Data-Based Choice of the Number of Pilot Stages for Plug-in Bandwidth Selection

The choice of the bandwidth is a crucial issue for kernel density estimation. Among all the data-dependent methods for choosing the bandwidth, the direct plug-in method has shown a particularly good performance in practice. This procedure is based on estimating an asymptotic approximation of the optimal bandwidth, using two “pilot” kernel estimation stages. Although two pilot stages seem to be enough for most densities, for a long time the problem of how to choose an appropriate number of stages has remained open. Here we propose an automatic (i.e., data-based) method for choosing the number of stages to be employed in the plug-in bandwidth selector. Asymptotic properties of the method are presented and an extensive simulation study is carried out to compare its small-sample performance with that of the most recommended bandwidth selectors in the literature.     

Kernel Survival Function Estimation Based on Doubly Censored Data

This work concerns the estimation of a smooth survival function based on doubly censored data. We establish strong consistency and asymptotic normality for a kernel estimator. Moreover, we also obtain an asymptotic expression for the mean integrated squared error, which yields an optimum bandwidth in terms of readily estimable quantities.     

On testing more IFRA ordering

In this paper, first we consider the problem of testing that two unknown distributions are identical against the alternative that one is more IFRA than the other and propose a new test that is asymptotically normal and consistent. Next, we prove that beta family of distributions is ordered according to more IFRA ordering. The empirical power of the proposed test is simulated for some specific families of distributions like beta, gamma and Weibull that are ordered with respect to more IFRA order. Finally, we apply our test to some real data sets in the context of reliability.     

Testing the adequacy of a linear model VIA critical smoothing

An important problem for fitting local linear regression is the choice of the smoothing parameter. As the smoothing parameter becomes large, the estimator tends to a straight line, which is the least squares fit in the ordinary linear regression setting. This property may be used to assess the adequacy of a simple linear model. Motivated by Silverman's (1981) work in kernel density estimation, a suitable test statistic is the critical smoothing parameter where the estimate changes from nonlinear to linear, while linearity or non- linearity requires a more precise judgment. We define the critical smoothing parameter through the approximate F-tests by Hastie and Tibshirani (1990). To assess the significance, the “wild bootstrap” procedure is used to replicate the data and the proportion of bootstrap samples which give a nonlinear estimate when using the critical bandwidth is obtained as the p-value. Simulation results show that the critical smoothing test is useful in detecting a wide range of alternatives.     

Shape-Constrained Kernel-Weighted Least Squares: Estimating Production Functions for Chilean Manufacturing Industries

ABSTRACT

In this article, we examine a novel way of imposing shape constraints on a local polynomial kernel estimator. The proposed approach is referred to as shape constrained kernel-weighted least squares (SCKLS). We prove uniform consistency of the SCKLS estimator with monotonicity and convexity/concavity constraints and establish its convergence rate. In addition, we propose a test to validate whether shape constraints are correctly specified. The competitiveness of SCKLS is shown in a comprehensive simulation study. Finally, we analyze Chilean manufacturing data using the SCKLS estimator and quantify production in the plastics and wood industries. The results show that exporting firms have significantly higher productivity.     

Nonparametric maximum likelihood estimators for ar and ma time series

The problem of estimating the parameters of moving average or autoregressive time series is studied when the error distribution is completely unknown. Four nonparametric maximum likelihood estimators (NPMLE) are presented for this purpose. These estimators are compared with the classical moment and least squares estimators in a simulation study. The behavior of these NPMLEs is much better than the classical ones, suggesting that they should be used extensively when no parametric information is known in advance about the error distribution. An application of these estimators to coal mining accidents data is also included.     

On the superposition of overlapping Poisson processes and nonparametric estimation of their intensity function

It is shown that, when measuring time in the Total Time on Test scale, the superposition of overlapping realizations of a nonhomogeneous Poisson process is also a Poisson process and is sufficient for inferential purposes. Hence, many nonparametric procedures which are available when only one realization is observed can be easily extended for the overlapping realizations setup. These include, for instance, the constrained maximum likelihood estimator of a monotonic intensity and bootstrap confidence bands based on Kernel estimates of the intensity. The kernel estimate proposed here performs the smoothing in the Total Time on Test scale and it is shown to behave approximately as a usual kernel estimate but with a variable bandwidth which is inversely proportional to the number of realizations at-risk. Likewise, bootstrap samples can be obtained from the single realization of the superimposed process. The methods are illustrated using a real data set consisting of the failure histories of 40 electrical power transformers.     

Semiparametric Estimation of Stochastic Production Frontier Models

This article extends the linear stochastic frontier model proposed by Aigner, Lovell, and Schmidt to a semiparametric frontier model in which the functional form of the production frontier is unspecified and the distributions of the composite error terms are of known form. Pseudolikelihood estimators of the parameters characterizing the two error terms of the model are constructed based on kernel estimation of the conditional mean function. The Monte Carlo results show that the proposed estimators perform well in finite samples. An empirical application is presented. Extensions to a partially linear frontier function and to more flexible one-sided error distributions than the half-normal are discussed     

Asymptotic Theory for the Multiscale Wavelet Density Derivative Estimator

Abstract

In this paper we consider the wavelet-based estimation of density derivatives. The multiscale density derivative estimator is proposed which is constructed by using a number of scaling functions. Asymptotic theory is developed in which asymptotic expressions for the bias, the variance and the mean integrated squared error are included. In addition, asymptotic normality of the proposed estimator is proved. Theoretical and numerical comparisons with the usual kernel-based estimators are also reported.