Chin-Diew Lai

Continuous bivariate distributions

Univariate distributions. - Bivariate copulas. - Distributions expressed as copulas. - Concepts of stochastic dependence. - Measures of dependence. - Constructions of bivariate distributions.- Bivariate distributions constructed by conditional approach. - Variables in common method. - Bivariate gamma and related distributions. - Simple forms of the bivariate density function. - Bivariate exponentional and related distributions. - Bivariate normal distribution. - Bivariate extreme value distributions. - Elliptically symmetric bivariate distributions and other symmetric distributions. - Simulation of bivariate observations.

A modified Weibull distribution

A new lifetime distribution capable of modeling a bathtub-shaped hazard-rate function is proposed. The proposed model is derived as a limiting case of the Beta Integrated Model and has both the Weibull distribution and Type 1 extreme value distribution as special cases. The model can be considered as another useful 3-parameter generalization of the Weibull distribution. An advantage of the model is that the model parameters can be estimated easily based on a Weibull probability paper (WPP) plot that serves as a tool for model identification. Model characterization based on the WPP plot is studied. A numerical example is provided and comparison with another Weibull extension, the exponentiated Weibull, is also discussed. The proposed model compares well with other competing models to fit data that exhibits a bathtub-shaped hazard-rate function.

Reliability analysis using an additive Weibull model with bathtub-shaped failure rate function

Lifetime distributions for many components usually have a bathtub-shaped failure rate in practice. However, there are very few practical models to model this type of failure rate function. In this paper we study a simple model based on adding two Weibull survival functions. Some simplifications of the model are also presented. The graphical estimation technique based on the conventional Weibull plot is demonstrated to be useful in this case.

A flexible Weibull extension

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 parameters are clearly demarcated to separate these classes. It thus provides an alternative to many existing life distributions. Details of parameter estimation are provided through a Weibull-type probability plot and maximum likelihood. We also derive explicit formulas for the turning points of the failure rate function in terms of its parameters. This, combined with the parameter estimation procedures, will allow empirical estimation of the turning points for real data sets, which provides useful information for reliability policies.

On Recent Generalizations of the Weibull Distribution

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 this topic are also discussed.

CONTROL CHART FOR MULTIVARIATE ATTRIBUTE PROCESSES

Many industrial processes are multivariate in nature since the quality of a product depends on more than one variable. Multivariate control procedures can be used to capture the relationship between the variables and to provide more sensitive control than that provided by the application of univariate control procedures on each variable. Much has been done on the multivariate variable processes, such as embodied in control procedures based on Hotellings T 2 statistic. However, little work has been done to deal with the control of multivariate attribute processes, which is very important in practical production processes. In this paper, we develop a Shewhart-type control chart to deal with multivariate attribute processes, which is called the multivariate np chart (MNP chart). The control chart uses the weighted sum of the counts of nonconforming units with respect to all the quality characteristics as the plotted statistics. It enhances the efficiency of identifying the critical assignable cause when an ...

On nonhomogeneous models for volcanic eruptions

Recently, a special nonhomogeneous Poisson process known as the Weibull process has been proposed by C-H. Ho for fitting historical volcanic eruptions. Revisiting this model, we learn that it possesses some undesirable features which make it an unsatisfactory tool in this context. We then consider the entire question of a nonstationary model in the light of availability and completeness of data. In our view, a nonstationary model is unnecessary and perhaps undesirable. We propose the Weibull renewal process as an alternative to the simple (homogeneous) Poisson process. For a renewal process the interevent times are independent and distributed identically with distribution function F where, in the Weibull renewal process, F has the Weibull distribution, which has the exponential as a special situation. Testing for a Weibull distribution can be achieved by testing for exponentiality of the data under a simple transformation. Another alternative considered is the lognormal distribution for F. Whereas the homogeneous Poisson process represents purely random (memoryless) occurrences, the lognormal distribution corresponds to periodic behavior and the Weibull distribution encompasses both periodicity and clustering, which aids us in characterizing the volcano. Data from the same volcanoes considered by Ho were analyzed again and we determined there is no reason to reject the hypothesis of Weibull interevent times although the lognormal interevent times were not supported. Prediction intervals for the next event are compared with Hos nonhomogeneous model and the Weibull renewal process seems to produce more plausible results.

Statistical analysis of New Zealand volcanic occurrence data

Abstract The Poisson process is considered to provide a good fit to many volcanoes for forecasting eruptions. However, several exceptions exist, usually where the intensity is non-stationary. The nonhomogeneous Poisson model (the Weibull process) caters for a monotonically varying intensity, but often results in physically unrealistic behaviour. A model which allows for more general behaviour than the Poisson process is the Weibull renewal model. This paper considers data from two New Zealand volcanoes, Mt. Ruapehu and Mt. Ngauruhoe, and fits the Poisson and Weibull renewal models. We conclude that a simple Poisson process fits Ngauruhoe very well. The behaviour of Mt. Ruapehu is considerably more complex, although quite reasonable forecasts can still be obtained from the renewal models. An interesting feature of our analysis is that there seems to be no correlation between the observed eruption sequences of these two closely neighbouring volcanoes.

First‐Order Integer Valued AR Processes with Zero Inflated Poisson Innovations

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 manifestation of overdispersion is that the incidence of zero counts is greater than expected from a Poisson model. In this paper, we introduce a new stationary first-order integer valued autoregressive process with zero inflated Poisson innovations. We derive some structural properties such as the mean, variance, marginal and joint distribution functions of the process. We consider estimation of the unknown parameters by conditional or approximate full maximum likelihood. We use simulation to study the limiting marginal distribution of the process and the performance of our fitting algorithms. Finally, we demonstrate the usefulness of the proposed model by analyzing some real time series on animal health laboratory submissions.

On the Effect of Redundancy for Systems with Dependent Components

Parallel redundancy is a common approach to increase system reliability and mean time to failure. When studying systems with redundant components, it is usually assumed that the components are independent; however, this assumption is seldom valid in practice. In the case of dependent components, the effectiveness of adding a component may be quite different from the case of independent components. In this paper we investigate how the degree of correlation affects the increase in the mean lifetime for parallel redundancy when the two components are positively quadrant dependent. A number of bivariate distributions that can be used in the modeling of dependent components are compared. Various bounds are also derived. The results are useful in reliability analysis as well as for designers who are required to take into account the possible dependence among the components.