Reliability inference for a general lower-truncated family of distributions under records

Based on record values, point and interval estimators are proposed in this paper for the parameters of a general lower-truncated family of distributions. Maximum likelihood and bias-corrected estimators are obtained for unknown model parameters. Based on a sufficient and complete statistic, the bias-corrected estimator is also shown to be uniformly minimum variance unbiased estimator. Different exact confidence intervals and exact confidence regions are constructed for the both model and truncated parameters, and other confidence interval estimates based on asymptotic distribution theory and bootstrap approaches are obtained as well. Finally, two real-life examples and a numerical study are presented to illustrate the performance of our methods.     

Bootstrap confidence intervals of generalized process capability index Cpyk for Lindley and power Lindley distributions

One of the indicators for evaluating the capability of a process is the process capability index. In this article, bootstrap confidence intervals of the generalized process capability index (GPCI) proposed by Maiti et al. are studied through simulation, when the underlying distributions are Lindley and Power Lindley distributions. The maximum likelihood method is used to estimate the parameters of the models. Three bootstrap confidence intervals namely, standard bootstrap (SB), percentile bootstrap (PB), and bias-corrected percentile bootstrap (BCPB) are considered for obtaining confidence intervals of GPCI. A Monte Carlo simulation has been used to investigate the estimated coverage probabilities and average width of the bootstrap confidence intervals. Simulation results show that the estimated coverage probabilities of the percentile bootstrap confidence interval and the bias-corrected percentile bootstrap confidence interval get closer to the nominal confidence level than those of the standard bootstrap confidence interval. Finally, three real datasets are analyzed for illustrative purposes.     

On compounded geometric distributions and their applications

Here, we introduce two-parameter compounded geometric distributions with monotone failure rates. These distributions are derived by compounding geometric distribution and zero-truncated Poisson distribution. Some statistical and reliability properties of the distributions are investigated. Parameters of the proposed distributions are estimated by the maximum likelihood method as well as through the minimum distance method of estimation. Performance of the estimates by both the methods of estimation is compared based on Monte Carlo simulations. An illustration with Air Crash casualties demonstrates that the distributions can be considered as a suitable model under several real situations.     

Estimation for the parameters of the generalized half-normal distribution

As an applicable and flexible lifetime model, the two-parameter generalized half-normal (GHN) distribution has been received wide attention in the field of reliability analysis and lifetime study. In this paper maximum likelihood estimates of the model parameters are discussed and we also proposed corresponding bias-corrected estimates. Unweighted and weighted least squares estimates for the parameters of the GHN distribution are also presented for comparison purpose. Moreover, the likelihood ratio test is provided as complementary. Simulation study and illustrative examples are provided to compare the performance of the proposed methods.     

AN EMPIRICAL LIKELIHOOD APPROACH FOR NON‐GAUSSIAN VECTOR STATIONARY PROCESSES AND ITS APPLICATION TO MINIMUM CONTRAST ESTIMATION

We develop the empirical likelihood approach for a class of vector‐valued, not necessarily Gaussian, stationary processes with unknown parameters. In time series analysis, it is known that the Whittle likelihood is one of the most fundamental tools with which to obtain a good estimator of unknown parameters, and that the score functions are asymptotically normal. Motivated by the Whittle likelihood, we apply the empirical likelihood approach to its derivative with respect to unknown parameters. We also consider the empirical likelihood approach to minimum contrast estimation based on a spectral disparity measure, and apply the approach to the derivative of the spectral disparity. This paper provides rigorous proofs on the convergence of our two empirical likelihood ratio statistics to sums of gamma distributions. Because the fitted spectral model may be different from the true spectral structure, the results enable us to construct confidence regions for various important time series parameters without assuming specified spectral structures and the Gaussianity of the process.     

Estimation of a process capability index for inverse gaussian distribution

In this article, interval estimates of Clements' process capability index are studied through bootstrapping when the underlying distribution is Inverse Gaussian. The standard bootstrap, the percentile bootstrap, and the bias-corrected percentile bootstrap confidence intervals are compared.     

Estimation of Hazard, Density and Survivor Functions for Randomly Censored Data

Maximum likelihood estimation and goodness-of-fit techniques are used within a competing risks framework to obtain maximum likelihood estimates of hazard, density, and survivor functions for randomly right-censored variables. Goodness-of- fit techniques are used to fit distributions to the crude lifetimes, which are used to obtain an estimate of the hazard function, which, in turn, is used to construct the survivor and density functions of the net lifetime of the variable of interest. If only one of the crude lifetimes can be adequately characterized by a parametric model, then semi-parametric estimates may be obtained using a maximum likelihood estimate of one crude lifetime and the empirical distribution function of the other. Simulation studies show that the survivor function estimates from crude lifetimes compare favourably with those given by the product-limit estimator when crude lifetimes are chosen correctly. Other advantages are discussed.     

General results for the beta Weibull distribution

In this paper, we study some mathematical properties of the beta Weibull (BW) distribution, which is a quite flexible model in analysing positive data. It contains the Weibull, exponentiated exponential, exponentiated Weibull and beta exponential distributions as special sub-models. We demonstrate that the BW density can be expressed as a mixture of Weibull densities. We provide their moments and two closed-form expressions for their moment-generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and two entropies. The density of the BW-order statistics is a mixture of Weibull densities and two closed-form expressions are derived for their moments. The estimation of the parameters is approached by two methods: moments and maximum likelihood. We compare the performances of the estimates obtained from both the methods by simulation. The expected information matrix is derived. For the first time, we introduce a log-BW regression model to analyse censored data. The usefulness of the BW distribution is illustrated in the analysis of three real data sets.     

Simulation of right and left truncated gamma distributions by mixtures

We study the properties of truncated gamma distributions and we derive simulation algorithms which dominate the standard algorithms for these distributions. For the right truncated gamma distribution, an optimal accept–reject algorithm is based on the fact that its density can be expressed as an infinite mixture of beta distribution. For integer values of the parameters, the density of the left truncated distributions can be rewritten as a mixture which can be easily generated. We give an optimal accept–reject algorithm for the other values of the parameter. We compare the efficiency of our algorithm with the previous method and show the improvement in terms of minimum acceptance probability. The algorithm proposed here has an acceptance probability which is superior to e/4.     

The poisson-inverse gaussian disiribuiion as a model for species abundance

An inverse Gaussian mixture of Poisson distributions(the P-IG distribution) is considered as a model for species abundance data,, Minimum chi-square and maximum likelihood methods of estimation for the zero-truncated P-IG distribution are developed, Ihe performance of the P-IG distribution is illustrated and discussed for several well-known sets of insect abundance data.     

Point estimation of the stress–strength reliability parameter for parallel system with independent and non-identical components

In this article, the estimation problem of the multicomponent stress–strength reliability parameter is considered where the stress and the strength systems have arbitrary fixed numbers of independent and non-identical parallel components. It is assumed that the distribution functions of the stress and the strength components satisfy the proportional reversed hazard rate model. The study is done in more details when the baseline distributions are exponential. Maximum likelihood and uniformly minimum variance unbiased estimators are obtained and compared. Also, Bayes and empirical Bayes estimators are discussed and Monte Carlo simulations are carried out to compare their performances.     

Minimum Hellinger distance estimation for randomized play the winner design

Response-adaptive designs in clinical trials incorporate information from prior patient responses in order to assign better performing treatments to the future patients of a clinical study. An example of a response adaptive design that has received much attention in recent years is the randomized play the winner design (RPWD). Beran [1977. Minimum Hellinger distance estimates for parametric models. Ann. Statist. 5, 445–463] investigated the problem of minimum Hellinger distance procedure (MHDP) for continuous data and showed that minimum Hellinger distance estimator (MHDE) of a finite dimensional parameter is as efficient as the MLE (maximum likelihood estimator) under a true model assumption. This paper develops minimum Hellinger distance methodology for data generated using RPWD. A new algorithm using the Monte Carlo approximation to the estimating equation is proposed. Consistency and asymptotic normality of the estimators are established and the robustness and small sample performance of the estimators are illustrated using simulations. The methodology when applied to the clinical trial data conducted by Eli-Lilly and Company, brings out the treatment effect in one of the strata using the frequentist techniques compared to the Bayesian argument of Tamura et al [1994. A case study of an adaptive clinical trialin the treatment of out-patients with depressive disorder. J. Amer. Statist. Assoc. 89, 768–776].     

Efficient estimation of the PDF and the CDF of the inverse Rayleigh distribution

In this paper, we consider the estimation of the probability density function and the cumulative distribution function of the inverse Rayleigh distribution. In this regard, the following estimators are considered: uniformly minimum variance unbiased estimator, maximum likelihood (ML) estimator, percentile estimator, least squares estimator and weighted least squares estimator. To do so, analytical expressions are derived for the mean integrated squared error. As the result of simulation studies and real data applications indicate, when the sample size is not very small the ML estimator performs better than the others.     

Estimation procedures for grouped data – a comparative study

Interval-censored data are very common in the reliability and lifetime data analysis. This paper investigates the performance of different estimation procedures for a special type of interval-censored data, i.e. grouped data, from three widely used lifetime distributions. The approaches considered here include the maximum likelihood estimation, the minimum distance estimation based on chi-square criterion, the moment estimation based on imputation (IM) method and an ad hoc estimation procedure. Although IM-based techniques are extensively used recently, we show that this method is not always effective. It is found that the ad hoc estimation procedure is equivalent to the minimum distance estimation with another distance metric and more effective in the simulation. The procedures of different approaches are presented and their performances are investigated by Monte Carlo simulation for various combinations of sample sizes and parameter settings. The numerical results provide guidelines to analyse grouped data for practitioners when they need to choose a good estimation approach.     

Robust minimum distance inference based on combined distances

The minimum disparity estimators proposed by Lindsay (1994) for discrete models form an attractive subclass of minimum distance estimators which achieve their robustness without sacrificing first order efficiency at the model. Similarly, disparity test statistics are useful robust alternatives to the likelihood ratio test for testing of hypotheses in parametric models; they are asymptotically equivalent to the likelihood ratio test statistics under the null hypothesis and contiguous alternatives. Despite their asymptotic optimality properties, the small sample performance of many of the minimum disparity estimators and disparity tests can be considerably worse compared to the maximum likelihood estimator and the likelihood ratio test respectively. In this paper we focus on the class of blended weight Hellinger distances, a general subfamily of disparities, and study the effects of combining two different distances within this class to generate the family of “combined” blended weight Hellinger distances, and identify the members of this family which generally perform well. More generally, we investigate the class of "combined and penal-ized" blended weight Hellinger distances; the penalty is based on reweighting the empty cells, following Harris and Basu (1994). It is shown that some members of the combined and penalized family have rather attractive properties     

Evaluation and comparison of estimations in the generalized exponential-Poisson distribution

The uniformly minimum variance unbiased, maximum-likelihood, percentile and least-squares estimators of the probability density function and the cumulative distribution function are derived for the generalized exponential-Poisson distribution. This model has shown to be useful in reliability and lifetime data modelling, especially when the hazard rate function has a bathtub shape. Simulation studies are also carried out to show that the maximum-likelihood estimator is better than the uniformly minimum variance unbiased estimator (UMVUE) and that the UMVUE is better than others.     

Simex approaches to measurement error in roc studies

This paper explores the estimation of the area under the ROC curve when test scores are subject to errors. The naive approach that ignores measurement errors generally yields inconsistent estimates. Finding the asymptotic bias of the naive estimator, Coffin and Sukhatme (1995, 1997) proposed bias-corrected estimators for parametric and nonparametric cases. However, the asymptotic distributions of these estimators have not been developed because of their complexity. We propose several alternative approaches, including the SIMEX procedure of Cook and Stefanski (1994). We also provide the asymptotic distributions of the SIMEX estimators for use in statistical inference. Small simulation studies illustrate that the SIMEX estimators perform reasonably well when compared to the bias-corrected estimators.     

A MARKOV CHAIN MODEL FOR MAGAZINE EXPOSURE

Klotz's (1973) Markov chain model for dependent Bernoulli trials is applied to magazine exposure distributions. Simple parameter estimates are derived and are shown to compare well with the maximum likelihood estimates. The Markov model is fitted to forty magazines from a large print media survey and compares favourably with the most popular non-proprietary magazine model, the beta-binomial model. In addition, the Markov model is used to simulate magazine exposure distributions.     

Weibull lagged effect step-stress model with competing risks

In many survival analysis studies, failure can come from one of several competing risks. Additionally, where survival times are lengthy, researchers can increase stress levels to cause units to fail faster. One type of accelerated testing is a step-stress test where the increase is presented in quantum jumps at predetermined time points. If the impact of the increase is not immediately attained, an interim lag period is modeled. In this article, we propose a two-competing risk step-stress model with a lag period where each independent risk follows a Weibull lifetime distribution, the interim lag period is linear, and the attainment point is assumed known. We obtain the maximum likelihood estimators and the observed information matrix; we construct confidence intervals and provide estimates of coverage probabilities using large sample theory, percentile bootstrap, and bias-corrected accelerated (BCa) bootstrap methods.     

Hellinger distance as a penalized log likelihood

The present paper studies the minimum Hellinger distance estimator by recasting it as the maximum likelihood estimator in a data driven modification of the model density. In the process, the Hellinger distance itself is expressed as a penalized log likelihood function. The penalty is the sum of the model probabilities over the non-observed values of the sample space. A comparison of the modified model density with the original data provides insights into the robustness of the minimum Hellinger distance estimator. Adjustments of the amount of penalty leads to a class of minimum penalized Hellinger distance estimators, some members of which perform substantially better than the minimum Hellinger distance estimator at the model for small samples, without compromising the robustness properties of the latter.