Quantifications of random variables

• Main Author: Yan, Xiaosong, 1963-
• Other Authors: Memorial University of Newfoundland. Dept. of Mathematics and Statistics
• Format: Thesis
• Language: English
• Published: 1996
• Subjects: Functions of real variables; Least squares; Order statistics; Random variables; Ewing; Newfoundland studies; University of Newfoundland
• Online Access: http://collections.mun.ca/cdm/ref/collection/theses3/id/204755

Thesis (Ph.D.)--Memorial University of Newfoundland, 1996. Mathematics and Statistics Bibliography: leaves 144-152 Order restricted inference is an important field in statistical science. The utilization of ordering informations can increase the efficiency of statistical inference procedures in several senses, see Ayer, Brunk, Ewing. Reid. and Silverman (1955), Robertson and Wright (1974), Barlow and Ubhaya (1971), Lee (1981) and Kelly (1989). -- In this thesis we review some basic theories about the least squares regressions, particularly the isotonic regressions. We give a simplified proof of an iterative procedure proposed by Dykstra (1983) for least squares problems. -- We investigate the properties of the orderings of real-valued functions from several aspects. Some definitions are extended and their properties are generalized. We also show that the concept of closed convex cones and their duals is important in estimating procedures as well as in testing procedures. We demonstrate that some seemingly different problems have actually the same likelihood ratio test statistics and critical regions. -- We observe that the orders of real-valued functions and the orders of random variables are closely related and statistical inference regarding these two orders behave similarly. A class of bivariate quantifications are defined based on these two orders. This bivariate notion has direct interpretation and appealing properties. More important, it characterizes a degree of positive dependence among random variables and therefore makes it possible to study the positive dependence of random variables by using the theories of the orders of real-valued functions and the orders of random variables. -- We consider several estimation problems under order restrictions. We propose an algorithm that finds the nonparametric maximum likelihood estimates of a stochastically bounded survival function in finite steps, usually two or three steps. Simulation study shows that in general, utilizing the prior knowledge of a lower bound and an upper bound may reduce the point-wise MSE's and the amount of reduction in MSE's could be substantial for small and moderate sample sizes for a pair of sharp bounds. We obtain the estimates of a multinomial parameter under various order constraints for a general multinomial estimation procedure defined by Cressie and Read (1984). -- We also consider the problem of simulating tail probabilities with a known stochastic bound. The proposed procedure may increase the efficiency of simulation significantly.