Semiparametric Estimation Approaches for Variant Dimension Reduction Models

To characterize the conditional distribution in a more general form, two variant semiparmaetric regressions are considered.
In the first scenario, we present more flexible semiparametric linear-index regression models for distribution regressions.
Such a model formulation captures varying effects of covariates over the support of a response variable in distribution, offers an alternative perspective on dimension reduction, and covers a lot of widely used parametric and semiparameteric regression models.
A more feasible pseudo likelihood approach is reasonably proposed for the mixed case with both varying and invariant coefficients.
In addition, the estimator are effectively computed through a simple and easily implemented algorithm.
Theoretically, its uniform consistency and asymptotic Gaussian process are established by justifying some properties on Banach spaces.
Under the monotonicity of distribution in linear-index, an alternative estimation procedure is further developed based on a varying accuracy measure.
In this research direction, other important achievements include showing the convergence of an iterative computation procedure and establishing the large sample properties of the resulting estimator from the asymptotic recursion relation for the estimators.
As a consequence of our general theoretical frameworks, it is convenient to construct confidence bands for the parameters of interest and tests for the hypotheses of various qualitative structures in distribution.
Generally, the developed estimation and inference procedures perform quite well in the conducted simulations and are demonstrated to be useful in reanalyzing data from the studies of house-price in Boston and World Values Survey.

In the second scenario of this thesis, we consider the sufficient dimension reduction model, which is a well-known exploratory model for describing the conditional distribution of interest.
With the associated counting process of a response, a simple and easily implemented semiparametric approach is developed to estimate the central subspace and underlying regression function.
Different from the existing sufficient dimension reduction approaches, two essential elements, basis and structural dimension, of the central subspace and the optimal bandwidth of a kernel distribution estimator can be simultaneously estimated through a cross-validation version of the pseudo sum of integrated squares.
One attractive merit of this estimation technique is that it allows a response to be discrete and some of covariates to be discrete or categorical.
Further, the uniform consistency of the cross-validation optimization function and the consistency of the resulting estimators are derived under very mild conditions.
Meanwhile, we establish the asymptomatic normality of the central subspace estimator with an estimated rather than exact structural dimension.
It is also demonstrated by our extensive numerical experiments that the developed approach dramatically outperforms the semiparametric competitors.
In addition, the applicability and practicality of the proposal are highlighted through data from previous studies.
Overall speaking, our methodology is computationally efficient in estimating the central subspace and the conditional distribution, highly flexible in adapting diverse types of a response and covariates, and practically feasible to obtain an asymptotically optimal bandwidth estimator.