PARTITIONING PROBLEMS IN EXTREME VALUE STATISTICS AND APPLICATION

Abstract

As n tends to infinity, zn, the largest or smallest value of a sample sized n has, asymptotically, one of the three types of extreme value distribution, namely, the Type I, Type II and Type III extreme value distributions. In real life, any correlation between the observations is difficult to detect and determine. When the sample size is large, we could assume that observations are independent. However, collecting data of large sample sizes is very time consuming. All these problems were solved in 1967 when Maritz and Munro presented a Generalized Extreme Value distribution (abbreviated GEV) that could serve extreme values for small as well as large samples. Its distribution function is given by G(x) - exp [ [ 1 - x – u /a k ] 1/k ] where u ? is the location parameter, a> 0 is the scale parameter or a measure of <lispers ion of X and k ? is the shape parameter. When k tends to zero, this distribution function converges to Type I. It is of Type II if k is negative and of Type III if k is positive. A computer program, which is written in Fortran, is implemented for the whole process of GEV-statistical analysis of a set of observations, using the moment estimates of the parameters as initial values for the evaluation of their maximum likelihood estimates. The way to collect extreme values is to gather m x n observations or m sets of observations, each of size n, and extract the maxima (or minima) from each sample of n observations, thereby giving us a set of m extreme values. Here, n is called the block size. Our task is to determine the optimum block size that minimises the mean-squared-error when predicting the maximum of T (- M x n) future observations by fitting an extreme value distribution to a ·collected set of m (- t/n ) past maximas where t is the total number of independent, identically distributed observations collected. The variation in the estimated location and scale parameters or the normalizing constants, (an, bn), will be used as a basis for determining the optimum block size. We investigate cases when underlying distributions are the Exponential (?) and Gamma (r, ?) distributions. We found that the optimum block size is independent of ? for Exponential distributions and varies with r and ? for Gamma distributions.