Hon Keung Tony Ng

Estimation of parameters from progressively censored data using EM algorithm

EM algorithm is used to determine the maximum likelihood estimates when the data are progressively Type II censored. The method is shown to be feasible and easy to implement. The asymptotic variances and covariances of the ML estimates are computed by means of the missing information principle. The methodology is illustrated with two popular models in lifetime analysis, the lognormal and Weibull lifetime distributions.

Optimal Progressive Censoring Plans for the Weibull Distribution

In this article we compute the expected Fisher information and the asymptotic variance–covariance matrix of the maximum likelihood estimates based on a progressively type II censored sample from a Weibull distribution by direct calculation as well as the missing-information principle. We then use these values to determine the optimal progressive censoring plans. Three optimality criteria are considered, and some selected optimal progressive censoring plans are presented according to these optimality criteria. We also discuss the construction of progressively censored reliability sampling plans for the Weibull distribution. Three illustrative examples are provided with discussion.

Modified moment estimation for the two-parameter Birnbaum--Saunders distribution

The maximum likelihood estimators and a modification of the moment estimators of a two-parameter Birnbaum-Saunders distribution are discussed. A simple bias-reduction method is proposed to reduce the bias of the maximum likelihood estimators and the modified moment estimators. The jackknife technique is also used to reduce the bias of these estimators. Monte Carlo simulation is used to compare the performance of all these estimators. The probability coverages of confidence intervals based on inferential quantities associated with all these estimators are evaluated using Monte Carlo simulations for small, moderate and large sample sizes. Two illustrative examples and some concluding remarks are finally presented.

Point and Interval Estimation for a Simple Step-Stress Model with Type-II Censoring

In reliability and life-testing experiments, the researcher is often interested in the effects of extreme or varying stress levels, such as temperature, voltage, and load, on the lifetimes of experimental units. Accelerated testing allows the experimenter to increase these stress levels to obtain information on the parameters of the life distributions more quickly than would be possible under normal operating conditions. A special class of accelerated tests are step-stress tests that allow the experimenter to increase the stress levels at fixed times during the experiment. In this article, we consider the simple step-stress model under Type-II censoring. We derive the maximum likelihood estimators (MLEs) of the parameters assuming a cumulative exposure model with lifetimes being exponentially distributed. The exact distributions of the MLEs of parameters are obtained through the use of conditional moment-generating functions. We also derive confidence intervals for the parameters using these exact distributions, asymptotic distributions, and the parametric bootstrap method, and assess their performance through a Monte Carlo simulation study.

Point and interval estimation for Gaussian distribution, based on progressively Type-II censored samples

The likelihood equations based on a progressively Type-II censored sample from a Gaussian distribution do not provide explicit solutions in any situation except the complete sample case. This paper examines numerically the bias and mean square error of the MLE, and demonstrates that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic s-normality are unsatisfactory, and particularly so when the effective sample size is small. Therefore, this paper suggests using unconditional simulated percentage points of these pivotal quantities for constructing s-confidence intervals. An approximation of the Gaussian hazard function is used to develop approximate estimators which are explicit and are almost as efficient as the MLE in terms of bias and mean square error; however, the probability coverages of the corresponding pivotal quantities based on asymptotic s-normality are also unsatisfactory. A wide range of sample sizes and progressive censoring schemes are used in this study.

Parameter estimation for a modified Weibull distribution, for progressively type-II censored samples

In this paper, the estimation of parameters based on a progressively Type-II censored sample from a modified Weibull distribution is studied. The likelihood equations, and the maximum likelihood estimators are derived. The estimators based on a least-squares fit of a multiple linear regression on a Weibull probability paper plot are compared with the MLE via Monte Carlo simulations. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLE are derived. Approximate confidence intervals for the parameters are constructed based on the s-normal approximation to the asymptotic distribution of MLE, and log-transformed MLE. The coverage probabilities of the individual s-normal-approximation confidence intervals for the parameters are examined numerically. Some recommendations are made from the results of a Monte Carlo simulation study, and a numerical example is presented to illustrate all of the methods of inference developed here.

Erratum to Point and interval estimation for the two-parameter Birnbaum-Saunders distribution based on Type-II censored samples

The maximum likelihood estimators, based on Type-II censored samples, of a two-parameter Birnbaum-Saunders distribution are discussed. We propose a simple bias-reduction method to reduce the bias of the maximum likelihood estimators. We also discuss a Monte Carlo EM-algorithm for the determination of the maximum likelihood estimators. Monte Carlo simulation is used to compare the performance of all these estimators. The probability coverages of confidence intervals based on inferential quantities associated with these estimators are evaluated using Monte Carlo simulations for small, moderate, and large sample sizes, for various degrees of censoring. Two illustrative examples and some concluding remarks are finally presented.

Testing the Ratio of Two Poisson Rates

In this paper we compare the properties of four different general approaches for testing the ratio of two Poisson rates. Asymptotically normal tests, tests based on approximate p -values, exact conditional tests, and a likelihood ratio test are considered. The properties and power performance of these tests are studied by a Monte Carlo simulation experiment. Sample size calculation formulae are given for each of the test procedures and their validities are studied. Some recommendations favoring the likelihood ratio and certain asymptotic tests are based on these simulation results. Finally, all of the test procedures are illustrated with two real life medical examples.

Statistical estimation for the parameters of Weibull distribution based on progressively type-I interval censored sample

In this paper, the estimation of parameters based on a progressively type-I interval censored sample from a two-parameter Weibull distribution is studied. Different methods of estimation are discussed. They include the maximum likelihood estimation, method of moments, estimation based on Weibull probability plot, mid-point approximation method and one-step approximation method. The estimation procedures are discussed in details and compared via Monte Carlo simulations in terms of their biases and mean square errors. Some recommendations are made from the simulation results and a numerical example is presented to illustrate all of the methods of estimation developed here.

A validation study of type 2 diabetes-related variants of the TCF7L2, HHEX, KCNJ11, and ADIPOQ genes in one endogamous ethnic group of North India.

The aim of this study was to validate the single nucleotide polymorphisms (SNPs) of four candidate genes (TCF7L2, HHEX, KCNJ11, and ADIPOQ) related to type 2 diabetes (T2D) in an endogamous population of north India; the Aggarwal population, having 18‐clans. This endogamous population model was heavily supported by recent land mark work and we also verified the homogeneity of this population by clan‐based stratification analysis. Two SNPs (rs4506565; rs7903146) in TCF7L2 were found to be significant (p‐value = 0.00191; p‐value = 0.00179, respectively), and odds ratios of 2.1 (dominant‐model) and 2.0 (recessive‐model) respectively, were obtained for this population. The TTT haplotype in the TCF7L2 gene was significantly associated with T2D. Waist‐Hip ratio (WHR), systolic blood pressure (SBP), and age were significant covariates for increasing risk of T2D. Single‐SNP, combined‐SNPs and haplotype analysis provides clear evidence that the causal mutation is near to or within the significant haplotype (TTT) of the TCF7L2 gene. In spite of a culturally‐learned sedentary lifestyle and fat‐enriched dietary habits, WHR rather than body‐mass‐index emerged as a robust predictor of risk for T2D in this population.