SUMMARY A general method of estimating parameters in continuous univariate distributions is proposed. It is especially suited to cases where one of the parameters is an unknown shifted origin. This occurs, for example, in the three-parameter lognormal, gamma and Weibull models. For such distributions it is known that maximum likelihood (ML) estimation can break down because the likelihood is unbounded and this can lead to inconsistent estimators. Properties of the proposed method are described. In particular it is shown to give consistent estimators with asymptotic efficiency equal to ML estimators when these exist. Moreover it gives consistent, asymptotically efficient estimators in situations where ML fails. Examples are given including numerical ones showing the advantages of the method.

Estimating Parameters in Continuous Univariate Distributions with a Shifted Origin

Although a vast amount of literature exists on the selection of an appropriate probability distribution for annual maximum floodflows, few studies have examined which probability distributions are most suitable to fit sequences of annual minimum streamflows. Probability plots have been used widely in hydrology as a graphical aid to assess the goodness of fit of alternative distributions. Recently, probability-plot correlation-coefficient (PPCC) tests were introduced to test the normal, two-parameter lognormal and Gumbel hypotheses. Those procedures are extended here to include both regional and at-site tests for the two-parameter Weibull and lognormal distributional hypotheses. In theory, PPCC-hypothesis tests can only be developed for two-parameter distributions that exhibit a fixed shape. Nevertheless, the PPCC is a useful goodness-of-fit statistic for comparing three-parameter distributions. The PPCC derived from fitting the two- and three-parameter lognormal, two- and three-parameter Weibull, and log-Pearson type III distributions to sequences of annual minimum seven-day low flows at 23 sites in Massachusetts are compared. How the PPCC can be used to discriminate among both competing distributional hypotheses for the distributions of fixed shape and competing parameter-estimation procedures for the distributions with variable shape is described. An approximate regional PPCC test is developed and used to show that there is almost no evidence to contradict the hypothesis that annual minimum seven-day low flows in Massachusetts are two-parameter lognormal.

Low-Flow Frequency Analysis Using Probability-Plot Correlation Coefficients

This study develops maximum likelihood estimators for the three parameter Weibull distribution based on various left, and right, censored data situations. For the case of single censoring from the left and progressive censoring from the right, the developed estimation procedure involves the simultaneous solution of two iterative equations compared to the arduous task of solving three simultaneous iterative equations as outlined by Harter and Moore [6]. For the two parameter Weibull [4], the above case results in an estimation procedure involving only one iterative equation. The asymptotic variance-covariance matrix of the estimators is given. Pivotal functions are provided whose distributions can be used for confidence interval estimation and for tests of hypotheses. An example is included which illustrates the estimation procedure involving left censored samples.

Maximum Likelihood Estimation for the Three Parameter Weibull Distribution Based on Censored Samples

Reliability analysis : Optimal inspection and maintenance schedules of failing systems

MIL-STD-781B gives sampling plans (sequential and fixed length) for reliability tests under the assumption of a constant failure rate. Using Monte Carlo techniques, the authors compare s-expected time to a decision and producer and consumer risks for some of these plans. It is shown that plans which assume an exponential distribution are not robust to departures from that assumption. A simple modification of these plans for use when life has a Weibull distribution with known shape parameter not equal to one, and an adaptive test procedure for use when life has a Weibull distribution with unknown shape parameter are proposed. The modified plans for a Weibull distribution with known shape parameter have the same designated producer and consumer risks, but different s-expected time to a decision than the corresponding exponential plans. Using Monte Carlo techniques, the authors determine s-expected time to a decision and producer and consumer risks for various forms of the adaptive procedure.

An Evaluation of Exponential and Weibull Test Plans

An iterative procedure is given for joint maximum likelihood estimation of the three parameters of the Weibull population. For this population, the likelihood function is forned and the maximum likelihood equations are obtained. Simultaneous solution of these equations yields joint maximum likelihood estimators for the three parameters. The proposed iterative procedure is a modified quasilinearization algorithm for solving nonlinear equations with bounded variables. Numerical results are given in which the parameters are estimated from real and from simulated life test data drawn from the Weibull population.

D. Royland Wingo

Papers

Maximum Likelihood Estimation of the Parameters of the Weibull Distribution by Modified Quasilinearization