Asymptotic properties of the generalized semi-parametric MLE in linear regression

Consider the semi-parametric linear regression model, Y = beta'X + e, with sample size n, where e has an unknown cdf F-o. The semi-parametric MLE (SMLE), of 3 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{beta(n) = beta if n is large} = 1. Simulation suggests that under some discontinuous distributions, beta(n) = beta even for n = 50. In contrast the least squares estimator (LSE), beta(n), satisfies P{beta(n) not equal beta 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 axe also mentioned.