Aaron Childs

Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution

Chen and Bhattacharyya (1988,Comm. Statist. Theory Methods,17, 1857–1870) derived the exact distribution of the maximum likelihood estimator of the mean of an exponential distribution and an exact lower confidence bound for the mean based on a hybrid censored sample. In this paper, an alternative simple form for the distribution is obtained and is shown to be equivalent to that of Chen and Bhattacharyya (1988). Noting that this scheme, which would guarantee the experiment to terminate by a fixed timeT, may result in few failures, we propose a new hybrid censoring scheme which guarantees at least a fixed number of failures in a life testing experiment. The exact distribution of the MLE as well as an exact lower confidence bound for the mean is also obtained for this case. Finally, three examples are presented to illustrate all the results developed here.

Exact Likelihood Inference for an Exponential Parameter Under Progressive Hybrid Censoring Schemes

The purpose of this chapter is to propose two types of progressive hybrid censoring schemes in life-testing experiments and develop exact inference for the mean of the exponential distribution. The exact distribution of the maximum likelihood estimator and an exact lower confidence bound for the mean lifetime are obtained under both types of progressive hybrid censoring schemes. Illustrative examples are finally presented.

Maximum likelihood estimation of Laplace parameters based on general type-II censored examples

In this paper, we derive the maximum likelihood estimators of the parameters of a Laplace distribution based on general Type-II censored samples. The resulting explicit MLEs turn out to be simple linear functions of the order statistics. We then examine the asymptotic variance of the estimates by calculating the elements of the Fisher information matrix.

Exact distribution of the MLEs of the parameters and of the quantiles of two-parameter exponential distribution under hybrid censoring

Epstein [Truncated life tests in the exponential case, Ann. Math. Statist. 25 (1954), pp. 555–564] introduced a hybrid censoring scheme (called Type-I hybrid censoring) and Chen and Bhattacharyya [Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods 17 (1988), pp. 1857–1870] derived the exact distribution of the maximum-likelihood estimator (MLE) of the mean of a scaled exponential distribution based on a Type-I hybrid censored sample. Childs et al. [Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math. 55 (2003), pp. 319–330] provided an alternate simpler expression for this distribution, and also developed analogous results for another hybrid censoring scheme (called Type-II hybrid censoring). The purpose of this paper is to derive the exact bivariate distribution of the MLE of the parameter vector of a two-parameter exponential model based on hybrid censored samples. The marginal distributions are derived and exact confidence bounds for the parameters are obtained. The results are also used to derive the exact distribution of the MLE of the pth quantile, as well as the corresponding confidence bounds. These exact confidence intervals are then compared with parametric bootstrap confidence intervals in terms of coverage probabilities. Finally, we present some numerical examples to illustrate the methods of inference developed here.

Some approximations to the multivariate hypergeometric distribution with applications to hypothesis testing

In this paper, we will examine some approximations to the multivariate hypergeometric distribution by continuous random variables. The continuous random variables will be chosen so as to have the same range of variation, means, variances and covariances as their discrete counterparts. We then show how these approximations can be used in testing hypotheses about the parameters of the multivariate hypergeometric distribution.

Higher Order Moments of Order Statistics From the Power Function Distribution and Edgeworth Approximate Inference

In this paper, we first derive exact explicit expressions for the triple and quadruple moments of order statistics from the power function distribution. Also, we present recurrence relations for single, double, triple and quadruple moments of order statistics from the power function distribution. These relations will enable one to find all moments (of order up to four) of order statistics for all sample sizes in a simple recursive manner. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. We then derive approximate confidence intervals for the parameters of the power function distribution using the Edgeworth approximation. Finally, we extend the recurrence relations to the case of the doubly truncated power function distribution.

Conditional Inference for the Parameters of Pareto Distributions when Observed Samples are Progressively Censored

In this paper we develop procedures for obtaining confidence intervals for the location and scale parameters of a Pareto distribution as well as upper and lower γ probability tolerance intervals for a proportion β when the observed samples are progressively censored. The intervals are exact, and are obtained by conditioning on the observed values of the ancillary statistics. Since the procedures assume that the shape parameter ν is known, a sensitivity analysis is also carried out to see how the procedures are affected by changes in ν.

Series approximations for moments of order statistics using MAPLE

In this paper, we present a series of Maple procedures that will approximate the means, variances, and covariances of order statistics from any continuous population using series approximations in the form of David and Johnsons (Biometrika 41 (1954) 228-240) approximation. These procedures will allow one to improve upon the accuracy of David and Johnsons approximation by extending the series to include higher order terms.

Unified scheme for testing for outliers in linear models

Mickey et al. [Mickey, M.R., Dunn, O.J. and Clark, V., 1967, Note on use of stepwise regression in detecting outliers. Computers and Biomedical Research, 1, 105–111.] proposed a test for discordancy in linear models, which compares the sum of squares of residuals based on the complete model to that of a model obtained by deleting the potential outliers. John [John, J.A., 1978, Outliers in factorial experiments. Applied Statistics, 27, 111–119.] proposed a test procedure which treats the outliers as missing values and involves obtaining estimates of those missing values. In this article, we show that the two procedures are in fact equivalent. We also compare the performance of two methods of implementing these procedures. Owing to the wide variety of possible designs, tables of critical values are not readily available for testing for outliers in linear models. Therefore, we provide a program that will allow the user to input a design matrix, along with some data, and will output a p-value for testing for a specified number of outliers.

DIRICHLET-RELATED PROBABILITY PROBLEMS AND NEW TOOLS FOR SOLVING THEM

ABSTRACT In this paper we primarily consider waiting time problems under three different sampling rules. SR1 is the usual sampling with replacement, SR2 is without replacement, and SR3 is also with replacement, but uses no repetitions. We develop a new methodology for solving a wide variety of waiting time problems under each of the three sampling rules. A connection between waiting time problems under SR2 and SR3 is established which enables one to simultaneously solve waiting time problems under both of these sampling rules. The methods are illustrated with a large number of examples.