Ping Shing Chan

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.

Start-Up Demonstration Tests with Rejection of Units upon Observing d Failures

The probability generating function of number of trials for the start-up demonstration test with rejection upon d failures is derived. The exact distribution of the number of trials is obtained. Some recurrence relations for the probabilities are also established. The average length of the test is derived. Some illustrative examples are finally presented.

Order statistics from extreme value distribution, i: tables of means, variances and covariances

Let X1, X2, …, Xn be a random sample of size n from an extreme value distribution and X1:n less than or equal X2:n less than or equal … less than or equal Xn:n be the order statistics ob-tained from this sample. Tables of the means, variances, and covariances of the order statistics for samples of size n are given for n = 1(1)15(5)30. The computational formulae and procedure used and some checks employed are explained.

On the normal record values and associated inference

In this note, we discuss the record values arising from a normal N([mu], [delta]2) distribution. After computing the means, variances and covariances of the record values, we determine the best linear unbiased estimators (BLUEs) of [mu] and [delta] based on the first n record values. Using these BLUEs, we then develop a prediction interval for a future record value and also propose a test for spuriosity of a current record value. Finally, we present an illustrative example to explain the inference procedures developed in this note.

Relations for single and product moments of record values from Gumbel distribution

In this paper some recurrence relations between the moments of record values from the Gumbel distribution are established. It is shown that using these recurrence relations, all the single and product moments of all record values can be obtained in a very simple recursive process.

Statistical Inference of Type-II Progressively Hybrid Censored Data with Weibull Lifetimes

In this article, we discuss the maximum likelihood estimators and approximate maximum likelihood estimators of the parameters of the Weibull distribution with two different progressively hybrid censoring schemes. We also present the associated expressions of the expected total test time and the expected effective sample size which will be useful for experimental planning purpose. Finally, the efficiency of the point estimation of the parameters based on the two progressive hybrid censoring schemes are compared and the merits of each censoring scheme are discussed.

Estimation for the scaled half logistic distribution under Type II censoring

Abstract We derive the best linear unbiased estimator (BLUE) based on doubly Type-II censored samples for the scaled half logistic distribution. Next, we derive the best linear unbiased estimator and the asymptotic best linear unbiased estimator based on k optimally selected order statistics and show that the asymptotic result provides very close approximation to the finite sample result even for a sample of size as small as 20. The maximum likelihood estimator (MLE) based on either complete or Type-II censored samples does not exist in explicit form. We determine its unbiasing factor and variance through Monte Carlo simulations employing a numerical iterative procedure. We derive an approximate maximum likelihood estimator (AMLE) which has an explicit form and is almost as efficient as the MLE and the BLUE. We illustrate all these methods of estimation with two examples.

Optimal allocation of redundancies in series systems

It is of great interest for the problem of how to allocate redundancies in a system so as to optimize the system performance in reliability engineering and system security. In this paper, we consider the problems of optimal allocation of both active and standby redundancies in series systems in the sense of various stochastic orderings. For the case of allocating one redundancy to a series system with two exponential components, we establish two likelihood ratio order results for active redundancy case and standby redundancy case, respectively. We also discuss the case of allocating K active redundancies to a series system and establish some new results. The results developed here strengthen and generalize some of the existing results in the literature. Specifically, we give an answer to an open problem mentioned in Hu and Wang [T. Hu, Y. Wang, Optimal allocation of active redundancies in r-out-of-n systems, Journal of Statistical Planning and Inference 139 (2009) 3733–3737]. Numerical examples are provided to illustrate the theoretic results established here.

Recurrence relations for moments of record values from generalized extreme value distribution

In this paper some recurrence relations between the moments of record values from the generalized extreme value distribution are established. It is shown that using these recurrence relations, all the single and product moments of all record values can be obtained in a simple recursive manner.