ROBUST KERNEL ESTIMATOR FOR DENSITIES OF UNKNOWN

Results on nonparametric kernel estimators of density differ according to the assumed degree of density smoothness; it is often assumed that the density function is at least twice differentiable. However, there are cases where non-smooth density functions may be of interest. We provide asymptotic results for kernel estimation of a continuous density for an arbitrary bandwidth/kernel pair. We also derive the limit joint distribution of kernel density estimators coresponding to different bandwidths and kernel functions. Using these reults, we construct an estimator that combines several estimators for different bandwidth/kernel pairs to protect against the negative consequences of errors in assumptions about order of smoothness. The results of a Monte Carlo experiment confirm the usefulness of the combined estimator. We demonstrate that while in the standard normal case the combined estimator has a relatively higher mean squared error than the standard kernel estimator, both estimators are highly accurate. On the other hand, for a non-smooth density where the MSE gets very large, the combined estimator provides uniformly better results than the standard estimator.

NON AND SEMI-PARAMETRIC ESTIMATION IN MODELS WITH UNKNOWN SMOOTHNESS

Many asymptotic results for kernel-based estimators were established under some smoothness assumption on density. For cases where smoothness assumptions that are used to derive unbiasedness or asymptotic rate may not hold we propose a combined estimator that could lead to the best available rate without knowledge of density smoothness. A Monte Carlo example confirms good performance of the combined estimator.

Kernel estimators : testing and bandwidth selection in models of unknown smoothness

Semiparametric and nonparametric estimators are becoming indispensable tools in applied econometrics. Many of these estimators depend on the choice of smoothing bandwidth and kernel function. Optimality of such parameters is determined by unobservable smoothness of the model, that is, by differentiability of the distribution functions of random variables in the model. In this thesis we consider two estimators of this class: the smoothed maximum score estimator for binary choice models and the kernel density estimator.We present theoretical results on the asymptotic distribution of the estimators under various smoothness assumptions and derive the limiting joint distributions for estimators with different combinations of bandwidths and kernel functions. Using these nontrivial joint distributions, we suggest a new way of improving accuracy and robustness of the estimators by considering a linear combination of estimators with different smoothing parameters. The weights in the combination minimize an estimate of the mean squared error. Monte Carlo simulations confirm suitability of this method for both smooth and non-smooth models.For the original and smoothed maximum score estimators, a formal procedure is introduced to test for equivalence of the maximum likelihood estimators and these semiparametric estimators, which converge to the true value at slower rates. The test allows one to identify heteroskedastic misspecifications in the logit/probit models. The method has been applied to analyze the decision of married women to join the labour force

Adapting kernel estimation to uncertain smoothness

For local and average kernel based estimators, smoothness conditions ensure that the kernel order determines the rate at which the bias of the estimator goes to zero and thus allows the econometrician to control the rate of convergence. In practice, even with smoothness the estimation errors may be substantial and sensitive to the choice of the bandwidth and kernel. For distributions that do not have sufficient smoothness asymptotic theory may importantly differ from standard; for example, there may be no bandwidth for which average estimators attain root-n consistency. We demonstrate that non-convex combinations of estimators computed for different kernel/bandwidth pairs can reduce the trace of asymptotic mean square error relative even to the optimal kernel/bandwidth pair. Our combined estimator builds on these results. To construct it we provide new general estimators for degree of smoothness, optimal rate and for the biases and covariances of estimators. We show that a bootstrap estimator is consistent for the variance of local estimators but exhibits a large bias for the average estimators; a suitable adjustment is provided

Non and semi-parametric estimation in models with unknown smoothness

Many asymptotic results for kernel-based estimators were established under some smoothness assumption on density. For cases where smoothness assumptions that are used to derive unbiasedness or asymptotic rate may not hold we propose a combined estimator that could lead to the best available rate without knowledge of density smoothness. A Monte Carlo example confirms good performance of the combined estimator

Robust kernel estimator for densities of unknown smoothness

Results on nonparametric kernel estimators of density differ according to the assumed degree of density smoothness; it is often assumed that the density function is at least twice differentiable. However, there are cases where non-smooth density functions may be of interest. We provide asymptotic results for kernel estimation of a continuous density for an arbitrary bandwidth/kernel pair. We also derive the limit joint distribution of kernel density estimators corresponding to different bandwidths and kernel functions. Using these results, we construct an estimator that combines several estimators for different bandwidth/kernel pairs to protect against the negative consequences of errors in assumptions about order of smoothness. The results of a Monte Carlo experiment confirm the usefulness of the combined estimator. We demonstrate that while in the standard normal case the combined estimator has a relatively higher mean squared error than the standard kernel estimator, both estimators are highly accurate. On the other hand, for a non-smooth density where the MSE gets very large, the combined estimator provides uniformly better results than the standard estimator

Rates of expansions for functional estimators

In this paper, we summarize results on convergence rates of various kernel based non- and semiparametric estimators, focusing on the impact of insufficient distributional smoothness, possibly unknown smoothness and even non-existence of density. In the presence of a possible lack of smoothness and the uncertainty about smoothness, methods of safeguarding against this uncertainty are surveyed with emphasis on nonconvex model averaging. This approach can be implemented via a combined estimator that selects weights based on minimizing the asymptotic mean squared error. In order to evaluate the finite sample performance of these and similar estimators we argue that it is important to account for possible lack of smoothness

Smoothness: bias and effciency of nonparametric kernel estimators

For kernel-based estimators, smoothness conditions ensure that the asymptotic rate at which the bias goes to zero is determined by the kernel order. In a finite sample, the leading term in the expansion of the bias may provide a poor approximation. We explore the relation between smoothness and bias and provide estimators for the degree of the smoothness and the bias. We demonstrate the existence of a linear combination of estimators whose trace of the asymptotic mean squared error is reduced relative to the individual estimator at the optimal bandwidth. We examine the finite-sample performance of a combined estimator that minimizes the trace of the MSE of a linear combination of individual kernel estimators for a multimodal density. The combined estimator provides a robust alternative to individual estimators that protects against uncertainty about the degree of smoothness