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Chapter-1
Introduction
Chapter 1: Growth and Development of the Reliability Inference
CHAPTER 1
GROWTH AND DEVELOPMENT OF THE
RELIABILITY INFERENCE
1.1. INTRODUCTION
Due to rapid advancement in technology, more and more products are
introduced to the market, consumers now have the luxury of demanding high quality
and long life in the products they purchase. In such a high demanding and competitive
market, one way by which manufacturers attract consumers to their products is by
providing warranties on product lifetimes. In order to design cost effective warranty, a
manufacturer must have sound knowledge about product failure time distributions. To
gain this knowledge, life testing and reliability experiments are carried out before
(and while) products are put on the market. Reliability technology has a potentially
wide range of application areas. Some of these areas are engineering design,
safety/risk analysis, environmental protection, quality/reliability management,
verification and optimization of maintenance and operation. In last few years of the
twentieth century, there have been remarkable progress in the application of reliability
principles in the industries and government departments in almost all developed and
developing countries. The growth and development of the reliability theory have
strong links with quality control and its development. Dodge and Romig (1929) and
Shewhart (1931) laid down the theoretical basis for utilizing statistical methods in
quality control of industrial products. Today, reliability has become a catchword in
many firms. Many statistical methods have been developed to get precise estimate of
reliability function during the last few decades.
The reliability function R(t) is defined as the probability of failure-free
operation until time t. Thus, if the random variable (rv) X denotes the lifetime of an
item, then R(t)=P(X>t). Another measure of reliability under stress-strength set-up is
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Chapter 1: Growth and Development of the Reliability Inference
the probability P = P(X>Y), which represents the reliability of an item of random
strength X subject to random stress Y. The problem of making inference about ‘P’ is
of particular interest in engineering statistics and in biostatistics. For example, in a
reliability study, let X be the strength of a system and Y be the stress applied to the
system. Then ‘P’ measures the chance that the system fails. Alternatively, in a clinical
study, let X be the response of a control group and Y be the response of a treatment
group. Then ‘P’ measures the effectiveness of the treatment, for example, if Y
represents a patient’s remaining years of life if treated with drug A and X represents
the patient’s remaining years of life if treated with drug B. If choice is left to the
patient, person’s deliberations will center on whether P(X>Y) is less than or greater
than 1/2 (Ali and Woo, 2005a, 2005b).
1.2. CLASSICAL INFERENTIAL PROCEDURES IN RELIABILITY THEORY
Various authors have adopted classical inferential procedures in the field of
life-testing and reliability theory. Cohen (1951) and Harter (1969) obtained local
maximum likelihood estimators (LMLES). Harter and Moore (1966) and later Calitz
(1973) noted that these LMLES appear to possess most of the desirable properties
ordinarily associated with MLE. Epstein and Sobel (1953) derived the MLE of the
scale parameter of the one-parameter exponential distribution in case of censoring
from right. Epstein and Sobel (1953, 1954) and Epstein (1960) extended the foregoing
analysis to the two-parameter exponential distribution with and without censoring.
Cohen (1960a, b) considered the estimation problems in Poisson distribution. Harter
and Moore (1966) obtained numerical approximations to the MLES for the
parameters of generalized gamma distribution. Lemon (1975) obtained MLES for
three parameter Weibull distribution based on censored samples. Farewell and
Prentice (1977) obtained regression estimators for the generalized gamma distribution.
Cohen and Whitten (1980) first proposed the modified moment estimators (MMES)
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Chapter 1: Growth and Development of the Reliability Inference
for log-normal distribution. For comparison between MLE, moment estimator (ME)
and MME one can refer to Cohen and Whitten (1986) and Cohen, Whitten and Ding
(1985). Grubbs (1971) determined an approximate confidence bound for the
reliability function for two-parameter negative exponential distribution. Exact
conditional confidence interval estimates have been obtained for type II censored data
from Weibull and extreme value distributions by Lawless (1978). Hager, Bain and
Antle (1971) studied generalized gamma distribution and gave some inferential
procedures for it. Johnson and Kotz (1970a, b) gave a comprehensive account of
estimation for homogenous population assumed to have the parametric models like
exponential, Weibull, gamma, etc. Mann, Schafer and Singpurwalla (1974)
summarized on industrial life testing point of view, estimation procedures for
exponential, Weibull, log-normal and many other distributions, both for single sample
and two sample problems, with censoring. Engelhardt and Bain (1974) dealt with
point estimation for two-parameter Weibull distribution.
Tyagi and Bhattacharya (1989a) obtained the uniformly minimum variance
unbiased estimators (UMVUES) of various parametric functions of Maxwell failure
distribution. Chaturvedi and Rani (1998) proposed generalized Maxwell failure
distribution and derived the UMVUE of the reliability function. Chaturvedi and
Surinder (1999) revisited the problem of obtaining UMVUE of the reliability function
of one-parameter exponential distribution under type I and II censorings. Tyagi, Singh,
Kumar and Bhattacharya (1993) derived the UMVUE of the reliability function for
inverted Erlang failure distribution. Classical inferences for Weibull process have
been developed by Muralidharan (2002).
Discrete distributions have played
important role in reliability theory. Kumar and Bhattacharya (1989) considered
negative binomial distribution as the lifetime model and obtained UMVUES of the
mean life and reliability function. For historical development in reliability theory
using Weibull distribution, one may refer to Rinne (2009).
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Chapter 1: Growth and Development of the Reliability Inference
1.3. BAYESIAN INFERENTIAL PROCEDURES IN RELIABILITY THEORY
Many researchers have considered the problems of estimation of parameters and
reliability function in the literature under censoring and complete sample case. For a
brief review on Bayesian reliability estimation, one may refer to Martz and Waller
(1982). Bhattacharya (1967), first time considered the problems of estimating the
parameter and reliability function of one parameter exponential distribution under
type II censoring under squared error loss function with uniform and beta priors.
Similar estimators for other priors have been obtained by Bhattacharya and Kumar
(1986) and Bhattacharya and Tyagi (1988). Harris and Singpurwalla (1968) derived
the Bayes estimators for the reliability function of exponential and Weibull
distributions using uniform and gamma priors. Canfield (1970) considered an
asymmetric loss function for the Bayesian estimation of the reliability function under
beta priors. For Weibull distribution, Soland (1969) derived the Bayes estimator of
reliability function using a discrete prior distribution for the shape parameter. In the
case of gamma distribution, Canavos and Tsokos (1971) derived Bayes estimator of
reliability, restricting the scale parameter as integer-valued. Lwin and Singh (1974)
revisited the problem and relaxed the restriction on scale parameter and derived Bayes
estimator of reliability function under exponential, inverse Gaussian and a certain
three-parameter prior distributions when both the parameters were unknown. Tsokos
(1972a) and Canavos and Tsokos (1973), using Monte-Carlo simulation, found that
the Bayes estimators of reliability in case of uniform, exponential and gamma priors,
have uniformly smaller mean squared-error than minimum variance unbiased
estimators. Tsokos (1972b) and Tsokos and Canavos (1972) also considered the
Bayesian estimation of reliability using uniform, exponential and beta priors. Martz
and Lian (1977) obtained the Bayes estimator of reliability of Weibull distribution
using a piecewise linear prior distribution. For log-normal distribution, Padgett and
Tsokos (1977) studied the mean squared-error performance of Bayes estimator of
reliability function compared to MLE and found that the Bayes estimator was superior
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Chapter 1: Growth and Development of the Reliability Inference
to it. Padgett (1981) obtained the Bayes estimator of reliability function of inverse
Gaussian distribution. Tyagi and Bhattacharya (1989b) considered the Bayesian
estimation of reliability function of Maxwell distribution. Chaturvedi and Rani (1998)
extended their results to the generalized Maxwell distribution. Kyriakoussis and
Papadopoulos (1993) derived Bayes estimators of reliability function of the zerotruncated binomial and negative binomial distributions. Chaturvedi and Sharma (2008)
generalized their results and obtained Bayes estimators of positive and negative
powers of the parameter of zero-truncated negative binomial distribution. The
recurrence relations were provided for these estimators. Alvandi (1990) considered
Bayes estimation of the parameter of a Poisson distribution under LINEX (linear in
exponential) loss function and gamma prior. Bhattacharya and Tiwari (1992) and
Tiwari and Badrinathi (1992) proposed hierarchical Bayesian approach to reliability
analysis. Pensky and Singh (1999) considered empirical Bayes squared-error loss
estimation of mean lifetime, variance and reliability function for failure-time
distributions belonging to an exponential family. Balakrishnan and Ma (2002) dealt
with empirical Bayes estimation and testing for a location parameter for the family of
gamma distribution. Nassar and Eissa (2004) performed Bayesian analysis for the
exponentiated Weibull model for complete and type II censoring. Liang (2005)
considered empirical Bayes testing problem for the reliability of an exponential
distribution. Farsipour and Zakerzadeh (2005) considered the estimation of scale
parameter under SELF (squared-error loss function) with restriction to the principle of
invariance and risk unbiasedness.
1.4. INFERENTIAL PROCEDURES FOR P(X>Y)
Estimation of P=P(X>Y) for several distributions of the exponential family are
discussed by Chao (1982). Kotz, Lumelskii and Pensky (2003) have given
comprehensive review on estimation of stress strength reliability over the last four
decades. The pioneering work of the estimation of ‘P’ has been done by Owen,
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Chapter 1: Growth and Development of the Reliability Inference
Craswell and Hanson (1964). Govindarajulu (1967), Church and Harris (1970),
Downton (1973), Gupta and Gupta (1990) and Woodward and Kelley (1977),
considered the cases when X and Y are normally distributed. Tong (1974), Kelley
Kelly and Schucany (1976) and Sathe and Shah (1981) considered the estimation of
stress-strength reliability for independent exponential random variables. Awad and
Gharraf (1986) considered the estimation of ‘P’ for the family of Burr XII
distributions. Enis and Geisser (1971) derived the estimator of ‘P’ when X and Y
follow exponential distributions. MLE of ‘P’ when X and Y have bivariate
exponential distribution has been considered by Awad, Azzam and Hamadan (1981).
Constantine, Karson and Tse (1986) and Ismail, Jeyaratnam and Panchapakesan,
(1986) considered estimation of ‘P’ for the family of gamma distributions. Simonoff,
Hochberg and Reiser (1985) provided alternative estimation procedures for ‘P’ in
discretized data. They obtained both MLE and UMVUE of ‘P’ and studied their
performances extensively. Basu and Tarmast (1987) considered the problem of the
Bayesian estimation of ‘P’. Basu and Ebrahimi (1991) obtained the same in complete
sample case, using SELF and LINEX loss function. McCool (1991) obtained MLE of
a particular transformation of ‘P’ and obtained its confidence interval. Chaturvedi and
Surinder (1999) revisited the problem of estimation of ‘P’ for exponential case under
type I and type II censorings and derived UMVUES using a simpler technique. Maiti
(1995) considered the estimation of ‘P’ for geometric distributions. Chaturvedi and
Tomer (2002) considered classical and Bayesian estimation procedures for negative
binomial distribution from a different approach. Kundu and Gupta (2006) proposed an
approximate maximum likelihood estimator of ‘P’ for Weibull random variables and
obtained asymptotic distribution of the MLE and based on it constructed asymptotic
confidence interval. Chaturvedi, Tiwari and Kumar (2007) considered the problem of
estimating stress-strength reliability from Bayesian view point for binomial and
Poisson distribution. Surles and Padgett (1998, 2001) and Raqab and Kundu (2005)
considered estimation of P when X and Y had Burr type X distribution. Kundu and
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Chapter 1: Growth and Development of the Reliability Inference
Gupta (2006) proposed an approximate maximum likelihood estimator of ‘P’ for
Weibull random variables and obtained asymptotic distribution of the MLE and based
on it constructed asymptotic confidence interval. Kundu and Gupta (2005) also
considered estimation of ‘P’ for the family of generalized exponential distributions.
Ali and Woo (2005a, 2005b) considered estimation of ‘P’ for the Levy and
p-dimensional Rayleigh distributions. Saracoglu, Kaya, and Abd-Elfattah (2009)
discussed on comparative study on estimators for stress-strength reliability in the
Gompertz case when one parameter is known.
1.5.
SEQUENTIAL INFERENTIAL PROCEDURES IN RELIABILITY THEORY
Following Wald’s (1947) sequential probability ratio test (SPRT), Oakland
(1950) derived SPRT for testing between two simple hypotheses concerning the mean
of the negative binomial distribution assuming dispersion parameter as known.
The zero-truncated negative binomial, binomial and Poisson distributions have played
important role in the literature [see Johnson and Kotz (1969; p. 73, 104,137)]. Taylor,
Woiwod and Perry (1978) found that dispersion parameter varies with the mean of the
negative binomial distribution. Foster, Tollefson and Steffey (1982) estimated an
average reduction in sampling time of 55% by using sequential sampling. Sequential
testing procedures for some discrete distributions have been developed by Foster,
Tollefson and Steffey (1982), Fowler (1983), Fowler and Lynch (1987) and others.
McAuslane, Ellis and Allen (1987) found dispersion parameter values for first year
corn fields substantially different from fields with at least two years of corn. Fowler
and Lynch (1987) applied SPRT for sampling plans in insect pest management.
Hubbard and Allen (1991) studied the robustness of the SPRT for the mean of a
negative binomial distribution when the dispersion parameter has undergone a change.
Kyriakoussis and Papadopoulos (1993) have discussed various situations when the
zero-truncated negative binomial and binomial distributions are of great importance to
scientists. Paul-Moraes, Burkness, Hunt, Wright, Hein and Hutchison (2011)
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Chapter 1: Growth and Development of the Reliability Inference
developed binomial sampling plans using Wald's SPRT for controlling western bean
cutworm. Subramaniam and Arumugam (2006) applied SPRT on negative binomial
distribution for predicting average sample number required for different population
density of cotton aphids.
Epstein and Sobel (1955) developed SPRT for testing simple null hypothesis
against simple alternative, for the scale parameter of an exponential distribution.
Harter and Moore (1976) conducted Monte Carlo study to investigate the robustness
of exponential SPRT when the underlying distribution is a Weibull with shape
parameter other than one. Montagne and Singpurwalla (1985) generalized the results
of Harter and Moore (1976) from Weibull to a class of distributions having an
increasing (decreasing) failure rate. They obtained inequalities for OC and ASN
functions in order to demonstrate the robustness. Phatarfod (1971) developed SPRT
for testing composite hypothesis for the shape parameter of the gamma distribution.
For some fixed sample size results concerning the robustness of testing and estimation
procedures related to exponential distribution, one may refer to Barlow and Proschan
(1967) and Hager, Bain and Antle (1971). Chaturvedi, Kumar and Kumar (2000)
developed SPRTS for the parameters of a family of continuous distributions. Pandit
and Gudaganavar (2010) developed SPRT for scale parameter of gamma and
exponential distribution. Kharin (2011) studied the robustness for Bayesian sequential
testing procedures of composite hypothesis.
1.6. ROAD MAP OF THE THESIS
In Chapter 2 of the thesis, a family of lifetime distributions is proposed.
UMVUES of the reliability function R(t) and ‘P’ are derived under complete sample.
MLES for R(t) and ‘P’ are also developed and extended for the case when all the
parameters are unknown. These estimators are derived by using the MLE and
UMVUE of powers of parameter. It is worth mentioning here that, in order to
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Chapter 1: Growth and Development of the Reliability Inference
estimate ‘P’, in the literature, the authors have considered the case when X and Y
follow the same distributions, may be with different parameters. We have generalized
the result to the case when X and Y may follow any distribution from the proposed
family of distributions. In order to examine the performance of MLE and UMVUE of
reliability function and ‘P’, we have conducted a simulation experiments using
bootstrap re-sampling technique over different sample sizes. For the case when all the
parameters are unknown, performance of the MLE of reliability function and ‘P’ are
also evaluated.
In Chapter 3, we propose a family of lifetime distributions. The UMVUES and
MLES of powers of parameter, reliability function and ‘P’ are derived under type I
and type II censorings. In order to obtain these estimators, the major role is played by
the estimators of the powers of the parameter(s) and the functional forms of the
parametric functions to be estimated are not needed. Performance of the developed
estimators for reliability function and ‘P’ have been examined using bootstrap resampling technique. Performance of the maximum likelihood estimator developed for
‘P’ under type I censoring has also been examined when all the parameters are
unknown and variables under study follow different distributions. We also consider
the problems of constructing tests and confidence intervals.
In Chapter 4, for the family of lifetime distributions considered in Chapter 3,
Bayes estimators are derived for the powers of the parameter, reliability function and
‘P’ under squared-error loss function and general entropy loss function. Type II
censoring is considered. Deviating from the conventional methods of obtaining Bayes
estimators of the reliability function and ‘P’, Bayes estimators of powers of parameter
are utilized to obtain Bayes estimator of the probability density function at a specified
point. This estimator is subsequently used to obtain Bayes estimators of reliability
function and ‘P’. Thus, in all the estimation problems, the major role is played by the
estimators of the powers of the parameter. Expressions for the risks, posterior risks
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Chapter 1: Growth and Development of the Reliability Inference
and Bayes risks of various estimators are provided. Performance of the developed
estimators have been examined on real data set.
Chapter 5 of the thesis deals with the problems of study of robustness of SPRTS
for the family of distributions considered in Chapter 2. SPRTS are developed for
testing simple hypotheses (versus simple alternatives).
The robustness of the
proposed SPRTS is studied when the distribution under consideration has undergone
a change. SPRT for testing the composite hypotheses is also developed. In order to
calculate the roots of the equation needed to evaluate OC and ASN functions, we give
a method of choosing the initial values in order to apply Newton-Raphson method.
A discussion on the numerical findings is provided.
Finally, in Chapter 6, we have developed SPRTS for the parameters of zerotruncated negative binomial and binomial distributions. The robustness of the SPRTS
is studied. We have proposed SPRT for the parameter of zero-truncated Poisson
distribution. The zero-truncated Poisson distribution is derived as a limiting form of
zero-truncated negative binomial and binomial distributions and this result is used to
‘assess’ the OC and ASN functions of zero-truncated Poisson distribution with the
help of OC and ASN functions of zero-truncated negative binomial and binomial
distributions.
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