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INEQUALITIES FOR FUNCTIONS OF ORDER STATISTICS UNDER AN ADDITIVE AND A MULTIPLICATIVE MODEL
Abstract
This dissertation contains inequalities concerning random variables of the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where: (a) X(,(1)) (LESSTHEQ) ... (LESSTHEQ) X(,(n)) are the order statistics of a random vector X = (X(,1),...,X(,n)) under various additive or multiplicative models, (b) (phi)(,i), i = 1,...,n are real-valued functions satisfying certain monotonicity, and convexity or concavity conditions, and (c) c(,i), i = 1,...,n is an increasing or decreasing sequence of constants. Inequalities are first obtained under the additive (or location) model X = Y + (delta) and then extended to the model X = Y + Z, where (delta) is a real vector and Z is a random vector. Similarly, inequalities are obtained under the multiplicative (or scale) models X = (delta)*Y and X = Z*Y, where "*" indicates componentwise multiplication. The inequalities obtained in this dissertation are in terms of the ordered components of Y, (delta), and Z. Majorization is an important tool in the derivation of these inequalities. One use of these inequalities is to extend the applicability of the large number of known results for random vectors with i.i.d. components to random vectors with dependent and/or heterogeneously distributed components. Several applications are included by way of illustration.
Subject Area
Mathematics
Recommended Citation
SMITH, NORMAN LOWELL, "INEQUALITIES FOR FUNCTIONS OF ORDER STATISTICS UNDER AN ADDITIVE AND A MULTIPLICATIVE MODEL" (1980). ETD collection for University of Nebraska-Lincoln. AAI8021358.
https://digitalcommons.unl.edu/dissertations/AAI8021358