Asit P. Basu

Bayesian approach to life testing and reliability estimation using asymmetric loss function

Abstract In this paper we derive Bayes estimators of the mean lifetime and the reliability function in the exponential life testing model. The loss functions used are asymmetric to reflect that, in most situations of interest, overestimating is more harmful than underestimating. A number of prior distributions have been considered and Bayesian estimates have been compared with corresponding estimates with squared error loss function.

The Power Law Process: A Model for the Reliability of Repairable Systems

The power law process, often misleadingly called the Weibull process, is a useful and simple model for describing the failure times of repairable systems. We present elementary properties of the power law process, such as point estimation of unknown par..

Identifiability of the multinormal and other distributions under competing risks model

Let X1, X2 ,..., Xp be p random variables with joint distribution function F(x1 ,..., xp). Let Z = min(X1, X2 ,..., Xp) and I = i if Z = Xi. In this paper the problem of identifying the distribution function F(x1 ,..., xp), given the distribution Z or that of the identified minimum (Z, I), has been considered when F is a multivariate normal distribution. For the case p = 2, the problem is completely solved. If p = 3 and the distribution of (Z, I) is given, we get a partial solution allowing us to identify the independent case. These results seem to be highly nontrivial and depend upon Liouvilles result that the (univariate) normal distribution function is a nonelementary function. Some other examples are given including the bivariate exponential distribution of Marshall and Olkin, Gumbel, and the absolutely continuous bivariate exponential extension of Block and Basu.

Identifiability of distributions under competing risks and complementary risks model

Let X1,X2,…,Xp be p random variables with cdfs F1(x),F2(x),…,Fp(x)respectively. Let U = min(X1,X2,…,Xp) and V = max(X1,X2,…,Xp).In this paper we study the problem of uniquely determining and estimating the marginal distributions F1,F2,…,Fp given the distribution of U or of V. First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X1s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X1s are dependent is considered next. Finally the problem of estimation is considered.

Nonparametric Accelerated Life Testing

This paper considers the problem of nonparametric accelerated life testing by extending the results of Shaked & Singpurwalla in two directions. First we solve the case of censored data. Next we extend the methods to the case of competing risks. A s-consistent estimate of the failure distribution at use-stress is given for both cases.

Bayesian analysis of incomplete time and cause of failure data

Abstract For series systems with k components it is assumed that the cause of failure is known to belong to one of the 2 k − 1 possible subsets of the failure-modes. The theoretical time to failure due to k causes are assumed to have independent Weibull distributions with equal shape parameters. After finding the MLEs and the observed information matrix of ( λ 1 , …, λ k , β), a prior distribution is proposed for ( λ 1 , …, λ k ), which is shown to yield a scale-invariant noninformative prior as well. No particular structure is imposed on the prior of β. Methods to obtain the marginal posterior distributions of the parameters and other parametric functions of interest and their Bayesian point and interval estimates are discussed. The developed techniques are illustrated using a numerical example.

An investigation of Kendall's τ modified for censored data with applications

Abstract To accommodate testing for independence in bivariate data subject to censoring, several modifications of Kendalls τ are discussed. An extensive computer simulation is done to investigate power properties of these modifications under alternatives of the bivariate normal or bivariate exponential types. The statistics are then applied to available heart pacemaker patient survival data.

Statistical Analysis of Nonnormal Data

This title uses an applications approach to provide thorough coverage of nonnormal data. It contains a large number of nonparametric techniques such as regression, BIBD, split plots and two-way layouts with interaction that are not available in other books. It features a section on survival analysis. Each procedure is illustrated with numerical data from actual situations. This title includes a ready-to-use program diskette.

Testing whether survival function is harmonic new better than used in expectation

SummaryStatistical procedures to test that a life distribution is exponential against the alternative that it is harmonic new better than used in expectation (HNBUE) are considered.

Estimating the intensity function of a weibull process at the current time: failure truncated case

The Weibull process, a nonhomogeneous Poisson process with intensity function is often used to model the failure times of complex systems which are repaired after failure. Point estimation of the value of the intensity function at the current time is frequently an important problem and in this paper, two classes of estimators of this quantity are proposed when the data are failure truncated. Several members of these classes are suggested as estimators. Expressions for the bias and the mean squared error of these estimators are derived and are evaluated for several values of β and for samples sizes of 5, 10,20 and 40. Some estimators have smaller mean squared error than the conditional MLE for a wide range of the parameters.