A distribution-free least squares estimator for censored linear regression models
A censored regression model in which \(y=\max (0,x\beta +u)\) is observed, is defined. Here, x is a random vector of explanatory variables, independent of the random disturbance term u. u has an unknown distribution function F. The vector \(\beta\) contains the parameters to be estimated. The author describes a method for estimating simultaneously \(\beta\) and F by minimizing the sum of squares of the differences between the observed values of y and the corresponding expected values of y according to the estimated parameter vector and distribution function. Consistency of the estimators of \(\beta\) and F is shown under a number of regularity conditions. Unfortunately, the conditions referred to contain some unconventional and, at least theoretically, very restrictive ingredients: the parameter space is assumed to be finite and the distribution functions for x and u are assumed to have bounded support. In order to investigate the estimators' finite-sample properties a small Monte Carlo study is conducted. In the simulations x is a scalar and x and u both follow a beta distribution. Four different cases with symmetric and skewed distributions on x and u are considered. Two different sample sizes (50 and 100) are used and each simulation consists of 100 replications. The results of the study are, as the author remarks, not conclusive. One important reason for this is that the simulation results are only compared with theoretical values from various quantile estimators. Although the proposed estimator for \(\beta\) seems to show a smaller standard deviation than that for the best quantile estimator, it should be remembered that the results for the proposed eStein) is presented, defined by: \({\hat \Sigma}=n^{-1}S\) unless Roy's maximum root test accepts the hypothesis that all \(\xi_ i\) are 0 in which case \({\hat \Sigma}=(n+k)^{-1}(S+XX')\). A similar testimator is found under \(L_ 2.\) The results are used for simultaneous estimation of \(\mu\) nd \(\Sigma\) in a sample from a p-variate N(\(\mu\),\(\Sigma)\) population. Estimation of \(| \Sigma |\) is handled in a similar way. Monte Carlo results on the improvement of the estimators are presented. Some results on sequential estimation are also given.
distribution-free least squares estimator
scientific article; zbMATH DE number 4032811