APPLICATIONS OF MATHEMATICS, Vol. 57, No. 5, pp. 531-542, 2012

A study of the number of solutions of the system of the log-likelihood equations for the 3-parameter Weibull distribution

George Tzavelas

G. Tzavelas, Department of Statistics and Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou Street, 185 34 Piraeus, Greece, e-mail: tzafor@unipi.gr

Abstract: The maximum likelihood estimators of the parameters for the 3-parameter Weibull distribution do not always exist. Furthermore, computationally it is difficult to find all the solutions. Thus, the case of missing some solutions and among them the maximum likelihood estimators cannot be excluded. In this paper we provide a simple rule with help of which we are able to know if the system of the log-likelihood equations has even or odd number of solutions. It is a useful tool for the detection of all the solutions of the system.

Keywords: Weibull distribution, Hessian matrix, maximum likelihood estimator, stationary value

Classification (MSC 2010): 62F10, 62F99


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