Relative Error Estimation and Efficient Estimation of Censored Linear Regression Model.

摘要

Multiplicative regression model or accelerated failure time model, which becomes linear regression model after logarithmic transformation, is useful in analyzing data with positive responses, such as stock prices or life times, that are particularly common in economic/financial or biomedical studies. Least squares or least absolute deviation are among the most widely used criterions in statistical estimation for linear regression model. However, in many practical applications, especially in treating, for example, stock price data, the size of relative error, rather than that of error itself, is the central concern of the practitioners. The first part of this thesis offers an alternative to the traditional estimation methods based on relative errors for multiplicative regression models. We prove consistency and asymptotic normality and provide an inference approach via random weighting. We also specify the error distribution, with which the proposed estimation based on relative errors is efficient. Supportive evidence is shown in simulation studies. Application is illustrated in an analysis of stock returns in Hong Kong Stock Exchange.;In linear regression or accelerated failure time model, the method of efficient estimation, with or without censoring, has long been overlooked. The second part of this thesis proposes a one-step efficient estimation method based on counting process martingale, which has several advantages: it avoids the multiple root problem, the initial estimator is easily available, and it is easy to implement numerically with a built-in inference procedure. The requirement on bandwidth is rather loose. A simple and effective data-driven bandwidth selection method is provided. The resulting estimator is proved to be semiparametric efficient with the same asymptotic variance as the efficient estimator when the error distribution is assumed to be known up to a location shift. The asymptotic properties of the proposed method are justified and the asymptotic variance matrix of the regression coefficients is provided in a closed form. Numerical studies with supportive evidence are presented. Applications are illustrated with the well-known PBC data and the Colorado Plateau uranium miners data.