Convex Transformations of Random Variables.

Abstract

MOMENT INEQUALITIES 2.1 Preliminaries 2.2 Convex transfo11ns CON1"ENTS 2.3 Antisymmetrical concave-convex transforms 3 ORDER STATISTICS 3.1 Notation 3.2 Large sample properties 4 TWO WEAK-ORDER RELATIONS FOR DISTRIBUTION FUNCTIONS 4.1 A weak ordering and an equivalence for the class 4.2 Properties of c-ordered pairs of distributions 4.3 Examples of c-ordering 4.3.1 c-Comparison with the rectangular distribution .,i,c. 1 * x-1 4.3.2c-Comparison with F (x) and F (x) -----X X 4.3.3c-Comparison with the exponential distribution 4.3.4The maximal c-chain of gamma distributions Page 1 6 9 17 23 25 47 50 54 55 55 4.4 A weak ordering for a class of symmetric distributions 64 4.5 Properties of s-ordered pairs of distributions 67 4.6 Examples of s-ordering 4.6.1 s-Comparison with the rectangular distribution 70 4.6.2The s-chain of symmetric beta distributions 71 4.6.3s-Comparison of normal and logistic distributions 4.7 Generalization to other distributions 5 s-COMPARISON WITH SYMMETRIC INVERSE BETA DISTRIBUTIONS 5.1 Symmetric inverse beta distributions 5.2 Inequalities for gamma and beta functions 5.3 Small sample inequalities 6 5.3.1 CAUCHY's distribution 5.3.2Symmetric beta distributions APPLICATIONS TO HYPOTHESIS TESTING AND ESTIMATION 6.1 Comparison of normal scores and WILCOXON tests 6.2 STUDENT's test under non-standard conditions 6.3 Efficiency of median and mean REFERENCES 99 100 113 Chapter 1 * * *F, but that we wish to obtain information about F (Ex.) where x -1.:n is another random variable with distribution function F* with inverse