| In our real life, the problem with 0-1 discrete response has become an increasingly important theme, there are many applications in every respect of life. In my thesis, I introduced the binary model E(Y|X)= g(X) in different three cases:parametric model、 semiparametric model and nonparametric model, where Y is 0-1 response, X is p × 1 covariates and g is some function.Under parametric and semiparametric model, we assume that g is the function of X’s linear combination, the model corresponds to the special form of generalized linear model and single index model respectively. We consider the construction of a binary classification rule by minimizing the risk based on a convex loss as a surrogate to the 0-1 loss. Compared with the approach of directly estimating the conditional probability of the binary class label given a vector of covariates, our proposed convex surrogate minimization approach is computationally simpler and more efficient because of the convexity. We begin with a rigorous discussion of what type of convex surrogate is valid. When the conditional probability model for class label is parametric, we show that our proposed approach is either equivalent to the traditional maximum likelihood method or a substitute for computational saving. When the conditional probability model is semiparametric, we show how to apply convex surrogate minimization in conjuncture with kernel weighting, which results in an asymptotically valid classification rule. Some convergence rates are established and empirical simulation results are presented.Under nonparametric model, we have considered three problems:jointly estimating marginal quantiles of a multivariate binary data, constructing confidence interval for a scalar parameter in binary model and calculating simultaneous confidence interval for a scalar parameter in multivariate binary model. In the first one, a sufficient condition for an estimator that converges in probability under a multivariate version of Robbins-Monro procedure is provided. We propose an efficient procedure which incorporates the corre-lation structure of the multivariate distribution to improve the estimation especially for cases involving extreme marginal quantiles. Estimation efficiency of the proposed method is demonstrated by simulation in comparison with a general multivariate Robbins-Monro procedure and an efficient Robbins-Monro procedure that estimates the marginal quan-tiles separately. In the second one, we propose a nonparametric method of constructing confidence interval for a scalar parameter from stochastic approximation through the effi-cient Robbins-Monro procedure proposed by Joseph (2004). Unlike the bootstrap method where the number of resampling is fixed in advance, the proposed procedure iteratively searches the endpoints in an optimal way such that the convergence is fast and the cover-age is obtained accurately. Simulation and real data application illustrate its superiority over the usual Robbins-Monro procedure and common bootstrap methods. In the last one, we combine the solutions of the first two problems. Based on multivariate efficient Robbins-Monro sequences and permutation test, we construct a sequence to estimate si-multaneous confidence intervals for scalar variable in multivariate binary data effectively. Also, the simulation and real data application are showed that our method has better results than the bootstrap method. |
