Empirical bayes estimation of reliability characteristics for an exponential family

This paper considers empirical Bayes (EB) squared-error-loss estimations of mean lifetime, variance and reliability function for failure-time distributions belonging to an exponential family, which includes gamma and Weibull distributions as special cases. EB estimators are proposed when the prior distribution of the lifetime parameter is completely unknown but has a compact (known or unknown) support. Asymptotic optimality and rates of convergence of these estimators are investigated. The rates established here under the compact support restriction are better than the polynomial rates of convergence obtained previously.     

Stochastic models of failure in random environments

Stochastic models of failure modes of frequent occurrence in the engineering sciences are considered. The failure-producing stress environment is modelled as a stationary stochastic process. Using theoretical properties of the sample paths of these processes, failure-time distributions which belong to the Birnbaum-Saunders family are obtained. Several examples of particular engineering relevance are treated.     

UMVU estimation of the reliability function of the generalized life distributions

The problems of estimating the reliability function and P=P_rX > Y are considered for the generalized life distributions. Uniformly minimum variance unbiased estimators (UMVUES) of the powers of the parameter involved in the probabilistic model and the probability density function (pdf) at a specified point are derived. The UMVUE of the pdf is utilized to obtain the UMVUE of the reliability function and ‘P’. Our method of obtaining these estimators is quite simple than the traditional approaches. A theoretical method of studying the behaviour of the hazard-rate is provided.     

Generalized exponential distributions

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.     

On univariate and bivariate generalized gamma convolutions

This paper has two parts. In the first part some results for generalized gamma convolutions (GGCs) are reviewed. A GGC is a limit distribution for sums of independent gamma variables. In the second part, bivariate gamma distributions and bivariate GGCs are considered. New bivariate gamma distributions are derived from shot-noise models. The remarkable property hyperbolic complete monotonicity (HCM) for a function is considered both in the univariate case and in the bivariate case.     

Markov Chain Monte Carlo in small worlds

As the number of applications for Markov Chain Monte Carlo (MCMC) grows, the power of these methods as well as their shortcomings become more apparent. While MCMC yields an almost automatic way to sample a space according to some distribution, its implementations often fall short of this task as they may lead to chains which converge too slowly or get trapped within one mode of a multi-modal space. Moreover, it may be difficult to determine if a chain is only sampling a certain area of the space or if it has indeed reached stationarity. In this paper, we show how a simple modification of the proposal mechanism results in faster convergence of the chain and helps to circumvent the problems described above. This mechanism, which is based on an idea from the field of “small-world” networks, amounts to adding occasional “wild” proposals to any local proposal scheme. We demonstrate through both theory and extensive simulations, that these new proposal distributions can greatly outperform the traditional local proposals when it comes to exploring complex heterogenous spaces and multi-modal distributions. Our method can easily be applied to most, if not all, problems involving MCMC and unlike many other remedies which improve the performance of MCMC it preserves the simplicity of the underlying algorithm.     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

Regression modelling of interval-censored failure time data using the Weibull distribution

A method is described for fitting the Weibull distribution to failure-time data which may be left, right or interval censored. The method generalizes the auxiliary Poisson approach and, as such, means that it can be easily programmed in statistical packages with macro programming capabilities. Examples are given of fitting such models and an implementation in the GLIM package is used for illustration.     

Characterization of a mixture of gamma distributions via conditional finite moments

A necessary and sufficient condition that a continuous, positive random variable follow a gamma distribution is given in terms of any one of its conditional finite moments and an expression involving its failure rate. The results are then used to develop a characterization for a mixture of two gamma distributions. The general results about characterization of a mixture of gamma distributions yield several special cases that have appeared separately in recent literature, including characterization of a single exponential distribution, characterization of a single gamma distribution (in terms of either first or second moments) and a sufficient condition for a mixture of two exponential distributions (in terms of first moments). The condition in this last result is shown to be necessary also. Numerous other cases are possible, using different choices for distribution parameters along with a selection of the mixing parameter, for either individual or mixtures of distributions. Various characterizations can be expressed using higher order moments, too.     

Information geodesics for gamma models of communication clustering

Recently, Akyildiz called for further work on non-Poisson models for communication arrivals in distributed networks such as cellular phone systems. The basic ‘random’ model for stochastic events is the Poisson process; for events on a line this resuits in an exponential disiribuuon of intervals between events. Network designers and managers need too monotor and quantify call clustering in order to optimize resaurce usage; the natural reference state from which to measure departures is that arising from a Poisson, process of calls. Here we consider gamma distributions, which contain exponential distributions as a special case. The surface representing gamma models has a natural Riemannian information metric and we obtain some geodesic sprays for this metric. The exponential distributions form a 1-dimensional subspace of the 2-dimensional space of all gamma distributions, so we have an isometric embedding of the random model as a subspace of the gamma models. This geometry may provide an appropriate structure on which to represent clustering as quantifiable departures from randomness and on which to impose dynamic control algorithms to optimize traffic at receiving nodes in distributed communication networks. In practice, we may expect correlation between call arrival times and call duration, reflecting for example peaks of different users of internet services. This would give rise to a twisted product of two surfaces with the twisting controlled by the correlation. Though bivariate gamma models do exist, such as Kibble's, none has tractabie information geometry nor sufficiently general marginal gammas,but a simulation method of approach is suggested.     

A relationship between a family of waiting time distributions and the asymptotic residual waiting time distribution

For a class of renewal process waiting time distributions defined herein, one may describe the distribution of asymptotic residual waiting times. The relationship between the two distributions characterizes the class, which includes the gamma distribution. Possible consequences for hypothesis testing are discussed.     

Bimatrix variate gamma-beta distributions

Two bimatrix distributions with beta and gamma marginals are introduced. Various properties (including product moments of determinants and traces, entropies, marginal distributions) are derived. Parameter estimation by the method of maximum likelihood is discussed. The performance and efficiencies of the maximum likelihood estimators and the associated confidence intervals are assessed by simulation. The efficiencies are compared versus those for the maximum likelihood estimators and the associated confidence intervals based on matrix variate gamma distributions. A discussion of possible applications of the bimatrix distributions is given.     

Finding the maximum efficiency for multistage ranked-set sampling

Multistage ranked-set sampling (MRSS) is a generalization of ranked-set sampling in which multiple stages of ranking are used. It is known that for a fixed distribution under perfect rankings, each additional stage provides a gain in efficiency when estimating the population mean. However, the maximum possible efficiency for the MRSS sample mean relative to the simple random sampling sample mean has not previously been determined. In this paper, we provide a method for computing this maximum possible efficiency under perfect rankings for any choice of the set size and the number of stages. The maximum efficiency tends to infinity as the number of stages increases, and, for large numbers of stages, the efficiency-maximizing distributions are symmetric multi-modal distributions where the number of modes matches the set size. The results in this paper correct earlier assertions in the literature that the maximum efficiency is bounded and that it is achieved when the distribution is uniform.     

A new class of weighted exponential distributions

Introducing a shape parameter to an exponential model is nothing new. There are many ways to introduce a shape parameter to an exponential distribution. The different methods may result in variety of weighted exponential (WE) distributions. In this article, we have introduced a shape parameter to an exponential model using the idea of Azzalini, which results in a new class of WE distributions. This new WE model has the probability density function (PDF) whose shape is very close to the shape of the PDFS of Weibull, gamma or generalized exponential distributions. Therefore, this model can be used as an alternative to any of these distributions. It is observed that this model can also be obtained as a hidden truncation model. Different properties of this new model have been discussed and compared with the corresponding properties of well-known distributions. Two data sets have been analysed for illustrative purposes and it is observed that in both the cases it fits better than Weibull, gamma or generalized exponential distributions.     

Stochastic comparisons between extreme order statistics from scale models

Stochastic ordering relations between extreme order statistics from exponential, Weibull and gamma distributions have been studied extensively by many researchers in recent years. In this work, we obtain various ordering results for the comparisons of two extreme order statistics from scale models when one set of scale parameters majorizes the other. The new results obtained here are applied when the baseline distributions are exponentiated Weibull or generalized gamma distributions. In this way, we generalize and extend some results established recently in the literature.     

Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

A new generalized Lindley distribution

A new generalized Lindley distribution, based on weighted mixture of two gamma distributions, is proposed. This model includes the Lindley, gamma and exponential distributions as and other forms of Lindley distributions as special cases. Lindley distribution based on two gamma with two consecutive shape parameter is investigated in some details. Statistical and reliability properties of this model are derived. The size-biased, the length-biased and Lorenze curve are established. Estimation of the underlying parameters via the moment method and maximum likelihood has been investigated and their values are simulated. Finally, fitting this model to a set of real-life data is discussed.     

A likelihood ratio test to discriminate exponential–Poisson and gamma distributions

The exponential–Poisson (EP) distribution with scale and shape parameters β>0 and λ∈?, respectively, is a lifetime distribution obtained by mixing exponential and zero-truncated Poisson models. The EP distribution has been a good alternative to the gamma distribution for modelling lifetime, reliability and time intervals of successive natural disasters. Both EP and gamma distributions have some similarities and properties in common, for example, their densities may be strictly decreasing or unimodal, and their hazard rate functions may be decreasing, increasing or constant depending on their shape parameters. On the other hand, the EP distribution has several interesting applications based on stochastic representations involving maximum and minimum of iid exponential variables (with random sample size) which make it of distinguishable scientific importance from the gamma distribution. Given the similarities and different scientific relevance between these models, one question of interest is how to discriminate them. With this in mind, we propose a likelihood ratio test based on Cox's statistic to discriminate the EP and gamma distributions. The asymptotic distribution of the normalized logarithm of the ratio of the maximized likelihoods under two null hypotheses – data come from EP or gamma distributions – is provided. With this, we obtain the probabilities of correct selection. Hence, we propose to choose the model that maximizes the probability of correct selection (PCS). We also determinate the minimum sample size required to discriminate the EP and gamma distributions when the PCS and a given tolerance level based on some distance are before stated. A simulation study to evaluate the accuracy of the asymptotic probabilities of correct selection is also presented. The paper is motivated by two applications to real data sets.     

Testing equality of scale parameters against restricted alternatives for m≥3 gamma distributions with unknown common shape parameter

Shiue and Bain (1983) proposed an approximate F-test for the equality of the scale parameters of two gamma distributions with equal but unknown shape parameters. In this article, we propose a simple procedure to test equality of scale parameters of m≥3 gamma distributions against nonincreasing order. The test is based on Fisher's method of combining p-values. The actual size of the resulting test is investigated through Monte Carlo studies. Also asymptotic results are derived for the nominal test size. These can be used to obtain a test which achieves the desired size. The case of more general partial orders is discussed.     

On an exponential family

This paper considers a generalization of the exponential type distributions in the class of exponential families. A characterization and a method of generating an exponential family from a given family are given. In particular the generalized gamma, the generalized Poisson, the inverse Gaussian distributions belonging to this family are discussed. The approximations of the cumulative sums for the generalized gamma and the generalized Poisson by the Chi-square are considered. Some of the results are extended to the bivariate case.