Augustine Wong

Improved interval estimation for the two-parameter Birnbaum–Saunders distribution

Abstract An improved interval estimation for the two-parameter Birnbaum–Saunders distribution is discussed. The proposed method is based on the recently developed higher-order likelihood-based asymptotic procedure. The probability coverages of confidence intervals are based on the proposed method and those procedures discussed in Ng et al. (Comput. Statist. Data Anal., (2003)) are evaluated using Monte Carlo simulations for small and moderate sample sizes. Two real life examples and some concluding remarks are also presented.

Small sample asymptotic inference for the coefficient of variation: normal and nonnormal models

In applied statistics, the coefficient of variation is widely calculated and interpreted even when the sample size of the data set is very small. However, confidence intervals for the coefficient of variation are rarely reported. One of the reasons is the exact confidence interval for the coefficient of variation, which is given in Lehmann (Testing Statistical Hypotheses, 2nd Edition, Wiley, New York, 1996), is very difficult to calculate. Various asymptotic methods have been proposed in literature. These methods, in general, require the sample size to be large. In this article, we will apply a recently developed small sample asymptotic method to obtain approximate confidence intervals for the coefficient of variation for both normal and nonnormal models. These small sample asymptotic methods are very accurate even for very small sample size. Numerical examples are given to illustrate the accuracy of the proposed method.

Regression analysis, nonlinear or nonnormal : Simple and accurate p values from likelihood analysis

We develop simple approximations for the p values to use with regression models having linear or nonlinear parameter structure and normal or nonnormal error distribution; computer iteration then gives confidence intervals. Both frequentist and Bayesian versions are given. The approximations are derived from recent developments in likelihood analysis and have third-order accuracy. Also, for very small and medium-sized samples, the accuracy can typically be high. The likelihood basis of the procedure seems to provide the grounds for this general accuracy. Examples are discussed, and simulations record the distributional accuracy.

Practical Small-Sample Asymptotics for Distributions Used in Life-Data Analysis

Fraser and Lawless discussed exact conditional intervals for the parameters and quantiles of the location-scale model when complete data are used. Moreover, Lawless extended the exact method to failure-censored (Type II censored) data. Nevertheless, the exact intervals are difficult to obtain in practice and are unavailable under time censoring (Type I censoring). As a consequence, approximate large-sample intervals are widely used. In this article, a likelihood-based third-order procedure is developed. The method does not require explicit nuisance parameterization and can be easily implemented into algebraic computational packages. Numerical examples are presented to show the accuracy of the method even when the sample size is small.

Simple and accurate inference for the mean of the gamma model

The two-parameter gamma model is widely used in reliability, environmental, medical and other areas of statistics. It has a two-dimensional sufficient statistic, and a two-dimensional parameter which can be taken to describe shape and mean. This makes it closely comparable to the normal model, but it differs substantially in that the exact distribution for the minimal sufficient statistic is not available. Some recently developed asymptotic theory is used to derive an approximation for observed levels of significance and confidence intervals for the mean parameter of the model. The approximation is as easy to apply as first-order methods, and substantially more accurate.

Inference for bounded parameters

The estimation of the signal frequency count in the presence of background noise has been widely discussed in recent physics literature, and Mandelkern [Stat. Sci. 17, 149 (2002)] brings the central issues to the statistical community, leading in turn to extensive discussion by statisticians. The primary focus however of Mandelkern and the accompanying discussion is on the construction of a confidence interval. We argue that the likelihood function and p-value function provide a comprehensive presentation of the information available from the model and the data. This is illustrated for Gaussian and Poisson models with lower bounds for the mean parameter.

A note on inference for the mean parameter of the gamma distribution

The two parameter gamma distribution with mean [mu] and shape [tau] is widely used in reliability and life data analysis. Unlike the normal distribution, which also has two parameters describing the location and the scale, inference for the mean parameter of the gamma distribution is much more complicated (Jensen, 1986) and consequently less well developed. In this paper, a method of averaging is proposed to obtain confidence intervals for the mean parameter of the gamma distribution at an arbitrary level of significance. Numerical examples showed that this method is not only simple but also very accurate.

Higher Accuracy for Bayesian and Frequentist Interference: Large Sample Theory for Small Sample Likelihood.

Recent likelihood theory produces

A unified confidence interval for reliability-related quantities of two-parameter Weibull distribution

p

Approximate studentization for Pareto distribution with application to censored data

-values that have remarkable accuracy and wide applicability. The calculations use familiar tools such as maximum likelihood values (MLEs), observed information and parameter rescaling. The usual evaluation of such