
The exponential distribution is often proposed as a model in life testing and reliability<br />
because of its simplicity and the ease with which estimates can be calculated. [2] deals with<br />
inference procedures for the one-parameter exponential model. Inference for the two-parameter<br />
exponential model has been studied by [9], [10], [12] and many others, based on complete<br />
samples and type-II censored data.<br />
In reliability studies, due to time limitations and/or other restrictions on data collection,<br />
several lifetimes of units put on test may not be observed. In addition, sometimes the lowest<br />
and/or highest few observations in a sample could be due to some negligence or some other<br />
extraordinary reasons. In such cases, it is convenient to remove those outlying observations.<br />
Type-II censored samples are considered here, whereby, in an ordered sample of size n , a<br />
known number of observations is missing at either end (single censoring) or both ends (double<br />
censoring). Doubly censored samples have been considered, by authors, like [1] and [11]. They<br />
used maximum likelihood estimator to construct interval estimation for survivor function of one<br />
parameter exponential distribution under double type-II censoring. Using the intervals<br />
estimation for survivor functions at every lifetime develops confidence bands for survivor<br />
function. This band allows us to see the region in which the survivor function lies.<br />
Bootstrap method is a computer-based method for assigning measures of accuracy to<br />
statistical estimates, especially to calculate the confidence interval. The aim of using bootstrap<br />
method is to gain the best estimation from minimal data [5].<br />
105
[6] used bootstrap method to construct interval estimation for one parameter exponential<br />
distribution under double type-II censoring. In [7] bootstrap method was utilised to construct the<br />
interval estimation for survivor function for one parameter exponential distribution under<br />
double type-II censoring. In this paper the focus is to make comparison of the confidence bands<br />
for survivor function obtained by the conventional method and bootstrap percentile method.<br />
Methodology<br />
An example of real data is analysed to illustrate the procedure. The data is an air quality<br />
date extracted from the Malaysian Data Report 2000 obtained from the Department of<br />
Environment, Ministry of Science, Technology and Environment. The confidence band for the<br />
survivor function was first constructed by the traditional approach. From the bootstrap’s<br />
repeated samples, the convergence condition is determined. This will be fallowed by developing<br />
the confidence band for the survivor function. The S-Plus software was used in the development<br />
of the programme.<br />
Theory<br />
The actual survival time of an individual, t , can be regarded as the value of a variable T ,<br />
which can take any non-negative value. The survivor function, S t<br />
, is defined to be the<br />
probability that the survival time is greater than or equal to t , and so:<br />
S t P T t 1 F t . (1)<br />
<br />
The survivor function can therefore be used to represent the probability that an individual<br />
survives from the time origin to some time beyond t [3].<br />
The essential element in lifetime data analysis is the presence of a nonnegative response,<br />
t , with appreciable dispersion and often with censoring. Due to sampling methods or the<br />
occurrence of some competing risk of removal from the study, several lifetimes of individuals<br />
may be censored. By censored data we mean that, in a potential sample of size n , a known<br />
number of observations is missing at either end (single censoring) or both ends (double<br />
censoring). The type of censoring just described is often called type-II censoring [8].<br />
Suppose some initial observations are censored in addition to some final observations<br />
being censored. Out of the n components put to test, suppose the experimenter fails to observe<br />
the first r and the last s , observations are then said to be double type-II censoring.<br />
t r 1: n tr<br />
2:<br />
n ... tns:<br />
n.<br />
(2)<br />
One Parameter Exponential Distribution<br />
One parameter exponential distribution has probability density function [10]:<br />
1 t <br />
f t<br />
; =<br />
exp<br />
; 0<br />
(3)<br />
<br />
Here, is the expected lifetime. Then it is simple exercise to derive the maximum likelihood<br />
estimation of the as [1]:<br />
ns<br />
i r<br />
t i n s t n s n T<br />
ˆ 1 : : <br />
(4)<br />
n s r n s r<br />
T is sometimes referred to as the total observed lifetime or the total time on test, since it is the<br />
total of the observed lifetimes for all n individuals. Confidence intervals on are obtained<br />
106
from the exact sampling distribution of the following quantity related to the total time on test T<br />
:<br />
ns<br />
2n<br />
s r<br />
ir<br />
t i : n s t n<br />
s : n n<br />
s rˆ<br />
1<br />
2<br />
2<br />
<br />
~ 2n-s-r<br />
<br />
<br />
<br />
(5)<br />
That is, the quantity n<br />
s r/<br />
<br />
2 n<br />
s r<br />
degree of freedom. A two-sided, equal-tail, <br />
2 is exactly distributed as a chi-squared variable with<br />
1 level confidence interval can<br />
therefore be constructed from the probability statement:<br />
<br />
<br />
<br />
2<br />
2 n s r ˆ<br />
2<br />
Pr <br />
<br />
<br />
1<br />
<br />
(6)<br />
2 n-s-r<br />
; / 2<br />
2 n-s-r<br />
; 1<br />
/ 2<br />
<br />
<br />
<br />
2<br />
Here 2n sr;<br />
q<br />
is the q-quantile of the<br />
The 1 <br />
confidence intervals for is:<br />
2T<br />
ˆ<br />
2T<br />
min <br />
2<br />
2<br />
2<br />
;1<br />
<br />
n-s-r -<br />
/ 2<br />
2n-s-r<br />
;<br />
2<br />
distribution, at n<br />
s r<br />
/ 2<br />
ˆ<br />
<br />
max<br />
Survivor function on one parameter exponential distribution is:<br />
S<br />
t<br />
t<br />
-1<br />
t<br />
<br />
Ft<br />
f t<br />
dt<br />
1<br />
exp<br />
t/<br />
dt<br />
exp<br />
t/<br />
<br />
2 degree of freedom.<br />
1 1 <br />
(8)<br />
0 0<br />
The 1 <br />
confidence for survivor function is:<br />
exp t / ˆ<br />
S t exp t ˆ<br />
(9)<br />
<br />
min /<br />
max<br />
(7)<br />
Bootstrap Percentile Method<br />
In setting up of the bootstrap method to find the confidence intervals and estimating<br />
significance levels, the method consists of approximating the distribution of a function of the<br />
observations and the underlying distribution, such as the pivot, denoted by Efron as the<br />
bootstrap distribution of this quantity. This distribution is obtained by replacing the unknown<br />
distribution by the empirical distribution of the data in the definition of the statistical function,<br />
and then resampling the data to obtain a Monte Carlo distribution for the resulting random<br />
variable [5].<br />
Bootstrap method is a computer-based method for assigning measures of accuracy to statistical<br />
estimates, especially to calculate the confidence interval. Bootstrap itself comes from the phrase<br />
“pull oneself up by one’s Bootstrap” which means to stand up by one’s own feet and do with<br />
minimal resources. The minimal resource is a minimum data, data which are free from certain<br />
assumption or data with no assumption at all about the population distribution. The aim of using<br />
bootstrap method is to gain the best estimation from minimal observation [4].<br />
The Bootstrap’s percentile procedures for the interval estimation for survivor function of<br />
exponential distribution under double type-II censoring are as follows:<br />
1 n s r to every observation of type-II censoring,<br />
1. give an equal opportunity <br />
2. take n<br />
s r<br />
sample with replication,<br />
3. do step 2 until B times in order to get an “independent bootstrap replications”,<br />
ˆ 1 2<br />
, ˆ B<br />
, ..., ˆ , and search for convergence condition. Calculate:<br />
107
S<br />
b<br />
*b *b<br />
t<br />
<br />
t / <br />
exp with<br />
ns<br />
r<br />
4. define the confidence interval at the level <br />
b<br />
b<br />
i<br />
t s t<br />
b<br />
i n n s n<br />
ˆ 1 : :<br />
<br />
, (10)<br />
n s r<br />
1 of the bootstrap percentile for<br />
survivor function of one and two parameters exponential distribution under double type<br />
II censoring as:<br />
<br />
S<br />
b<br />
<br />
/<br />
2 b<br />
<br />
1<br />
/ 2<br />
t , S t<br />
<br />
Expressions (11) refer to the ideal bootstrap situation where the bootstrap replications<br />
b<br />
are infinite. So if B = 2000 and = 0.05, <br />
/<br />
2<br />
b<br />
is the 50th and <br />
1 / 2<br />
S t<br />
1950th ordered value of the replications.<br />
S t<br />
(11)<br />
is the<br />
5. confidence bands for survivor function are developed using the intervals estimation for<br />
survivor functions at every lifetime.<br />
Results And Discussion<br />
The data presented in air pollutant (special case: carbon monoxide) data report 2000<br />
Malaysia from Department of Environment, Ministry of Science, Technology and Environment.<br />
The first 22 observations in a random sample of 28 lifetimes from carbon monoxide (ppm/part<br />
per million)) on 1 st December 2000 are as follow:<br />
- , - , - , 0.1100, 0.4950, 0.5338, 0.6075,<br />
0.6150, 0.7029, 0.7350, 0.7871, 0.8650, 0.8925, 0.8938,<br />
0.9429, 0.9543, 0.9629, 1.0186, 1.0500, 1.0514, 1.0625,<br />
1.2171, 2.3050, 2.8038, 2.9275, - , - , - .<br />
These data are of double type-II censoring. Using Lilliefors test, the data are exponentially<br />
distributed. As an illustration we will use these data to construct confidence bands for the<br />
survivor function.<br />
Based on the one parameter exponential distribution under double type-II, the intervals<br />
estimation for survivor functions at every lifetime, are tabulated in Table 1.<br />
Table 1.<br />
The floor (F) and ceiling (C) for survivor functions to every lifetime at the level of<br />
99% and 95 % with traditional method<br />
lifetime<br />
Traditional method<br />
99% 95%<br />
F C F C<br />
0.1100 0.884833 0.960657 0.896491 0.954154<br />
0.4950 0.576601 0.834752 0.611585 0.809624<br />
0.5338 0.552245 0.823017 0.588462 0.796332<br />
0.6075 0.508779 0.801179 0.546921 0.771682<br />
0.6150 0.504552 0.798989 0.542861 0.769217<br />
0.7029 0.457555 0.773769 0.497472 0.740904<br />
108
0.7350 0.441505 0.764759 0.481860 0.730827<br />
0.7871 0.416646 0.750358 0.457557 0.714761<br />
0.8650 0.382064 0.729329 0.423486 0.691396<br />
0.8925 0.370554 0.722047 0.412074 0.683332<br />
0.8938 0.370018 0.721705 0.411542 0.682953<br />
0.9429 0.350351 0.708890 0.391952 0.668795<br />
0.9543 0.345937 0.705947 0.387539 0.665550<br />
0.9629 0.342643 0.703735 0.384242 0.663113<br />
1.0186 0.322058 0.689576 0.363560 0.647541<br />
1.0500 0.311004 0.681721 0.352395 0.638924<br />
1.0514 0.310520 0.681373 0.351905 0.638542<br />
1.0625 0.306710 0.678618 0.348047 0.635526<br />
1.2171 0.258252 0.641396 0.298499 0.594960<br />
2.3050 0.077002 0.431247 0.101303 0.374037<br />
2.8038 0.044212 0.359486 0.061722 0.302338<br />
2.9275 0.038529 0.343620 0.054585 0.286796<br />
Table 2.<br />
The floor (F) and ceiling (C) for survivor functions to every lifetime at the level of<br />
99% and 95 % bootstrap percentile method<br />
lifetime<br />
Bootstrap percentile method<br />
99% 95%<br />
F C F C<br />
0.1100 0.886015 0.943217 0.894880 0.939989<br />
0.4950 0.580075 0.768692 0.606653 0.756924<br />
0.5338 0.555834 0.753004 0.583346 0.740580<br />
0.6075 0.512543 0.724081 0.541513 0.710500<br />
0.6150 0.508331 0.721200 0.537428 0.707508<br />
0.7029 0.461474 0.688285 0.491785 0.673371<br />
0.7350 0.445461 0.676643 0.476101 0.661319<br />
0.7871 0.420645 0.658165 0.451703 0.642216<br />
0.8650 0.386095 0.631473 0.417536 0.614677<br />
0.8925 0.374589 0.622312 0.406102 0.605240<br />
0.8938 0.374053 0.621882 0.405569 0.604798<br />
0.9429 0.354383 0.605864 0.385953 0.588319<br />
0.9543 0.349966 0.602205 0.381536 0.584558<br />
0.9629 0.346670 0.599459 0.378237 0.581736<br />
1.0186 0.326064 0.581974 0.357552 0.563789<br />
1.0500 0.314992 0.572343 0.346394 0.553916<br />
1.0514 0.314507 0.571917 0.345905 0.553480<br />
1.0625 0.310690 0.568554 0.342050 0.550035<br />
1.2171 0.262094 0.523708 0.292615 0.504214<br />
2.3050 0.079186 0.293764 0.097555 0.273400<br />
2.8038 0.045742 0.225358 0.058955 0.206501<br />
2.9275 0.039922 0.211020 0.052033 0.192619<br />
Connecting the intervals estimation of every lifetime develops confidence bands for survivor<br />
function with traditional method and bootstrap percentile method.<br />
109
Comparison of Confidence Bands<br />
Figure 1 and 2 give the confidence bands for survivor function on one parameter exponential<br />
distribution under double type-II censoring using the traditional method and the bootstrap<br />
percentile method.<br />
1.00<br />
0.90<br />
0.80<br />
0.70<br />
0.60<br />
0.50<br />
0.40<br />
0.30<br />
0.20<br />
0.10<br />
0.00<br />
t<br />
traditional t method<br />
bootstrap method<br />
0.110.4950.5340.6080.6150.7030.7350.7870.8650.8930.8940.9430.9540.9631.0191.0501.051.0631.2172.3052.8042.928<br />
Figure 1.<br />
99% Confidence bands for survivor function<br />
1.00<br />
0.90<br />
0.80<br />
0.70<br />
0.60<br />
0.50<br />
0.40<br />
0.30<br />
0.20<br />
0.10<br />
0.00<br />
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<br />
traditional method<br />
bootstrap method<br />
t<br />
Figure 2.<br />
95% Confidence bands for survivor function<br />
From these figures 1 and 2 with 99% and 95% respectively, the width of the confidence regions<br />
for the survivor function with bootstrap percentile are narrower compared to the traditional<br />
method.<br />
Conclusion<br />
Using the intervals estimation for survivor functions at every lifetime develops the confidence<br />
bands for survivor function. Bootstrap percentile method proved to be more potential in<br />
constructing confidence bands for survivor function on one parameter exponential distribution<br />
under double type-II censoring than the traditional method. Therefore, bootstrap method can be<br />
used as an alternative method in constructing confidence bands.<br />
110
Acknowledgement<br />
The author would like to acknowledged the financial support from <strong>Indonesia</strong>n Directorate<br />
General of Higher Education under Hibah Bersaing 2009-2010.<br />
References<br />
[1] Balakrishnan. (1990). On the Maximum Likelihood Estimation of the Location and Scale<br />
Parameters of the Exponential Distribution Based on Multiply Type-II Censored<br />
Samples. Journal of Applied Statistics. 17:55-61.<br />
[2] Bartholomew, D. J. (1963). The Sampling Distribution of an Estimate Arising in Life<br />
Testing. Technometrics 5. 361-374.<br />
[3] Collett, D. (1996). Modelling Data In Medical Research. London: Chapman & Hall.<br />
[4] Efron, B. (1979). Bootstrap Method: Another Look at the Jackknife. The Annals of<br />
Statistics, 7(1), 1-26.<br />
[5] Efron, B. and Tibshirani, R. (1993). An Introduction to the Bootstrap. New York:<br />
Chapman & Hall.<br />
[6] Fauzy, A., Ibrahim, N. A., Daud, I. & Abu Bakar, M. R. (2003a). Interval Estimation for<br />
One and Two Parameters Exponential Distribution under Double Type-II Censoring with<br />
Bootstrap Percentile. Weekly Seminar to Institute for Mathematical Research. UPM.<br />
[7] Fauzy, A., Ibrahim, N. A., Daud, I. & Abu Bakar, M. R. (2003b). Interval Estimation for<br />
Survivor Function of Exponential Distribution under Double Type-II Censoring with<br />
Bootstrap Percentile. International Conference on Research and Education in<br />
Mathematics, Institute for Mathematical Research, UPM.<br />
[8] Fernandez, A. J. (2000). Estimation and Hypothesis Testing for Exponential Lifetime<br />
Models with Double Censoring and Prior Information. Journal of Economic and Social<br />
Research, 2(2), 1-17.<br />
[9] Hsieh, H. K. (1981). On Testing the Quality of Two Exponential Distributions.<br />
Technometrics 23. 263-269.<br />
[10] Lawless. (1982). Statistical Models and Methods for Lifetime Data. New York: John<br />
Wiley & Sons.<br />
[11] Raqab, M. Z. (1995). On the Maximum Likelihood Prediction of the Exponential<br />
Distribution Based on Double Type-II Censored Samples. Pakistan Journal of Statistics,<br />
11, 1-10.<br />
[12] Singh, N. (1985). A Simple and Asymptotically Optimal Test for the Equality of K<br />
Exponential Distribution Based on Type II Censored Samples. Comm. Statist. Theory<br />
Methods. 14.1615-1625.<br />
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