Statistica Sinica 13(2003), 311-325

ASYMPTOTIC PROPERTIES OF THE GENERALIZED

SEMI-PARAMETRIC MLE IN LINEAR REGRESSION

Qiqing Yu and George Y. C. Wong

SUNY, Binghamton and Strang Cancer Prevention Center

Abstract: Consider the semi-parametric linear regression model, $Y=\b^\prime \bfX+\epsilon$, with sample size $n$, where $\epsilon$ has an unknown cdf $F_o$. The semi-parametric MLE (SMLE) $\tb_n$ of $\b$ under this set-up, called the generalized SMLE or GSMLE, has neither been studied in the literature nor an algorithm for it. We begin with an algorithm for the GSMLE. It is then shown that if $F_o$ has a discontinuity point, P$\{\tb_n=\b$ if $n$ is large$\}=1$. Simulation suggests that under some discontinuous distributions, $\tb_n=\b$ even for $n= 50$. In contrast the least squares estimator (LSE), , satisfies P i.o.$\}=1$. We demonstrate via a real discontinuous data example that the GSMLE can be better than the LSE in applications. Properties of the GSMLE in the continuous case are also mentioned.

Key words and phrases: Algorithms, consistency, generalized likelihood, super efficiency.