DMS 96-26658 Shao The methods of maximum likelihood estimation and likelihood ratio test have constituted the cornerstone of statistical inference. In numerous cases, the maximum likelihood method generates estimates with optimal asymptotic properties and the likelihood ratio test is known to be powerful. However, there are many important practical situations where these methods break down. This research focuses on investigating non-regular likelihood problems including estimation problems where likelihood functions may be unbounded (e.g., efficient estimation of a unimodal distribution) as well as hypothesis testing problems where the likelihood ratio test statistics do not have chi-square type limiting distributions (e.g., testing homogeneity in Tukey models). This research involves generalizing the maximum product of spacings method, which is a natural way to rectify the problems caused by the unboundedness of likelihood functions and to preserve the essential asymptotic optimalities usually possessed by the maximum likelihood method. In addition, the research involves adapting and extending modern empirical process techniques to characterize the asymptotic behavior of the rescaled likelihood ratio statistic. Scientists and engineers often build models based upon various prior assumptions about the underlying conditions. Very often, these assumptions fail to hold in real applications. If the model is too sensitive to small departures from those prior assumptions, the results from such models will not be reliable, and may be misleading with far-reaching adverse consequences. It is, therefore, very important to develop efficient methods to check the validity of model assumptions, and to develop techniques that are not constrained by unrealistic assumptions. This research concerns efficient procedures for testing model validity and introduces "robust" statistical models which remain effective even when prior assumptions are not exactly true. Such proced ures have broad applications ranging from biotechnology to high-speed computing system. In particular, advancement in this research area would be crucial in fostering modern manufacturing, for it relates to developing optimal procedures for assessing reliability and quality control. In this respect, the research plays an integral role in the promotion of economic growth by improving manufacturing processes, thereby ensuring U.S. competitiveness in world markets.

Agency
National Science Foundation (NSF)
Institute
Division of Mathematical Sciences (DMS)
Type
Standard Grant (Standard)
Application #
9626658
Program Officer
Joseph M. Rosenblatt
Project Start
Project End
Budget Start
1996-07-01
Budget End
1999-06-30
Support Year
Fiscal Year
1996
Total Cost
$80,681
Indirect Cost
Name
Columbia University
Department
Type
DUNS #
City
New York
State
NY
Country
United States
Zip Code
10027