Estimation of the Weibull distribution with applications to tornado climatology
Some general properties of the Weibull distribution are discussed. The mathematical development of the distribution is linked to the family of extreme value distributions, and the origins in science are found to be related to survival analysis. Some generalizations of the distribution are noted, and a limited discussion of its numerous applications undertaken. One such application is the Weibull model of tornado intensity developed by Dotzek, Grieser, and Brooks (2003). In an attempt to improve this model, several methods for estimating the parameters of the Weibull distribution are discussed. Maximum likelihood estimation is found to be the best method of estimation for the two-parameter Weibull distribution with respect to the asymptotic estimator properties discussed. An existing algorithm to locate the maximum likelihood estimator for the three-parameter Weibull distribution is described, and the complexities of the three-parameter case investigated. It is known that the maximum likelihood estimates for the Weibull distribution display bias for small sample sizes. An equation is analytically derived to estimate this small sample bias in the two-parameter case, and numerical unbiasing procedures discussed. Simulated data are analyzed using the methods developed, and the asymptotic properties of the estimates discussed for the two-parameter case. The estimation procedures are then applied to actual tornado intensity data from the April 25th - 28th, 2011 tornado outbreak as well as the historic records for both Alabama and the United States as a whole. In all cases, the Weibull model is found to be appropriate as judged by the Chi-squared test at 5 percent significance.