Consider a semiparametric regression model Y = f(θ, X, ϵ), where f is a known function, θ is an unknown vector, ϵ consists of a random error and possibly of some unobserved variables, and the distribution F(·) of (ϵ, X) is unspecified. This article introduces, in a general setting, new methodology for estimating θ and F(·). The proposed method constructs a profile likelihood defined on random-level sets (a random sieve). The proposed method is related to empirical likelihood but is more generally applicable. Four examples are discussed, including a quadratic model, high-dimensional semiparametric regression, a nonparametric random-effects model, and linear regression with right-censored data. Simulation results and asymptotic analysis support the utility and effectiveness of the proposed method.
Bibliographical noteFunding Information:
Xiaotong Shen is Associate Professor, Department of Statistics, Ohio State University, Columbus, OH 43210. His research was supported in part by National Security Agency grant MDA904-98-1-0030. Jian Shi is Associate Research Scientist, Institute of Systems Science, Academia Sinica, Beijing, 10080, China. His research was supported in part by the Institute of Mathematical Sciences of the Chinese University of Hong Kong. Wing Hung Wong is Professor, Department of Statistics, University of California, Los Angeles, CA 90095. His research was supported in part by the Institute of Mathematical Sciences of the Chinese University of Hong Kong and National Science Foundation grant DMS97-03918. The authors would like to thank the associate editor and anonymous referees for very helpful comments and suggestions.
- Empirical likelihood
- General regression model
- Profile likelihood
- Random sieve likelihood