Extreme Order Statistics in Large Samples from Exponential Type Distributions and their Application to Fire Loss

Abstract

In some practical situations data may be available only for large values of the variable under consideration. Hence population characteristics have to be inferred from extreme (large) order statistics, the largest being a case of particular interest among these statistics. This problem is studied in this paper for large samples from exponential type parent distributions. The joint distributions of any two extreme order statistics and all extreme order statistics up to a given rank are derived using Gumbel’s heuristic methods. The theory is applied to large losses due to fires, assuming from physical considerations that the logarithm of loss follows a probability distribution of the exponential type. Possible uses of extreme observations in fire protection and fire insurance problems are suggested.