Cross-validation Bandwidth Matrices for Multivariate Kernel Density Estimation

Abstract.  The performance of multivariate kernel density estimates depends crucially on the choice of bandwidth matrix, but progress towards developing good bandwidth matrix selectors has been relatively slow. In particular, previous studies of cross-validation (CV) methods have been restricted to biased and unbiased CV selection of diagonal bandwidth matrices. However, for certain types of target density the use of full (i.e. unconstrained) bandwidth matrices offers the potential for significantly improved density estimation. In this paper, we generalize earlier work from diagonal to full bandwidth matrices, and develop a smooth cross-validation (SCV) methodology for multivariate data. We consider optimization of the SCV technique with respect to a pilot bandwidth matrix. All the CV methods are studied using asymptotic analysis, simulation experiments and real data analysis. The results suggest that SCV for full bandwidth matrices is the most reliable of the CV methods. We also observe that experience from the univariate setting can sometimes be a misleading guide for understanding bandwidth selection in the multivariate case.     

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.     

Hidden Second‐order Stationary Spatial Point Processes

In the existing statistical literature, the almost default choice for inference on inhomogeneous point processes is the most well‐known model class for inhomogeneous point processes: reweighted second‐order stationary processes. In particular, the K‐function related to this type of inhomogeneity is presented as the inhomogeneous K‐function. In the present paper, we put a number of inhomogeneous model classes (including the class of reweighted second‐order stationary processes) into the common general framework of hidden second‐order stationary processes, allowing for a transfer of statistical inference procedures for second‐order stationary processes based on summary statistics to each of these model classes for inhomogeneous point processes. In particular, a general method to test the hypothesis that a given point pattern can be ascribed to a specific inhomogeneous model class is developed. Using the new theoretical framework, we reanalyse three inhomogeneous point patterns that have earlier been analysed in the statistical literature and show that the conclusions concerning an appropriate model class must be revised for some of the point patterns.     

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.     

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.     

UNCONSTRAINED PILOT SELECTORS FOR SMOOTHED CROSS‐VALIDATION

Two of the most useful multivariate bandwidth selection techniques are the plug‐in and cross‐validation methods. The smoothed version of the cross‐validation method is known to reduce the variability of its non‐smoothed counterpart; however, it shares with the plug‐in choice the need for a pilot bandwidth matrix. Owing to the mathematical difficulties encountered in the optimal pilot choice, it is common to restrict this pilot matrix to be a scalar multiple of the identity matrix, at the expense of losing the flexibility afforded by the unconstrained approach. Here we show how to overcome these difficulties and propose a smoothed cross‐validation selector using an unconstrained pilot matrix. Our numerical results indicate that the unconstrained selector outperforms the constrained one in practice, and is a viable competitor to unconstrained plug‐in selectors.     

Kernel estimation of regression function gradient

Abstract

This paper is focused on kernel estimation of the gradient of a multivariate regression function. Despite the importance of this topic, the progress in this area is rather slow. Our aim is to construct a gradient estimator using the idea of local linear estimator for a regression function. The quality of this estimator is expressed in terms of the Mean Integrated Square Error. We focus on a choice of bandwidth matrix. Further, we present some data-driven methods for its choice and develop a new approach. The performance of presented methods is illustrated using a simulation study and real data example.     

Nonparametric Shrinkage Estimation for Aalen's Additive Hazards Model

Aalen's nonparametric additive model in which the regression coefficients are assumed to be unspecified functions of time is a flexible alternative to Cox's proportional hazards model when the proportionality assumption is in doubt. In this paper, we incorporate a general linear hypothesis into the estimation of the time‐varying regression coefficients. We combine unrestricted least squares estimators and estimators that are restricted by the linear hypothesis and produce James‐Stein‐type shrinkage estimators of the regression coefficients. We develop the asymptotic joint distribution of such restricted and unrestricted estimators and use this to study the relative performance of the proposed estimators via their integrated asymptotic distributional risks. We conduct Monte Carlo simulations to examine the relative performance of the estimators in terms of their integrated mean square errors. We also compare the performance of the proposed estimators with a recently devised LASSO estimator as well as with ridge‐type estimators both via simulations and data on the survival of primary billiary cirhosis patients.     

Autocovariance Estimation in Regression with a Discontinuous Signal and m‐Dependent Errors: A Difference‐Based Approach

We discuss a class of difference‐based estimators for the autocovariance in nonparametric regression when the signal is discontinuous and the errors form a stationary m‐dependent process. These estimators circumvent the particularly challenging task of pre‐estimating such an unknown regression function. We provide finite‐sample expressions of their mean squared errors for piecewise constant signals and Gaussian errors. Based on this, we derive biased‐optimized estimates that do not depend on the unknown autocovariance structure. Notably, for positively correlated errors, that part of the variance of our estimators that depend on the signal is minimal as well. Further, we provide sufficient conditions for ‐consistency; this result is extended to piecewise Hölder regression with non‐Gaussian errors. We combine our biased‐optimized autocovariance estimates with a projection‐based approach and derive covariance matrix estimates, a method that is of independent interest. An R package, several simulations and an application to biophysical measurements complement this paper.     

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

We consider the problem of parameter estimation for inhomogeneous space‐time shot‐noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality‐reducing techniques to estimate different parts of the model separately. We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes – possible overlapping in the projection process of clusters, which are clearly separated in the original space‐time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process. For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.     

Non‐parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

Abstract. We consider a stochastic process driven by diffusions and jumps. Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred. This allows us to determine a consistent non‐parametric estimator of the integrated volatility when the infinite activity jump component is Lévy. Jump size estimation and central limit results are proved in the case of finite activity jumps. Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data.     

Non‐Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise‐Deterministic Markov Processes

This paper presents a non‐parametric method for estimating the conditional density associated to the jump rate of a piecewise‐deterministic Markov process. In our framework, the estimation needs only one observation of the process within a long time interval. Our method relies on a generalization of Aalen's multiplicative intensity model. We prove the uniform consistency of our estimator, under some reasonable assumptions related to the primitive characteristics of the process. A simulation study illustrates the behaviour of our estimator.     

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

We consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.     

Inference in Semi‐Parametric Dynamic Models for Repeated Count Data

This paper deals with a longitudinal semi‐parametric regression model in a generalised linear model setup for repeated count data collected from a large number of independent individuals. To accommodate the longitudinal correlations, we consider a dynamic model for repeated counts which has decaying auto‐correlations as the time lag increases between the repeated responses. The semi‐parametric regression function involved in the model contains a specified regression function in some suitable time‐dependent covariates and a non‐parametric function in some other time‐dependent covariates. As far as the inference is concerned, because the non‐parametric function is of secondary interest, we estimate this function consistently using the independence assumption‐based well‐known quasi‐likelihood approach. Next, the proposed longitudinal correlation structure and the estimate of the non‐parametric function are used to develop a semi‐parametric generalised quasi‐likelihood approach for consistent and efficient estimation of the regression effects in the parametric regression function. The finite sample performance of the proposed estimation approach is examined through an intensive simulation study based on both large and small samples. Both balanced and unbalanced cluster sizes are incorporated in the simulation study. The asymptotic performances of the estimators are given. The estimation methodology is illustrated by reanalysing the well‐known health care utilisation data consisting of counts of yearly visits to a physician by 180 individuals for four years and several important primary and secondary covariates.     

Functional Mixed Effects Model for Small Area Estimation

Functional data analysis has become an important area of research because of its ability of handling high‐dimensional and complex data structures. However, the development is limited in the context of linear mixed effect models and, in particular, for small area estimation. The linear mixed effect models are the backbone of small area estimation. In this article, we consider area‐level data and fit a varying coefficient linear mixed effect model where the varying coefficients are semiparametrically modelled via B‐splines. We propose a method of estimating the fixed effect parameters and consider prediction of random effects that can be implemented using a standard software. For measuring prediction uncertainties, we derive an analytical expression for the mean squared errors and propose a method of estimating the mean squared errors. The procedure is illustrated via a real data example, and operating characteristics of the method are judged using finite sample simulation studies.     

On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimate

Empirical Bayes (EB) estimates in general linear mixed models are useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EB is that the overall estimate for a larger geographical area based on a (weighted) sum of EB estimates is not necessarily identical to the corresponding direct estimate such as the overall sample mean. Another difficulty is that EB estimates yield over‐shrinking, which results in the sampling variance smaller than the posterior variance. One way to fix these problems is the benchmarking approach based on the constrained empirical Bayes (CEB) estimators, which satisfy the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. In this paper, we treat the general mixed models, derive asymptotic approximations of the mean squared error (MSE) of CEB and provide second‐order unbiased estimators of MSE based on the parametric bootstrap method. These results are applied to natural exponential families with quadratic variance functions. As a specific example, the Poisson‐gamma model is dealt with, and it is illustrated that the CEB estimates and their MSE estimates work well through real mortality data.