Maximum spacing estimation

Abstract: The maximum likelihood method (ML method) works properly if each contri- bution to the likelihood function is bounded from above. This is the case for all discrete distributions but not for all mixtures of continuous distributions and then the ML method can break down. Since the ML method can be obtained by approximating the Kullback-Leibler information we raise the following question. Is it possible to obtain better methods than the ML method by approximating the Kullback-Leibler information in another way? Using spacings we can obtain an approximation of the Kullback-Leibler information such that each component in the approximation is bounded from above. In this paper we present the method, which we call the maximum spacing method, and show consistency of the maximum spacing estimate. Some examples are given, which show that the new method works also in situations where the ML method breaks down.

Abstract: The numerical technique of the maximum likelihood method to estimate the parameters of Gamma distribution is examined. A convenient table is obtained to facilitate the maximum likelihood estimation of the parameters and the estimates of the variance-covariance matrix. The bias of the estimates is investigated numerically. The empirical result indicates that the bias of both parameter estimates produced by the maximum likelihood method is positive.

Abstract: The three-parameter Weibull distribution is a commonly-used distribution for the study of reliability and breakage data. However, given a data set, it is difficult to estimate the parameters of the distribution and that, for many reasons: (1) the equations of the maximum likelihood estimators are not all available in closed form. These equations can be estimated using iterative methods. However, (2) they return biased estimators and the exact amount of bias is not known. (3) The Weibull distribution does not meet the regularity conditions so that in addition to being biased, the maximum likelihood estimators may also be highly variable from one sample to another (weak efficiency). The methods to estimate parameters of a distribution can be divided into three classes: a) the maximizing approaches, such as the maximum likelihood method, possibly followed by a bias-correction operation; b) the methods of moments; and c) a mixture of the previous two classes of methods. We found using Monte Carlo simulations that a mixed method was the most accurate to estimate the parameters of the Weibull distribution across many shapes and sample sizes, followed by the weighted maximum likelihood estimation method. If the shape parameter is known to be larger than 1, the maximum product of spacing method is the most accurate whereas in the opposite case, the mixed method is to be preferred. A test that can detect if the shape parameter is smaller than 1 is discussed and evaluated. Overall, the maximum likelihood estimation method was the worst, with errors of estimation almost twice as large as those of the best methods.