Max Engelhardt

Inferences on the Parameters of the Birnbaum-Saunders Fatigue Life Distribution Based on Maximum Likelihood Estimation

This paper provides tests of hypotheses and confidence intervals for the scale and shape parameters singly of the Birnbaum-Saunders fatigue life model, each with the other an unknown nuisance parameter. The procedure for each parameter is based on its maximum likelihood estimate, although in the case of the scale parameter it is necessary to consider conditional procedures.

Simplified Statistical Procedures for the Weibull or Extreme-Value Distribution

A method of finding confidence bounds on Weibull reliability, or tolerance limits for the Weibull or extreme-value distribution is presented. Inference procedures for the parameters are also discussed. Comparisons are made with some other available methods. New simplified estimators, of the parameters, for complete samples, are presented.

Some Complete and Censored Sampling Results for the Weibull or Extreme-Value Distribution

A simple, unbiased estimator, based on a censored sample, has been proposed by Rain [1] for the scale parameter of the Extreme-value distribution. This estimator was shown to have high efficiency and to be approximately distributed as a chi-square variable if substantial censoring occurs. Further small sample and asymptotic properties of this estimator are considered in this paper. The estimator is modified so that it is more applicable to the complete sample case and a close chi-square approximation is established for all cases. The estimator is also shown to be related to the maximum likelihood estimator.

Interval Estimation for the Two-parameter Double Exponential Distribution

Confidence intervals based on the maximum likelihood estimators are given for the location and scale parameters of the Double Exponential distribution. These intervals are obtained by determining the distribution of the pivotal quantities ( – θ)/ and /σ. Exact distributions are determined for n = 3 and n = .5, and approximate distributions are provided for larger n. The asymptotic distributions are also given and the accluacy of these approximations are investigated. The powers of the associated tests of hypotheses are given and tolerance limits for the population are also provided. Some possible applications are indicated.

Tests for an Increasing Trend in the Intensity of a Poisson Process: A Power Study

Abstract This article concerns a comparison of several tests for testing the hypothesis of a constant intensity against the alternative of an increasing intensity function in a nonhomogeneous Poisson process (NHPP). The study includes the well-known Laplace test statistic, the most powerful test for the shape parameter in a Poisson process with Weibull intensity, the likelihood ratio test against arbitrary NHPP alternatives, two nonparametric tests for trends based on Kendalls tau and Spearmans rho, and a test based on an F statistic. The powers of the tests are determined by Monte Carlo simulation against alternatives that are increasing at an exponential rate, a power rate (Weibull intensity), and a logarithmic rate. Alternatives that are step functions with one jump are also considered. In a few cases, the exact powers are also obtained analytically.

A Two-Moment Chi-Square Approximation for the Statistic Log

Abstract A two-moment chi-square approximation is derived for the statistic log , where and are, respectively, the arithmetic and geometric means of a sample from a gamma distribution. The proposed approximation is used for testing the shape parameter in a gamma distribution and for testing the homogeneity of normal population variances.

Tolerance Limits and Confidence Limits on Reliability for the Two-Parameter Exponential Distribution

Tolerance limits and confidence limits on reliability, which closely approximate exact limits. are proposed for the two-parameter exponential distribution. These approximations have the advantage that solutions to both the tolerance limit problem and the confidence limit problem can be written explicitly.

On Simple Estimation of the Parameters of the Weibull or Extreme-Value Distribution*

Simple, closed form approximations for maximum likelihood estimates of the parameters of the Weibull or extreme-value distribution are discussed. A method for the exact computation of constants required to calculate the estimates is presented, and simpler approximate methods are also provided. Some inference procedures for the parameters are also discussed.

Simple Approximate Distributional Results for Confidence and Tolerance limits for the Weibull Distribution Based on Maximum likelihood Estimators

One form of the generalized gamma distribution brings out an association between the normal distribution and the extreme-value distribution. This relationship suggests chi-squared, Students t and noncentral t approximations for certain functions of the maximum likelihood estimators of the parameters of the extreme-value distribution. The accuracy of these approximations is studied by comparing them to existing Monte Carlo simulation results. These approximations provide simple procedures for obtaining approximate confidence intervals for the parameters, tolerance limits, and confidence limits on reliability for the Weibull or extremevalue distribution.

Prediction Intervals for the Weibull Process

Exact prediction intervals, based upon maximum likelihood estimation, for the kth future observation of a Weibull process are derived. For k = 1, an exact, closed-form prediction limit is given. For k > 1, the exact solution, and a simpler approximate prediction limit are both given.