Often researchers find parametric models restrictive and sensitive to deviations from the parametric specifications; semi-nonparametric models are more flexible and robust, but lead to other complications such as introducing infinite dimensional parameter spaces that may not be compact. The method of sieves provides one way to tackle such complexities by optimizing an empirical criterion function over a sequence of approximating parameter spaces, called sieves, which are significantly less complex than the original parameter space. With different choices of criteria and sieves, the method of sieves is very flexible in estimating complicated econometric models. For example, it can simultaneously estimate the parametric and nonparametric components in semi-nonparametric models with or without constraints. It can easily incorporate prior information, often derived from economic theory, such as monotonicity, convexity, additivity, multiplicity, exclusion and non-negativity. This chapter describes estimation of semi-nonparametric econometric models via the method of sieves. We present some general results on the large sample properties of the sieve estimates, including consistency of the sieve extremum estimates, convergence rates of the sieve M-estimates, pointwise normality of series estimates of regression functions, root-n asymptotic normality and efficiency of sieve estimates of smooth functionals of infinite dimensional parameters. Examples are used to illustrate the general results.

. The nonparametric and the nuisance parameter approaches to consistently testing statistical models are both attempts to estimate topological measures of distance between a parametric and a nonparametric fit, and neither dominates in experiments. This topological unification allows us to greatly extend the nuisance parameter approach. How and why the nuisance parameter approach works and how it can be extended bears closely on recent developments in artificial neural networks. Statistical content is provided by viewing specification tests with nuisance parameters as tests of hypotheses about Banach-valued random elements and applying the Banach Central Limit Theorem and Law of Iterated Logarithm, leading to simple procedures that can be used as a guide to when computationally more elaborate procedures may be warranted. 1. Introduction In testing whether or not a parametric statistical model is correctly specified, there are a number of apparently distinct approaches one might take. T...

We develop a general theory which provides a unified treatment for the asymptotic normality and efficiency of the maximum likelihood estimates (MLE’s) in parametric, semiparametric and nonparametric models. We find that the asymptotic behavior of substitution estimates for estimating smooth functionals are essentially governed by two indices: the degree of smoothness of the functional and the local size of the underlying parameter space. We show that when the local size of the parameter space is not very large, the substitution standard (nonsieve), substitution sieve and substitution penalized MLE’s are asymptotically efficient in the Fisher sense, under certain stochastic equicontinuity conditions of the loglikelihood. Moreover, when the convergence rate of the estimate is slow, the degree of smoothness of the functional needs to compensate for the slowness of the rate in order to achieve efficiency. When the size of the parameter space is very large, the standard and penalized maximum likelihood procedures may be inefficient, whereas the method of sieves may be able to overcome this difficulty. This phenomenon is particularly manifested when the functional of interest is very smooth, especially in the semiparametric case.

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We consider likelihood based inference in a class of logistic models for case-control studies with a partially observed covariate. The likelihood is a combination of a nonparametric mixture, a parametric likelihood and an empirical likelihood. We prove the asymptotic normality of the maximum likelihood estimator for the regression slope, the asymptotic chisquared distribution of the likelihood ratio statistic, and the consistency of the observed information, in both the prospective and the retrospective model.

this paper we study the weak convergence of the sequence of stochastic processes fZn (t) : t 2 Tg. As a particular case, we consider sums of i.i.d. stochastic processes. Let fX j g

This paper considers the problem of consistent model specification tests using series estimation methods. The null models we consider in this paper all contain some nonparametric components. A leading case we consider is to test for an additive partially linear model. The null distribution of the test statistic is derived using a central limit theorem for Hilbert valued random arrays. The test statistic is shown to be able to detect local alternatives that approach the null models at the order of O p (n -1/2 ). We suggest to use the wild bootstrap method to approximate the critical values of the test. A small Monte Carlo simulation is reported to examine the finite sample performance of the proposed test. We also show

We consider multivariate density estimation with identically distributed observations. We study a density estimator which is a convex combination of functions in a dictionary and the convex combination is chosen by minimizing the L2 empirical risk in a stagewise manner. We derive the convergence rates of the estimator when the estimated density belongs to the L2 closure of the convex hull of a class of functions which satisfies entropy conditions. The L2 closure of a convex hull is a large non-parametric class but under suitable entropy conditions the convergence rates of the estimator do not depend on the dimension, and density estimation is feasible also in high dimensional cases. The variance of the estimator does not increase when the number of components of the estimator increases. Instead, we control the bias-variance trade-off by the choice of the dictionary from which the components are chosen.

We prove a uniform CLT for Markov chains for functions dominated by a function in a L -space. Using empirical process CLT's for stationary sequences satisfying a variety of mixing conditions one can obtain similar results. However, our conditions are less restrictive than those required for a mixing process application to these problems, and an example is given to these differences. 1

For n particles di using throughout R (or Rd), let n�t(A), A 2B, t 0, be the random measure that counts the number of particles in A at time t. Itisshown that for some basic models (Brownian particles with or without branching and di usion with a simple interaction) the processes f ( n�t ( ) ; E n�t ()) = p n: t 2 [0�M] � 2 CL (R)g, n 2 N, converge in law uniformly in (t �). Previous results consider only convergence in law uniform in t but not in. The methods used are from empirical process theory. 1