Mean square invariant forecasters for the weibull distribution

Abstract

Many techniques of forecasting are based upon extrapolation from time series. While such techniques have useful applications, they entail strong assumptions which are not expli$t& e%unciated_Furth_ermore, the time series approach is not based on an indigenous forecast principle. The first attack from the present point of viewwasinitiated by S. -. S. Wilks. _ _ _ Of particular interest over a wide range of operational situations in reliability, for example, is the behavior of the extremes of the Weibull and Gumbel distributions. Here we formulate forecasters for the minima of various forms of these distributions. The forecasters are determined for minimization in mean square of the distance. From n original observations the forecaster provides the minimum of the next m observations when the original distribution is maintained.For each of the forecasters developed, tables of efficiency have been calculated and included in the appendix. An explicit example has been included for one of the forecasters. Its performance has been demonstrated by the use of Monte Car10 technique. The results indicate that the forecaster can be used in practice with satisfactory results. ___- INTRODUCTIONA basic scientific problem which has many ramifications is that of estimating future process outcomes from an observed set of outcomes. This is a problem of long standing, that is, in essence, the fundamental problem of forecasting. This problem can be, and has been, approached in many ways. Yet it is not often recognized and expressed as a problem of forecasting. An early (the first) fundamental contribution to forecasting future process outcomes from a given set of, say, n process outcomes was made by S. S. Wilks [22, 231 in his pioneering papers on "Non-parametric tolerance intervals" in which he introduced a new statistical interval, referred to in the literature either as a tolerance or a prediction interval. It turns out that Wilks' results have great practical usefilness as well as simplicity and elegance which are hard to match.