On the asymptotic behavior of the periodograms of periodically correlated spatial processes: Periodicity detection

In this work, the asymptotic unbiasedness and the asymptotic uncorrelatedness of periodograms for the periodically correlated spatial processes are given. This will be done using the time dependent spectral representation of periodically correlated spatial processes and Cholesky factorization of the spectral density. A graphical method is also proposed to detect the period of periodically correlated spatial processes. In order to support the theory, a simulation study and a real data example are performed.     

A comparative study of some bias correction techniques for kernel- based density estimators

Methods for obtaining kernel-based density estimators with lower bias and mean integrated squared error than an estimator based on a standard Normal kernel function are described and discussed. Three main approaches are considered which are: firstly by using 'optimal' polynomial kernels as described, for example, by Gasser er a1 (1985); secondly by employing generalised jackknifing as proposed by Jones nd Foster (1993) and thirdly by subtracting an estimator of the principal asymptotic bias term from the original estimator. The emphasis in this initial discussion is on their asymptotic properties. The finite sample performance of those that have the best asymptotic properties are compared with two adaptive estimators, as well as the fixed Normal kernel estimator, in a simulation study.     

Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields

We investigate the asymptotic behaviour of binned kernel density estimators for dependent and locally non-stationary random fields converging to stationary random fields. We focus on the study of the bias and the asymptotic normality of the estimators. A simulation experiment conducted shows that both the kernel density estimator and the binned kernel density estimator have the same behavior and both estimate accurately the true density when the number of fields increases. We apply our results to the 2002 incidence rates of tuberculosis in the departments of France.     

Intensity Estimation for Spatial Point Processes Observed with Noise

Abstract.  This article introduces a kernel estimator of the intensity function of spatial point processes taking into account location errors. The asymptotic properties of the estimator are derived and a bandwidth selection procedure is described. A simulation study compares our results with that of the classical kernel estimator and shows that the edge-corrected deconvoluting kernel estimator is more appropriate.     

Estimation of integrated squared density derivatives from a contaminated sample

Summary. We propose a kernel estimator of integrated squared density derivatives, from a sample that has been contaminated by random noise. We derive asymptotic expressions for the bias and the variance of the estimator and show that the squared bias term dominates the variance term. This coincides with results that are available for non-contaminated observations. We then discuss the selection of the bandwidth parameter when estimating integrated squared density derivatives based on contaminated data. We propose a data-driven bandwidth selection procedure of the plug-in type and investigate its finite sample performance via a simulation study.     

Nonparametric prediction of spatial multivariate data

This paper investigates a nonparametric spatial predictor of a stationary multidimensional spatial process observed over a rectangular domain. The proposed predictor depends on two kernels in order to control both the distance between observations and that between spatial locations. The uniform almost complete consistency and the asymptotic normality of the kernel predictor are obtained when the sample considered is an alpha-mixing sequence. Numerical studies were carried out in order to illustrate the behaviour of our methodology both for simulated data and for an environmental data set.     

Iterated Bootstrap-t Confidence Intervals for Density Functions

Abstract.  Conventional bootstrap- t intervals for density functions based on kernel density estimators exhibit poor coverages due to failure of the bootstrap to estimate the bias correctly. The problem can be resolved by either estimating the bias explicitly or undersmoothing the kernel density estimate to undermine its bias asymptotically. The resulting bias-corrected intervals have an optimal coverage error of order arbitrarily close to second order for a sufficiently smooth density function. We investigated the effects on coverage error of both bias-corrected intervals when the nominal coverage level is calibrated by the iterated bootstrap. In either case, an asymptotic reduction of coverage error is possible provided that the bias terms are handled using an extra round of smoothed bootstrapping. Under appropriate smoothness conditions, the optimal coverage error of the iterated bootstrap- t intervals has order arbitrarily close to third order. Examples of both simulated and real data are reported to illustrate the iterated bootstrap procedures.     

Nonparametric spectrum estimation for spatial data

Smoothed nonparametric kernel spectral density estimates are considered for stationary data observed on a d-dimensional lattice. The implications for edge effect bias of the choice of kernel and bandwidth are considered. Under some circumstances the bias can be dominated by the edge effect. We show that this problem can be mitigated by tapering. Some extensions and related issues are discussed.     

Asymptotic properties of nonparametric regression for long memory random fields

The kernel estimator of spatial regression function is investigated for stationary long memory (long range dependent) random fields observed over a finite set of spatial points. A general result on the strong consistency of the kernel density estimator is first obtained for the long memory random fields, and then, under some mild regularity assumptions, the asymptotic behaviors of the regression estimator are established. For the linear long memory random fields, a weak convergence theorem is also obtained for kernel density estimator. Finally, some related issues on the inference of long memory random fields are discussed through a simulation example.     

Mixed-spectra analysis for stationary random fields

We consider a, discrete time, weakly stationary bidimensional process, for which the spectral measure is the sum of an absolutely continuous measure, a discrete measure of finite order and a finite number of absolutely continuous measures on several lines. In this paper we are interested in estimating the spectral density of the absolutely continuous measure and of the density on the lines. For this aim, by using the double kernel method, we construct consistent estimators of these densities and we study their asymptotic behaviors in term of the mean squared error with rate.     

Asymptotic properties of the kernel mode estimator under twice censorship model

In this article, we study the asymptotic properties of the kernel estimator of the mode and density function when the data are twice censored. More specifically, we first establish a strong uniform consistency over a compact set with a rate of the kernel density estimator and then we give the consistency with rate and asymptotic normality for the kernel mode estimator. An application to confidence bands is given.     

On the Estimation of the Density of a Directional Data Stream

Many directional data such as wind directions can be collected extremely easily so that experiments typically yield a huge number of data points that are sequentially collected. To deal with such big data, the traditional nonparametric techniques rapidly require a lot of time to be computed and therefore become useless in practice if real time or online forecasts are expected. In this paper, we propose a recursive kernel density estimator for directional data which (i) can be updated extremely easily when a new set of observations is available and (ii) keeps asymptotically the nice features of the traditional kernel density estimator. Our methodology is based on Robbins–Monro stochastic approximations ideas. We show that our estimator outperforms the traditional techniques in terms of computational time while being extremely competitive in terms of efficiency with respect to its competitors in the sequential context considered here. We obtain expressions for its asymptotic bias and variance together with an almost sure convergence rate and an asymptotic normality result. Our technique is illustrated on a wind dataset collected in Spain. A Monte‐Carlo study confirms the nice properties of our recursive estimator with respect to its non‐recursive counterpart.     

Varying kernel marginal density estimator for a positive time series

This paper analyses the large sample behaviour of a varying kernel density estimator of the marginal density of a non-negative stationary and ergodic time series that is also strongly mixing. In particular we obtain an approximation for bias, mean square error and establish asymptotic normality of this density estimator. We also derive an almost sure uniform consistency rate over bounded intervals of this estimator. A finite sample simulation shows some superiority of the proposed density estimator over the one based on a symmetric kernel.     

Infill asymptotics for adaptive kernel estimators of spatial intensity

We apply the Abramson principle to define adaptive kernel estimators for the intensity function of a spatial point process. We derive asymptotic expansions for the bias and variance under the regime that n independent copies of a simple point process in Euclidean space are superposed. The method is illustrated by means of a simple example and applied to tornado data.     

Bivariate uniform deconvolution

We construct a density estimator in the bivariate uniform deconvolution model. For this model, we derive four inversion formulas to express the bivariate density that we want to estimate in terms of the bivariate density of the observations. By substituting a kernel density estimator of the density of the observations, we then obtain four different estimators. Next we construct an asymptotically optimal convex combination of these four estimators. Expansions for the bias, variance, as well as asymptotic normality are derived. Some simulated examples are presented.     

A Convolution Estimator for the Density of Nonlinear Regression Observations

Abstract. The problem of estimating an unknown density function has been widely studied. In this article, we present a convolution estimator for the density of the responses in a nonlinear heterogenous regression model. The rate of convergence for the mean square error of the convolution estimator is of order n ^?1 under certain regularity conditions. This is faster than the rate for the kernel density method. We derive explicit expressions for the asymptotic variance and the bias of the new estimator, and further a data‐driven bandwidth selector is proposed. We conduct simulation experiments to check the finite sample properties, and the convolution estimator performs substantially better than the kernel density estimator for well‐behaved noise densities.     

A Note on a Specification Test for Time Series Models Based on Spectral Density Estimation

In a recent paper, Paparoditis [Scand. J. Statist. 27 (2000) 143] proposed a new goodness‐of‐fit test for time series models based on spectral density estimation. The test statistic is based on the distance between a kernel estimator of the ratio of the true and the hypothesized spectral density and the expected value of the estimator under the null and provides a quantification of how well the parametric density fits the sample spectral density. In this paper, we give a detailed asymptotic analysis of the corresponding procedure under fixed alternatives.     

A comparison of estimators of the geographical relative risk function

The geographical relative risk function is a useful tool for investigating the spatial distribution of disease based on case and control data. The most common way of estimating this function is using the ratio of bivariate kernel density estimates constructed from the locations of cases and controls, respectively. An alternative is to use a local-linear (LL) estimator of the log-relative risk function. In both cases, the choice of bandwidth is critical. In this article, we examine the relative performance of the two estimation techniques using a variety of data-driven bandwidth selection methods, including likelihood cross-validation (CV), least-squares CV, rule-of-thumb reference methods, and a new approximate plug-in (PI) bandwidth for the LL estimator. Our analysis includes the comparison of asymptotic results; a simulation study; and application of the estimators on two real data sets. Our findings suggest that the density ratio method implemented with the least-squares CV bandwidth selector is generally best, with the LL estimator with PI bandwidth being competitive in applications with strong large-scale trends but much worse in situations with elliptical clusters.     

Large Deviations Limit Theorems for the Kernel Density Estimator

We establish pointwise and uniform large deviations limit theorems of Chernoff-type for the non-parametric kernel density estimator based on a sequence of independent and identically distributed random variables. The limits are well-identified and depend upon the underlying kernel and density function. We derive then some implications of our results in the study of asymptotic efficiency of the goodness-of-fit test based on the maximal deviation of the kernel density estimator as well as the inaccuracy rate of this estimate     

Spectral Density Based Goodness-of-Fit Tests for Time Series Models

A new goodness-of-fit test for time series models is proposed. The test statistic is based on the distance between a kernel estimator of the ratio between the true and the hypothesized spectral density and the expected value of the estimator under the null. It provides a quantification of how well a parametric spectral density model fits the sample spectral density (periodogram). The asymptotic distribution of the statistic proposed is derived and its power properties are discussed. To improve upon the large sample (Gaussian) approximation of the distribution of the test statistic under the null, a bootstrap procedure is presented and justified theoretically. The finite sample performance of the test is investigated through a simulation experiment and applications to real data sets are given.