Asymptotic behaviour of the mean integrated squared error of kernel density estimators for dependent observations

Let X₁, X₂, … be a strictly stationary sequence of observations, and g be the joint density of (X₁, …, X_d) for some fixed d ? 1. We consider kernel estimators of the density g. The asymptotic behaviour of the mean integrated squared error of the kernel estimators is obtained under an assumption of weak dependence between the observations.     

Asymptotic normality of kernel type regression estimators for random fields

The asymptotic normality of the Nadaraya–Watson regression estimator is studied for α-mixing$\alpha \text{-}\mathrm{mixing}$ random fields. The infill-increasing setting is considered, that is when the locations of observations become dense in an increasing sequence of domains. This setting fills the gap between continuous and discrete models. In the infill-increasing case the asymptotic normality of the Nadaraya–Watson estimator holds, but with an unusual asymptotic covariance structure. It turns out that this covariance structure is a combination of the covariance structures that we observe in the discrete and in the continuous case.     

Note on the minimum mean integrated squared error of kernel estimates of a distribution function and its derivatives

An exact expiession for the minimum integrated squared error associated with the kernel distribution function and its derivatives is given. Furthermore, the virtual optimality of the Fourier integral estimate in density estimation, shown by Davis (1977), is extended to estimation of a distibution function and its derivatives.     

Bayesian bandwidth estimation for a functional nonparametric regression model with mixed types of regressors and unknown error density

We investigate the issue of bandwidth estimation in a functional nonparametric regression model with function-valued, continuous real-valued and discrete-valued regressors under the framework of unknown error density. Extending from the recent work of Shang (2013 Shang, H.L. (2013), ‘Bayesian Bandwidth Estimation for a Nonparametric Functional Regression Model with Unknown Error Density’, Computational Statistics &; Data Analysis, 67, 185–198. doi: 10.1016/j.csda.2013.05.006[Crossref], [Web of Science ®] , [Google Scholar]) [‘Bayesian Bandwidth Estimation for a Nonparametric Functional Regression Model with Unknown Error Density’, Computational Statistics &; Data Analysis, 67, 185–198], we approximate the unknown error density by a kernel density estimator of residuals, where the regression function is estimated by the functional Nadaraya–Watson estimator that admits mixed types of regressors. We derive a likelihood and posterior density for the bandwidth parameters under the kernel-form error density, and put forward a Bayesian bandwidth estimation approach that can simultaneously estimate the bandwidths. Simulation studies demonstrated the estimation accuracy of the regression function and error density for the proposed Bayesian approach. Illustrated by a spectroscopy data set in the food quality control, we applied the proposed Bayesian approach to select the optimal bandwidths in a functional nonparametric regression model with mixed types of regressors.     

Asymptotic properties for estimates of nonparametric regression model with martingale difference errors

Consider the nonparametric regression model with martingale difference errors. Nonparametric estimator g _n (x) of regression function g(x) will be introduced, and its asymptotic properties are studied. In particular, the pointwise and uniform convergence of g _n (x) and its asymptotic normality will be investigated. This extends the earlier work on independent random errors.     

Plug-in bandwidth choice for estimation of nonparametric part in partial linear regression models with strong mixing errors

Consider a regression model where the regression function is the sum of a linear and a nonparametric component. Assuming that the errors of the model follow a stationary strong mixing process with mean zero, the problem of bandwidth selection for a kernel estimator of the nonparametric component is addressed here. We obtain an asymptotic expression for an optimal band-width and we propose to use a plug-in methodology in order to estimate this bandwidth through preliminary estimates of the unknown quantities. Asymptotic optimality for the plug-in bandwidth is established.     

Convergence rate of the kernel regression estimator for associated and truncated data

This paper studies the behaviour of the kernel estimator of the regression function for associated data in the random left truncated model. The uniform strong consistency rate over a real compact set of the estimate is established. The finite sample performance of the estimator is investigated through extensive simulation studies.     

Asymptotic normality of a conditional hazard function estimate in the single index for quasi-associated data

Abstract

The main goal of this paper is to study the estimation of the conditional hazard function of a scalar response variable Y given a hilbertian random variable X in functional single-index model. We construct an estimator of this nonparametric function and we study its asymptotic properties, under quasi-associated structure. Precisely, we establish the asymptotic normality of the constructed estimator. We carried out simulation experiments to examine the behavior of this asymptotic property over finite sample data.     

On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Estimation of the derivative of a density and related functions

Let Y₁,…,Y _n, (Y₁ <Y₂<…<Y _n) be the order statistics of a random sample from a distribution F with density f on the realline. This paper gives a class of estimators of the derivativef'(x) of the density f at points x for which f has

a continuoussecond derivative. These estimators are based on spacings inthe order statistics Y_j+kn -y _j j = 1,…,n-k_n,k_n<n.     

Robust parameter estimation of regression model with AR(p) error terms

In this article, we consider a linear regression model with AR(p) error terms with the assumption that the error terms have a t distribution as a heavy-tailed alternative to the normal distribution. We obtain the estimators for the model parameters by using the conditional maximum likelihood (CML) method. We conduct an iteratively reweighting algorithm (IRA) to find the estimates for the parameters of interest. We provide a simulation study and three real data examples to illustrate the performance of the proposed robust estimators based on t distribution.     

Performance of Wald-type estimator for parametric component in partial linear regression with a mixture of Berkson and classical error models

This article discusses a consistent and almost unbiased estimation approach in partial linear regression for parameters of interest when the regressors are contaminated with a mixture of Berkson and classical errors. Advantages of the presented procedure are: (1) random errors and observations are not necessarily to be parametric settings; (2) there is no need to use additional sample information, and to consider the estimation of nuisance parameters. We will examine the performance of our presented estimate in a variety of numerical examples through Monte Carlo simulation. The proposed approach is also illustrated in the analysis of an air pollution data.     

Asymptotic distribution of data-driven smoothers in density and regression estimation under dependence

We consider automatic data-driven density, regression and autoregression estimates, based on any random bandwidth selector h/_T. We show that in a first-order asymptotic approximation they behave as well as the related estimates obtained with the “optimal” bandwidth h_T as long as h_T/h_T → 1 in probability. The results are obtained for dependent observations; some of them are also new for independent observations.     

Goodness‐of‐fit testing based on a weighted bootstrap: A fast large‐sample alternative to the parametric bootstrap

The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness‐of‐fit testing. The simplest way to carry out such goodness‐of‐fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data increase. An alternative resampling technique based on a fast weighted bootstrap is proposed in this paper, and is studied both theoretically and empirically. The outcome of this work is a generic and computationally efficient multiplier goodness‐of‐fit procedure that can be used as a large‐sample alternative to the parametric bootstrap. In order to approximately determine how large the sample size needs to be for the parametric and weighted bootstraps to have roughly equivalent powers, extensive Monte Carlo experiments are carried out in dimension one, two and three, and for models containing up to nine parameters. The computational gains resulting from the use of the proposed multiplier goodness‐of‐fit procedure are illustrated on trivariate financial data. A by‐product of this work is a fast large‐sample goodness‐of‐fit procedure for the bivariate and trivariate t distribution whose degrees of freedom are fixed. The Canadian Journal of Statistics 40: 480–500; 2012 © 2012 Statistical Society of Canada     

Theoretical evaluation of prediction error in linear regression with a bivariate response variable containing missing data

Methods for linear regression with multivariate response variables are well described in statistical literature. In this study we conduct a theoretical evaluation of the expected squared prediction error in bivariate linear regression where one of the response variables contains missing data. We make the assumption of known covariance structure for the error terms. On this basis, we evaluate three well-known estimators: standard ordinary least squares, generalized least squares, and a James–Stein inspired estimator. Theoretical risk functions are worked out for all three estimators to evaluate under which circumstances it is advantageous to take the error covariance structure into account.     

Testing for a Change of the Innovation Distribution in Nonparametric Autoregression: The Sequential Empirical Process Approach

We consider a nonparametric autoregression model under conditional heteroscedasticity with the aim to test whether the innovation distribution changes in time. To this end, we develop an asymptotic expansion for the sequential empirical process of nonparametrically estimated innovations (residuals). We suggest a Kolmogorov–Smirnov statistic based on the difference of the estimated innovation distributions built from the first ?ns?and the last n ? ?ns? residuals, respectively (0 ≤ s ≤ 1). Weak convergence of the underlying stochastic process to a Gaussian process is proved under the null hypothesis of no change point. The result implies that the test is asymptotically distribution‐free. Consistency against fixed alternatives is shown. The small sample performance of the proposed test is investigated in a simulation study and the test is applied to a data example.