Chin-Diew Lai

Massey University · Institute of Fundamental Sciences

In this paper, we introduce a new parametric distribution for modelling lifetime data with bathtub-shaped hazard rate, derived by representing the cumulative hazard as proportional to a generalized beta density. The distribution has a finite range, useful when there is a maximum possible lifetime, a common scenario when there are additional failure...

The correlation structure of Marshall–Olkin bivariate exponential distribution (BVE) is well known. However, we are unable to compute the correlation of Marshall–Olkin bivariate Weibull distribution analytically. Fortunately, bivariate observations from this family can be obtained easily through extensive simulations. As expected, the key factors t...

In this paper, we consider the testing problem whether burn-in is required based on the long-run average cost for a population with bathtub-shaped failure rate function. We propose a test based on kernel density estimation. We then apply our proposed test to two real data sets in the context of reliability.

We first review the basic properties of Marshall-Olkin bivariate exponential distribution (BVE) and then investigate its correlation structure. We provide the correct reasonings for deriving some properties of the Marshall-Olkin BVE and show that the correlation of the BVE is always smaller than that of its copula regardless of the parameters. The...

We introduce a new finite range life distribution with a hazard rate function of the form of a nonstandard beta density function. The hazard rate function is either increasing, or has a bathtub shape with a long flat middle interval. The proposed model has a simple structure. Several datasets from the reliability literature with known bathtub shape...

In lifetime modeling, it is common to treat failure data as being continuous, implying some degree of precision in measurement. Too often in practice, however, failures are either noted at regular inspection intervals, occur in a discrete process or are simply recorded in bins. In life testing experiments, it is sometimes impossible or inconvenient...

In probability theory and statistics, the Weibull distribution is a continuous probability distribution named after Waloddi Weibull who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin and Rammler (1933) to describe the size distribution of particles.

In this chapter we discuss several univariate statistical models that are derived from two or more independent Weibull distributions that interact with each other under some governing ‘operations’.

The Weibull distribution has been one of the most cited lifetime distributions in reliability engineering. Over the last decade, many generalizations and extensions of the Weibull have been proposed in order to provide more flexibility than the traditional version when it comes to modeling lifetime data in diverse fields. This book offers an update...

We consider various methods of constructing discrete lifetime models by discretizations of continuous lifetime models, in particular, through discretizations of their hazard rates. It is well known that there are at least two possible definitions of discrete hazard rates. While this rich source of supply is indeed a good thing, it could also become...

In this paper we consider the problem of testing H 0 : F is an exponential against H 1 : F is NWBUE and not exponential. We propose a new test statistic based on an inequality of moment and obtain its asymptotic distribution to compare our new test with another well known test given by Klefsjö (1989).

Lifetime (ageing) distributions play a fundamental role in reliability. We present a semi-unified approach in constructing them, and show that most of the existing distributions may arise from one of these methods. Generalizations/modifications of the Weibull distribution are often required to prescribe the nonmonotonic nature of the empirical haza...

We propose a new reliability model which is a generalization of the logistic frailty model first considered by Vaupel (1990) for fitting human mortality data. Essentially, a shape parameter is added to the existing model providing more flexibility in modeling lifetime data. Several model properties such as the survival function, density and the haz...

The first‐order nonnegative integer valued autoregressive process has been applied to model the counts of events in consecutive points of time. It is known that, if the innovations are assumed to follow a Poisson distribution then the marginal model is also Poisson. This model may however not be suitable for overdispersed count data. One frequent m...

Although failure data are usually treated as being continuous, they may have been collected in a discrete manner, or in fact be discrete in nature. Reliability models with bathtub-shaped hazard rate are fundamental to the concepts of burn-in and maintenance, but how well do they incorporate discrete data? We explore discrete versions of the additiv...

Using a new distribution capable of exhibiting all the possible modes of accelerating and decelerating mortality, we conduct a systematic investigation of late-life mortality in humans. We check the insensitivity of the distribution to age cutoffs in the data relative to the logistic mortality model and propose a method to forecast evolution in the...

A concise definition expressed in terms of the mortality (hazard) rate as well as the survival function is proposed for modeling human mortality decelerating to a plateau. Several mortality leveling-off aging distributions are considered and the question concerning the identification of the ‘onset of deceleration’ is discussed.

The classical integer valued first-order autoregressive (INAR(1)) model has been defined on the basis of Poisson innovations. This model has Poisson marginal distribution and is suitable for modeling equidispersed count data. In this paper, we introduce an modification of the INAR(1) model with geometric innovations (INARG(1)) for modeling overdisp...

Discrete life data arise in many practical situations and even for continuous data we may find cases where the data are presented in grouped form, so that a discrete model can be used. In this paper, we propose a new two-parameter discrete lifetime distribution for modeling this type of data. The distribution under consideration has some interestin...

The basic Weibull distribution is considered the most fundamental and basic lifetime distribution. Various extensions of the Weibull distribution have been proposed since the 1970s and are useful in the modeling of complex lifetime data that are beyond the capability of the basic Weibull. This article reviews the properties of the basic Weibull dis...

Mortality deceleration is the observed but yet to be understood phenomenon that the increase in the late-life death rate slows down after a certain species-related advanced age. Various definitions of onsets of mortality deceleration are examined. A new distribution based on the Strehler-Mildvan theory of aging takes on the required shapes. The app...

Since 1970’s, many extensions of the Weibull distribution have been proposed to enhance its capability to fit diverse lifetime data and Murthy et alMurthy et al. (2004) proposes a scheme to classify these distributions.

In this article, we study, the maximum likelihood as well as Bayes estimation on parameters of Mixture of Weibull with ‘nearly instantaneous failure’ as introduced in Lai et.al. (2007). For Maximum likelihood estimation, the EM algorithm is used. For Bayes estimation of parameters, we used three different algorithms namely, Population Monte Carlo m...

We propose some simple generalizations of the Erlang distribution and study their properties and their connections to other existing well known distributions. In particular, we examine the mortality rate functions and their shapes which informs us how the population ages under these models. With suitable selections of parameter values, we can achie...

The mean residual life (MRL), or “life expectancy”, encapsulates the entire residual life of the product, and is thus of great interest in many fields, particularly those concerned with warranties, replacement policies, burn-in, and insurance. Estimating the profile of the MRL, let alone constructing confidence bands for it, is a difficult problem....

A discrete analogue of the standard continuous Weibull distribution was proposed in the literature to meet the need of fitting discrete-time reliability and survival data sets. Its properties were studied and the methods of estimation of its parameters were also investigated by various authors. Analogous to its continuous counterpart, the discrete...

Lifetime modeling of physical systems and biological organisms involves the use of failure distribution functions. The shape of the hazard rate (HR) function associated with the distribution function characterizes the effect of age (and other factors) on the failure. Examples of failure (and survival) data indicate that the shapes are many and vari...

We construct a parametric “growth curve”, the average fruit weight versus days since mid bloom, for kiwifruit (Actinidia deliciosa). The growth curve takes the form of a double sigmoid multiplied by a “shrinkage curve” (to represent late shrinkage of fruit on the vine before harvest) and the parameters have been fitted satisfactorily to data from f...

In a system subject to both repairable and catastrophic (i.e., nonrepairable) failures, ‘mission success’ can be defined as operating for a specified time without a catastrophic failure. We examine the effect of a burn-in process of duration τ on the mission time x, and also on the probability of mission success, by introducing several functions an...

Dependence relations between two variables are studied extensively in probability and statistics. No meaningful statistical models can be constructed without some assumptions regarding dependence although in many cases one may simply assume the variables are not dependent, i.e., they are independent.

The vast majority of the bivariate exponential distributions arise in the reliability context one way or another. When we talk of reliability, we have in mind the failure of an item or death of a living organism. We especially think of time elapsing between the equipment being put into service and its failure. In the bivariate or multivariate conte...

In introductory statistics courses, one has to know why the (univariate) normal distribution is important—especially that the random variables that occur in many situations are approximately normally distributed and that it arises in theoretical work as an approximation to the distribution of many statistics, such as averages of independent random...

A feature common to all the distributions in this chapter is that H(x,y) is a simple function of the uniform marginals F(x) and G(y). These types of joint distributions are known as copulas, as mentioned in the last chapter, and will be denoted by C(u,v); the corresponding random variables will be denoted by U and V, respectively.

The univariate extreme-value distributions consist of types 1 (Gumbel), 2 (Fréchet), and 3. The three types can be transformed to each other. The type 3 distribution of (−X) is the usual Weibull distribution. In the bivariate context, marginals are of secondary interest compared with the dependence structure.

This chapter is devoted to describing a class of bivariate distributions whose contours of probability densities are ellipses; in particular, those ellipses with constant eccentricity. These distributions are generally known as elliptically contoured or elliptically symmetric distributions. A subclass of distributions with contours that are circles...

In Section 5.6, we outlined the construction of a bivariate p.d.f. as the product of a marginal p.d.f. and a conditional p.d.f., h(x,y)=f(x)g(y|x). This construction is easily understood, and has been a popular choice in the literature, especially when Y can be thought of as being caused by, or predicted from, X. Arnold et al. (1999, p. 1) contend...

The terms “trivariate reduction” or “variables in common” are used for schemes for constructing of pairs of r.v.’s that start with three (or more) r.v.’s and perform some operations on them to reduce the number to two.

Devroye (1986) has provided an exhaustive treatment on the generation of random variates. Gentle (2003) has also recently provided a state-of-the-art treatise on random number generation and Monte Carlo methods. For this reason, we provide here a brief review of this subject and refer readers to these two references for a comprehensive treatment. I...

A study of bivariate distributions cannot be complete without a sound background knowledge of the univariate distributions, which would naturally form the marginal or conditional distributions. The two encyclopedic volumes by Johnson et al. (1994, 1995) are the most comprehensive texts to date on continuous univariate distributions. Monographs by O...

The study of copulas is a growing field. The construction and properties of copulas have been studied rather extensively during the last 15 years or so. Hutchinson and Lai (1990) were among the early authors who popularized the study of copulas. Nelsen (1999) presented a comprehensive treatment of bivariate copulas, while Joe (1997) devoted a chapt...

Many of the bivariate gamma distributions considered in this chapter may be derived from the bivariate normal in some fashion, such as by marginal transformation. It is well known that a univariate chi-squared distribution can be obtained from one or more independent and identically distributed normal variables and that a chi-squared random variabl...

In this chapter, we review methods of constructing bivariate distributions. There is no satisfactory mathematical scheme for classifying the methods. Instead, we offer a classification that is based on loosely connected common structures, with the hope that a new bivariate distribution can be fitted into one of these schemes. We focus especially on...

A measure of dependence indicates in some particular manner how closely the variables X and Y are related; one extreme will include a case of complete linear dependence, and the other extreme will be complete mutual independence. Although it is customary in bivariate data analysis to compute a correlation measure of some sort, one number (or index)...

When one considers a bivariate distribution, it is perhaps common to think of a joint density function rather than a joint distribution function, and it is also conceivable that such a density may be simple in expression, while the corresponding distribution function may involve special functions, can be expressed only as an infinite series, and so...

While there is a need for more detailed information on health inequality to guide public health policy, the most complete and easily available data remain those in mortality tables. We investigate, via a comparative analysis of data from New Zealand on Māori and non-Māori mortality, whether more detailed information than raw life expectancy may be...

The presence of non-conforming components instead of, or in addition to, the usual assembly errors results in N- or W-shaped hazard rate (HR) functions rather than the usual bathtub (i.e. U-shaped) ones. Although there have been numerous models for bathtub-shaped HR functions, N- and W-shaped HR functions are usually modelled using mixtures of two...

A large number of methods for modeling lactation curves have been proposed - parametric and nonparametric, mathematically or biologically oriented. The most popular of these are methods that express the milk yield in terms of time via a parametric nonlinear functional equation. This is intuitive and allows for relatively easy mathematical and biolo...

Preface: " This volume, which is completely dedicated to continuous bivariate distributions, describes in detail their forms, properties, dependence structures, computation, and applications. It is a comprehensive and thorough revision of an earlier edition of 'Continuous Bivariate Distributions, Emphasizing Applications' by T.P. Hutchinson and C.D...

The presence of non-conforming components instead of, or in addition to, the usual assembly errors results in N- or W-shaped hazard rate (HR) functions rather than the usual bathtub (i.e. U-shaped) ones. Although there have been numerous models for bathtub- shaped HR functions, N- and W-shaped HR functions are usually modelled using mixtures of tw...

Random variables are rarely independent in practice and so many multivariate distributions have been proposed in the literature to give a dependence structure for two or more variables. In this book, we restrict ourselves to the bivariate distributions for two reasons: (i) correlation structure and other properties are easier to understand and the...

The design of a control chart is often based on the statistical measure of average run length (ARL). A longer in-control ARL is ensured by the design, but the variance run length distribution may also be large for such a design. In practical terms, the variability in false alarms and true signals may be large. If the sample size for plotting a poin...

The use of the Weibull distribution to model lifetimes of incandescent lamps was originally suggested by Leff (1990). Following this suggestion, Agrawal and Menon have offered and investigated, in a series of papers, an improved model constructed from physical considerations and laws of mathematical statistics. In the present paper we offer supplem...

The turning point of a hazard rate function is useful in assessing the hazard in the useful life phase and helps to determine and plan appropriate burn-in, maintenance, and repair policies and strategies. For many bathtub-shaped distributions, the turning point is unique, and the hazard varies little in the useful life phase. We investigate the per...

The addition of a constant ‘competing risk’ corresponding to an additional, usually less significant, source of failure, frequently improves the fit in reliability and survival analysis. This is often termed a ‘lift’, as the effect is to increase the hazard rate (HR) function by a constant, which does not, of course, change the shape and hence the...

We prove that R. E. Glaser’s [J. Am. Stat. Assoc. 75, 667–672 (1980; Zbl 0497.62017)] η-function of the Z. W. Birnbaum and S. C. Saunders distribution [J. Appl. Probab. 6, 319–327 (1969; Zbl 0209.49801)] is upside-down bathtub-shaped. This implies – as a consequence of Glaser’s general result – that the Birnbaum-Saunders hazard rate function is ups...

We have had our attention drawn to a series of papers which, had we been aware of them, would have been cited in our paper, Bebbington et al. (2007b). We would like to thank Dr. Tarynn M. Witten for providing us with a number of papers related to our research, and the accompanying commentary.

We are grateful to Professor Nadarajah for pointing out an erroneous note in our paper Bebbington et al. [1]. The sentence following equation (5) in that paper should read as follows: ‘‘The function (5) is monotonic in t and so the distribution under consideration belongs to the class postulated by Gurvich et al. [14].’’

This short communication first offers a clarification to a claim by Nadarajah & Kotz. We then present a short summary (by no means exhaustive) of some well-known, recent generations of Weibull-related lifetime models for quick information. A brief discussion on the properties of this general class is also given. Some future research directions on t...

A generalized Weibull model that allows instantaneous or early failures is modified so that the model can be expressed as a mixture of the uniform distribution and the Weibull distribution. Properties of the resulting distribution are derived; in particular, the probability density function, survival function, and the hazard rate function are ob...

To increase the reliability of modules, and thus of systems assembled from them, they are frequently constructed using parallel load-sharing components. Examples include jet engines, electrical power networks, and telecommunications networks. We consider the situation when the components operate independently, but when any one of them fails, the lo...

Finding optimal, or at least good, maintenance and repair policies is crucial in reliability engineering. Likewise, describing life phases of human mortality is important when determining social policy or insurance premiums. In these tasks, one searches for distributions to fit data and then makes inferences about the population(s). In the present...

Aging and mortality is usually modeled by the Gompertz-Makeham distribution, where the mortality rate accelerates with age in adult humans. The resulting parameters are interpreted as the frailty and decrease in vitality with age. This fits well to life data from 'westernized' societies, where the data are accurate, of high resolution, and show the...

An important problem in reliability is to define and estimate the optimal burn-in time. For bathtub shaped failure-rate lifetime distributions, the optimal burn-in time is frequently defined as the point where the corresponding mean residual life function achieves its maximum. For this point, we construct an empirical estimator and develop the corr...

The study of systems with dependent components from a reliability point of view is a very important topic. However, the majority of the articles study the case of independent components. In this article, we study how the dependency influences the performance of the system. We extend some comparison results obtained in the case of independent compon...

We propose a new two-parameter ageing distribution which is a generalization of the Weibull and study its properties. It has a simple failure rate (hazard rate) function. With appropriate choice of parameter values, it is able to model various ageing classes of life distributions including IFR, IFRA and modified bathtub (MBT). The ranges of the two...

This paper reports a service quality application of the control chart procedure. A framework for the monitoring award of grades to students by the teaching staff in a large university is presented. The procedure signals the presence of special cause variations, if any, in the award of grades. Implementation of the grade monitoring framework saved c...

A generalized Weibull model that allows instantaneous or early failures is modified so that the model can be expressed as a mixture of the uniform distribution and the Weibull distribution. Properties of the resulting distribution are derived; in particular, the probability density function, survival function, and the hazard rate function are ob...

A large number of methods for modeling lactation curves have been proposed - parametric and nonparametric, mathematically or biologically oriented. The most popular of these are probably methods that express the milk yield in terms of time via a parametric nonlinear functional equation. This is intuitive and allows for relatively easy mathematical...

Various techniques for constructing discrete bivariate distributions are scattered in the literature. We review these methods of construction and group them into some loosely defined clusters.

This paper addresses the problem of estimating the ratio of the means of independent normal variables in agricultural research. The first part of the research examines the distributional properties of the ratio of independent normal variables, both theoretically and using simulation. The second part of the research evaluates the relative merits of...

One of the most controversial issues in the aftermath of the Asian financial crisis has been the appropriate response of monetary policy to a sharp decline in the value of some currencies. In this paper, we empirically examine the effects on Asian exchange rates of sharply higher interest rates during the Asian financial crisis. Taking account of t...

We propose computationally tractable formal mathematical definitions for the 'useful period' of lifetime distributions with bathtub shaped hazard rate functions. Detailed analysis of the reduced additive Weibull hazard rate function illustrates its utility for identifying such useful periods. Examples of several other bathtub shaped hazard rate fun...

Weibull models are used to describe various types of observed failures of components and phenomena. They are widely used in reliability and survival analysis. In addition to the traditional two-parameter and three-parameter Weibull distributions in the reliability or statistics literature, many other Weibull-related distributions are available. The...

Ageing and dependence are two important characteristics in reliability and survival analysis, and they affect significantly the decision people make with regard to maintenance, repair/replacement, price setting, warranties, medical studies, and other areas. There are many papers published at different technical levels. This book aims at providing a...

First Page of the Article

This paper considers a sequence of independent counts, with each count arising from a mixture of binomial distributions; the mixing distribution is fixed but the number of trials varies from count to count. In this common situation, an estimate of the underlying mean binomial proportion is needed. Two estimators are in general use: the arithmetic a...

This paper addresses the problem of estimating a binomial proportion from several independent samples in agricultural research, where the arithmetic average is widely used. The penalties of using a suboptimal estimator, the arithmetic estimator, relative to the preferred best estimator, the weighted average, are theoretically and empirically invest...

Quadratic programming is concerned with minimizing a convex quadratic function subject to linear inequality constraints. The variables are assumed to be nonnegative. The unique solution of quadratic programming (QP) problem (QPP) exists provided that a feasible region is non-empty (the QP has a feasible space).A method for searching for the solutio...

The usual S 2 chart is constructed with equal false alarm probabilities for the lower and upper control limits. An alternative is the ARL-unbiased S 2 chart, where the ARL (average run length) curve attains its maximum when the common-cause variance is at its in-control value. We examine the ARL properties of these S 2 charts, and show that neither...

This paper considers a sequence of independent counts, with each count arising from a mixture of binomial distributions; the mixing distribution is fixed but the number of trials varies from count to count. In this common situation, an estimate of the underlying mean binomial proportion is needed. Two estimators are in general use: the arithmetic a...

System availability is a major performance measure for distributed systems. A typical type of distributed systems is the so called Homogeneously Distributed Software/Hardware System (HDSHS), for which identical copies of application software run on the same type of computers. When this type of systems is tested, the software faults detected are usu...

We propose a new charting procedure, combining two or more Exponentially Weighted Moving Averages (EWMAs), which we call a Composite EWMA (CEWMA) Control Chart. In the case of two EWMAs, the CEWMA chart corresponds to the usual combined Shewhart-EWMA chart, although our plotting procedure remain, we feel, an improvement in application and interpret...

The fundamental properties of a punctured normal distribution are studied. The results are applied to three issues concerning X/Y where X and Y are independent normal random variables with means μX and μY respectively. First, estimation of μX/μY as a surrogate for E(X/Y) is justified, then the reason for preference of a weighted average, over an ar...