Comparisons of methods of estimation for a pareto distribution of the first kind

• Published in: Communications in statistics. Theory and methods 2000-01, Vol.29 (4), p.859-878
• Main Authors: Hossain, Anwar M., Zimmer, William J.
• Format: Article
• Language: English
• Subjects: Decision theory; Distribution theory; Exact sciences and technology; least squares esti-malar; Mathematics; Maximum likelihood estimator; mean squared error; modified maximum likelihood estimator; Probability and statistics; Sciences and techniques of general use; Statistics
• ISSN: 0361-0926; 1532-415X
• DOI: 10.1080/03610920008832520

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters. Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.