Lee J. Bain

An exponential power life-testing distribution

An extreme-value type life-testing distribution is studied which has the property that the hazard function say assume a U-shaped form. Also the hazard function is exponentially increasing on the right. Some general properties of least squares type estimators are discussed for the case of location-scale parameter distributions, and these estimators are applied to the proposed model. Properties of the estimators are studied by Monta Carlo simulation, and procedures for interval estimation and tests of hypotheses for the parameters and reliability are provided. A numerical example is also considered.

Analysis for the Linear Failure-Rate Life-Testing Distribution

Some general comments are made concerning life-testing distributions with polynomial hazard functions, and some least squares type estimators are suggested as a possible method of parameter estimation. The linear hazard function case (h(t) = α + bt ) is considered in some detail. The maximum likelihood estimators of the parameters and reliability are studied for both complete and censored sampling, and the asymptotic variance covariance matrix is derived. In the linear case the simple least squares type estimators were compared to the maximum likelihood estimators by Monte Carlo simulation, and they were found to be fairly comparable to the maximllm likelihood estimators, being somewhat, better for small b/α2 and poorer for large b/α2. Percentage points were also determined by Monte Carlo simulation to mske possible tests of hypotheses for the parameters.

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.

INFERENTIAL PROCEDURES FOR THE GENERALIZED GAMMA DISTRIBUTION

Abstract The maximum likelihood estimators of the parameters of the generalized gamma distribution are shown to have the property that are distributed independently of a and b. Similar properties are also obtained for some Weibull-type statistics. These results are applied in particular to the problem of discriminating between the Weibull model and the generalized gamma model, and several test statistics are considered. In general, the Weibull model is quite flexible and, unless the sample size is quite large, is perhaps a preferable assumption because of the computational complexity and other difficulties encountered with the generalized gamma distribution.

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.

Estimation and Hypothesis Testing for the Parameters of a Bivariate Exponential Distribution

Abstract Statistical properties of the bivariate exponential distribution are investigated. Maximum likelihood and method of moments type estimates are obtained and compared with the estimates given by Arnold. The method of moments type estimates are easy to compute and highly efficient, whereas the maximum likelihood estimates are computationally inconvenient. The problem of testing for correlation in the bivariate exponential distribution is also investigated.

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.

Inferences Based on Censored Sampling From the Weibull or Extreme-Value Distribution

A simple, unbiased estimator, based on a censored sample, is proposed for the scale parameter of the extreme-value distribution. The exact distribution of the estimator is determined for the cases in which only the first two or only the first three ordered observations are available. The asymptotic distribution is derived, and an approximate distribution for small sample size is also provided. Interval estimation for the scale parameter is developed and a conservative interval estimate for reliability is also obtained.

Inferences Concerning the Mean of the Gamma Distribution

Abstract Approximate tests for the mean of a gamma distribution with both parameters unknown are derived. The power of these tests is close to the power of the corresponding uniformly most powerful tests for the mean when the shape parameter is assumed known. The tests are studied by Monte Carlo simulation, and limiting results are also provided as the shape parameter goes to zero or infinity, as well as for the large-sample case. A limited study indicating the nonrobustness of the standard t test for this case is also included.

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.