Applying medical survival data to estimate the three-parameter Weibull distribution by the method of probability-weighted moments

Abstract

The method of probability-weighted moments is used to derive estimators of parameters and quantiles of the three-parameter Weibull distribution. The properties of these estimators are studied. The results obtained are compared with those obtained by using the method of maximum likelihood. The Weibull probability distribution has numerous applications in various areas: for example, breaking strength, life expectancy, survival analysis and animal bioassay. Because of its useful applications, its parameters need to be evaluated precisely, accurately and efficiently. There is a rich literature available on its maximum likelihood estimation method. However, there is no explicit solution for the estimates of the parameters or the best linear unbiased estimates. Further, the Weibull parameters cannot be expressed explicitly as a function of the conventional moments and iterative computational methods are needed. The maximum likelihood methodology is based on large-sample theory and the method might not work well when samples are small or moderate in size. Others have proposed a class of moments called probability-weighted moments. This class seems to be interesting as a method for estimating parameters and quantiles of distributions which can be written in inverse form. Such distributions include the Gumbel, Weibull, logistic, Tukey's symmetric lambda, Thomas Wakeby, and Mielke's kappa. It has been illustrated that rather simple expressions for the parameters can be written in inverse form in terms of probability-weighted moments (PWMs) for most of these distributions. In this paper we define the PWM estimators of the parameters for the three-parameter Weibull distribution. We investigate the properties of these estimators in a medical application setting. We also examine the added influence that censored data may have on the estimates.