CONTRIBUTIONS TO ORDER STATISTICS by

Three different problems in order statistics are considered in this dissertation. The first problem deals with the recurrence relations between moments and other functions of order statistics. It is shown that recurrence relations valid for independent and identically distributed random variables continue to hold for exchangeable variables. In the second problem a method, based upon orthogonal polynomials, for obtaining bounds and approximations for the moments of order statistics is given. These bounds and approximations depend on the distribution function only through certain moments of order statistics in small samples. It is shown that for the Cauchy distribution bounds and approximations of all finite moments can be obtained. Finally, the problem of detecting a single outlier in a fixed effects linear regression model is considered in some detail. The various cases considered are: (i) known variance, (ii) external studentization and (iii) pooled studentization. In each case, oneand two-sided test statistics for detecting a single outlier are proposed. These statistics are maxima of suitably standardized or studentized weighted residuals. With the help of Bonferroni and other inequalities upper and lower limits for the true upper percentage points of the proposed statistics are developed and some tables are provided. Some measures of performance, appropriate for our purposes are also introduced and studied. Finally, a comparison between external and pooled studentization is made.