Sieve likelihood ratio statistics and Wilks phenomenon

Maximum likelihood ratio theory contributes tremendous success to parametric inferences, due to the fundamental theory of Wilks (1938). Yet, there is no general applicable approach for nonparametric inferences based on function estimation. Maximum likelihood ratio test statistics in general may not exist in nonparametric function estimation setting. Even if they exist, they are hard to nd and can not be optimal as shown in this paper. In this paper, we introduce the sieve likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks' phenomenon is unveiled. We demonstrate that the sieve likelihood statistics are asymptotically distribution free and follow 2-distributions under null hypotheses for a number of useful hypotheses and a variety of useful models including Gaussian white noise models, nonparametric regression models, varying coefficient models and generalized varying coefficient models. We further demonstrate that sieve likelihood ratio statistics are asymptotically optimal in the sense that they achieve optimal rates of convergence given by Ingster (1993). They can even be adaptively optimal in the sense of Spokoiny (1996) by using a simple choice of adaptive smoothing parameter. Our work indicates that the sieve likelihood ratio statistics are indeed general and powerful for nonparametric inferences based on function estimation

A loss function approach to model specification testing and its relative efficiency

The generalized likelihood ratio (GLR) test proposed by Fan, Zhang and Zhang [Ann. Statist. 29 (2001) 153-193] and Fan and Yao [Nonlinear Time Series: Nonparametric and Parametric Methods (2003) Springer] is a generally applicable nonparametric inference procedure. In this paper, we show that although it inherits many advantages of the parametric maximum likelihood ratio (LR) test, the GLR test does not have the optimal power property. We propose a generally applicable test based on loss functions, which measure discrepancies between the null and nonparametric alternative models and are more relevant to decision-making under uncertainty. The new test is asymptotically more powerful than the GLR test in terms of Pitman's efficiency criterion. This efficiency gain holds no matter what smoothing parameter and kernel function are used and even when the true likelihood function is available for the GLR test.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1099 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Specification testing

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A selective overview of nonparametric methods in financial econometrics

This paper gives a brief overview on the nonparametric techniques that are useful for financial econometric problems. The problems include estimation and inferences of instantaneous returns and volatility functions of time-homogeneous and time-dependent diffusion processes, and estimation of transition densities and state price densities. We first briefly describe the problems and then outline main techniques and main results. Some useful probabilistic aspects of diffusion processes are also briefly summarized to facilitate our presentation and applications.Comment: 32 pages include 7 figure

Pranab Kumar Sen: Life and works

In this article, we describe briefly the highlights and various accomplishments in the personal as well as the academic life of Professor Pranab Kumar Sen.Comment: Published in at http://dx.doi.org/10.1214/193940307000000013 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric checks for count data models: an application to demand for health care in Spain

This paper presents model specification checking procedures for count data regression models which are consistent in the direction of nonparametric alternatives. The discussion is motivated in the context of a model of demand for health care in Spain. The parameters of the regression model are estimated by maximum likelihood based on Poisson and Negative Binomial specifications as well as by ordinary least squares and semiparametric generalized least squares. However, our interest is not only centered on the estimation ofthe regression parameters, but also the conditional probabilities of counts. Therefore, the specification of the conditional distribution function of counts is the main focus of attention. A useful preliminary diagnosis tool consists of comparing the conditional probabilities estimates by nonparametric regression and by maximum likelihood methods based on alternative models. We present formal specification procedures based on new developed testing methods for regression model checking. The test statistics are based on marked empirical processes which are not distribution free, but their critical values are well approximated by bootstrap. Such tests are valid for testing the functional form of the conditional mean and conditional probabilities resulting from alternative distributional specifications. In our health care demand model, the linear exponential regression model with a Negative Binomial seems to be appropiate for this data set