Bandwidth selection procedures tor kernel density estimates

A crucial problem in kernel density estimates of a probability density function is the selection of the bandwidth. The aim of this study is to propose a procedure for selecting both fixed and variable bandwidths. The present study also addresses the question of how different variable bandwidth kernel estimators perform in comparison with each other and to the fixed type of bandwidth estimators. The appropriate algorithms for implementation of the proposed method are given along with a numerical simulation.The numerical results serve as a guide to determine which bandwidth selection method is most appropriate for a given type of estimator over a vide class of probability density functions, Also, we obtain a numerical comparison of the different types of kernel estimators under various types of bandwidths.     

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.     

Comparison of Kernel Density Estimators with Assumption on Number of Modes

A data-driven bandwidth choice for a kernel density estimator called critical bandwidth is investigated. This procedure allows the estimation to have as many modes as assumed for the density to estimate. Both Gaussian and uniform kernels are considered. For the Gaussian kernel, asymptotic results are given. For the uniform kernel, an argument against these properties is mentioned. These theoretical results are illustrated with a simulation study that compares the kernel estimators that rely on critical bandwidth with another one that uses a plug-in method to select its bandwidth. An estimator that consists in estimates of density contour clusters and takes assumptions on number of modes into account is also considered. Finally, the methodology is illustrated using environment monitoring data.     

DENSITY ESTIMATION FOR CLUSTERED DATA

The commonly used survey technique of clustering introduces dependence into sample data. Such data is frequently used in economic analysis, though the dependence induced by the sample structure of the data is often ignored. In this paper, the effect of clustering on the non-parametric, kernel estimate of the density, f(x), is examined. The window width commonly used for density estimation for the case of i.i.d. data is shown to no longer be optimal. A new optimal bandwidth using a higher-order kernel is proposed and is shown to give a smaller integrated mean squared error than two window widths which are widely used for the case of i.i.d. data. Several illustrations from simulation are provided.     

A note on kernel density estimation for non-negative random variables

Kernel-based density estimation algorithms are inefficient in presence of discontinuities at support endpoints. This is substantially due to the fact that classic kernel density estimators lead to positive estimates beyond the endopoints. If a nonparametric estimate of a density functional is required in determining the bandwidth, then the problem also affects the bandwidth selection procedure. In this paper algorithms for bandwidth selection and kernel density estimation are proposed for non-negative random variables. Furthermore, the methods we propose are compared with some of the principal solutions in the literature through a simulation study.     

Kernel Likelihood Inference for Time Series

Abstract.  This paper develops non-parametric techniques for dynamic models whose data have unknown probability distributions. Point estimators are obtained from the maximization of a semiparametric likelihood function built on the kernel density of the disturbances. This approach can also provide Kullback–Leibler cross-validation estimates of the bandwidth of the kernel densities. Confidence regions are derived from the dual-empirical likelihood method based on non-parametric estimates of the scores. Limit theorems for martingale difference sequences support the statistical theory; moreover, simulation experiments and a real case study show the validity of the methods.     

Asymptotically optimal bandwidth selection rules for the kernel density estimator with dependent observations

With mild restrictions placed on the kernel, kernel estimates of an unknown multivariate density are investigated when the observed data are dependent. A modified cross validation rule, the simple ‘leave-(2P + 1)-out’ version of simple cross validation, is considered for bandwidth selection. Under the mild assumption that the unknown density is bounded, this rule is shown to be asymptotically optimal under a geometric strong mixing condition. This strengthens recent results of Hart and Vieu (Ann. Statist., 18 (1990)). The results are then extended to bandwidth selection problems associated to the Gibbs sampler.     

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.     

Variable kernel density estimation

This paper considers the problem of selecting optimal bandwidths for variable (sample‐point adaptive) kernel density estimation. A data‐driven variable bandwidth selector is proposed, based on the idea of approximating the log‐bandwidth function by a cubic spline. This cubic spline is optimized with respect to a cross‐validation criterion. The proposed method can be interpreted as a selector for either integrated squared error (ISE) or mean integrated squared error (MISE) optimal bandwidths. This leads to reflection upon some of the differences between ISE and MISE as error criteria for variable kernel estimation. Results from simulation studies indicate that the proposed method outperforms a fixed kernel estimator (in terms of ISE) when the target density has a combination of sharp modes and regions of smooth undulation. Moreover, some detailed data analyses suggest that the gains in ISE may understate the improvements in visual appeal obtained using the proposed variable kernel estimator. These numerical studies also show that the proposed estimator outperforms existing variable kernel density estimators implemented using piecewise constant bandwidth functions.     

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.     

Simulations and computations of nonparametric density estimates for the deconvolution problem

The nonparametric density function estimation using sample observations which are contaminated with random noise is studied. The particular form of contamination under consideration is Y = X + Z, where Y is an observable random variableZ is a random noise variable with known distribution, and X is an absolutely continuous random variable which cannot be observed directly. The finite sample size performance of a strongly consistent estimator for the density function of the random variable X is illustrated for different distributions. The estimator uses Fourier and kernel function estimation techniques and allows the user to choose constants which relate to bandwidth windows and limits on integration and which greatly affect the appearance and properties of the estimates. Numerical techniques for computation of the estimated densities and for optimal selection of the constant are given.     

An optimal local bandwidth selector for kernel density estimation

The problem of selecting the bandwidth for optimal kernel density estimation at a point is considered. A class of local bandwidth selectors which minimize smoothed bootstrap estimates of mean-squared error in density estimation is introduced. It is proved that the bandwidth selectors in the class achieve optimal relative rates of convergence, dependent upon the local smoothness of the target density. Practical implementation of the bandwidth selection methodology is discussed. The use of Gaussian-based kernels to facilitate computation of the smoothed bootstrap estimate of mean-squared error is proposed. The performance of the bandwidth selectors is investigated empirically.     

Improved orthogonal polynomial density estimates

Density estimates that are expressible as the product of a base density function and a linear combination of orthogonal polynomials are considered in this paper. More specifically, two criteria are proposed for determining the number of terms to be included in the polynomial adjustment component and guidelines are suggested for the selection of a suitable base density function. A simulation study reveals that these stopping rules produce density estimates that are generally more accurate than kernel density estimates or those resulting from the application of the Kronmal–Tarter criterion. Additionally, it is explained that the same approach can be utilized to obtain multivariate density estimates. The proposed orthogonal polynomial density estimation methodology is applied to several univariate and bivariate data sets, some of which have served as benchmarks in the statistical literature on density estimation.     

Kernel density estimation under negative superadditive dependence and its application for real data

In this paper, the kernel density estimator for negatively superadditive dependent random variables is studied. The exponential inequalities and the exponential rate for the kernel estimator of density function with a uniform version, over compact sets are investigated. Also, the optimal bandwidth rate of the estimator is obtained using mean integrated squared error. The results are generalized and used to improve the ones obtained for the case of associated sequences. As an application, FGM sequences that fulfil our assumptions are investigated. Also, the convergence rate of the kernel density estimator is illustrated via a simulation study. Moreover, a real data analysis is presented.     

A Composite Likelihood Cross-validation Approach in Selecting Bandwidth for the Estimation of the Pair Correlation Function

Abstract.  A useful tool while analysing spatial point patterns is the pair correlation function (e.g. Fractals, Random Shapes and Point Fields, Wiley, New York, 1994). In practice, this function is often estimated by some nonparametric procedure such as kernel smoothing, where the smoothing parameter (i.e. bandwidth) is often determined arbitrarily. In this article, a data-driven method for the selection of the bandwidth is proposed. The efficacy of the proposed approach is studied through both simulations and an application to a forest data example.     

Nonparametric localized bandwidth selection for Kernel density estimation

As conventional cross-validation bandwidth selection methods do not work properly in the situation where the data are serially dependent time series, alternative bandwidth selection methods are necessary. In recent years, Bayesian-based methods for global bandwidth selection have been studied. Our experience shows that a global bandwidth is however less suitable than a localized bandwidth in kernel density estimation based on serially dependent time series data. Nonetheless, a di?cult issue is how we can consistently estimate a localized bandwidth. This paper presents a nonparametric localized bandwidth estimator, for which we establish a completely new asymptotic theory. Applications of this new bandwidth estimator to the kernel density estimation of Eurodollar deposit rate and the S&P 500 daily return demonstrate the effectiveness and competitiveness of the proposed localized bandwidth.     

Accuracy Analysis of the Percentile Method for Estimating Non Normal Manufacturing Quality

We consider asymmetric kernel estimates based on grouped data. We propose an iterated scheme for constructing such an estimator and apply an iterated smoothed bootstrap approach for bandwidth selection. We compare our approach with competing methods in estimating actuarial loss models using both simulations and data studies. The simulation results show that with this new method, the estimated density from grouped data matches the true density more closely than with competing approaches.     

On the estimation of the spectral density for continuous spatial processes

In this work, we study the asymptotic properties of smoothed nonparametric kernel spectral density estimators for the spatial spectral density. We consider the case of continuous stationary spatial processes under a shrinking asymptotic framework. Expressions for the bias and the covariance structure are obtained and the implications for the edge effect bias of the choice of the kernel, bandwidth and spacing parameter in the design are also discussed, both for tapered and untapered estimates. Results are illustrated with a simulation study.     

Density estimation using asymmetric kernels and Bayes bandwidths with censored data

We propose a modification to the regular kernel density estimation method that use asymmetric kernels to circumvent the spill over problem for densities with positive support. First a pivoting method is introduced for placement of the data relative to the kernel function. This yields a strongly consistent density estimator that integrates to one for each fixed bandwidth in contrast to most density estimators based on asymmetric kernels proposed in the literature. Then a data-driven Bayesian local bandwidth selection method is presented and lognormal, gamma, Weibull and inverse Gaussian kernels are discussed as useful special cases. Simulation results and a real-data example illustrate the advantages of the new methodology.