Estimation and Testing Methods for Monotone Transformation Models
This thesis deals with a general class of transformation models that contains many important semiparametric regression models as special cases. It develops a self-induced smoothing method for estimating the regression coefficients of these models, resulting in simultaneous point and variance estimations. The self-induced smoothing does not require bandwidth selection, yet provides the right amount of smoothness so that the estimator is asymptotically normal with mean zero (unbiased) and variance-covariance matrix consistently estimated by the usual sandwich-type estimator. An iterative algorithm is given for the variance estimation and shown to numerically converge to a consistent limiting variance estimator. The self-induced smoothing method is also applied to selecting the non-zero regression coefficients for the monotone transformation models. The resulting regularized estimator is shown to be root-n-consistent and achieve desirable sparsity and asymptotic normality under certain regularity conditions. The smoothing technique is used to estimate the monotone transformation function as well. The smoothed rank-based estimate of the transformation function is uniformly consistent and converges weakly to a Gaussian process which is the same as the limiting process for that without smoothing. An explicit covariance function estimate is obtained by using the smoothing technique, and shown to be consistent. The estimation of the transformation function reduces the multiple hypotheses testing problems for the monotone transformation models to those for linear models. A new hypotheses testing procedure is proposed in this thesis for linear models and shown to be more powerful than some widely-used testing methods when there is a strong collinearity in data. It is proved that the new testing procedure controls the family-wise error rate.
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