On the Asymptotic Distribution of Likelihood Ratio Test When Both Nuisance Parameters and Parameters of Interest Lie on the Boundary
Leonid Kopylev¹, Bimal Sinha²
¹Office of Research and Development, US Environmental Protection Agency, Washington, DC, United States; ²Mathematics and Statistics, University of Maryland at Baltimore County, Baltimore, MD, United States

This talk presents new results on statistical inference dealing with the asymptotic theory of likelihood ratio tests when both parameters of interest and nuisance lie on boundary of the parameter space. Following seminal paper by Self and Liang (1987), we derive a closed form solution for the case when one parameter of interest and one nuisance parameter lie on the boundary. The asymptotic distribution in this case is not always a mixture of several chi-square distributions (Kopylev and Sinha, 2010).

These results can be used in many applications, e.g. one-sided confidence intervals, often used in environmental risk assessment, and testing for random effects in genetics. The problem of boundary conditions received a lot of attenetion in applied literature, particularly in genetics. Contrary to the claim of some authors in genetics that use of chi-square distribution with degrees of freedom as in case of interior parameters will be too conservative when some parameters are on the boundary, we show that when nuisance parameters are on the boundary, that approach may often be anti-conservative.

Kopylev L. and Sinha B. (2010), “On the Asymptotic Distribution of Likelihood Ratio Test when Parameters Lie on the Boundary,” accepted, Sankhya: The Indian Journal of Statistics, Series B.

Self S.G. and Liang K-Y. (1987). “Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions,” Journal of American Statistical Association, 82, 605-610.

Disclaimer: The views expressed in this presentation are those of the authors and do not necessarily reflect the views or policies of the U.S. EPA.

Keywords: One-sided tests; Nuisance parameters on the boundary; Parameters of interest on the boundary

Biography: Leonid Kopylev is a Mathematical Statistician with the Office of Research and Development in the US Environmental Protection Agency. He received a Ph.D in Mathematical Statistics in 1997 from the University of Maryalnd at College Park and since then worked for NIH and EPA. His research interests include survival analysis and better statistical methods for quantitative risk assessments. His presentation is devoted to the topic that every statistician encounters in practice at least once, but almost never encounters in graduate school – what happens when regularity conditions stated in every textbook do not hold, i.e. parameters presumed in the interior of the parameters space are on the boundary.