Semiparametric Density Deconvolution

Abstract.  A new semiparametric method for density deconvolution is proposed, based on a model in which only the ratio of the unconvoluted to convoluted densities is specified parametrically. Deconvolution results from reweighting the terms in a standard kernel density estimator, where the weights are defined by the parametric density ratio. We propose that in practice, the density ratio be modelled on the log-scale as a cubic spline with a fixed number of knots. Parameter estimation is based on maximization of a type of semiparametric likelihood. The resulting asymptotic properties for our deconvolution estimator mirror the convergence rates in standard density estimation without measurement error when attention is restricted to our semiparametric class of densities. Furthermore, numerical studies indicate that for practical sample sizes our weighted kernel estimator can provide better results than the classical non-parametric kernel estimator for a range of densities outside the specified semiparametric class.     

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.     

Kernel Density Estimation with Generalized Binning

We propose kernel density estimators based on prebinned data. We use generalized binning schemes based on the quantiles points of a certain auxiliary distribution function. Therein the uniform distribution corresponds to usual binning. The statistical accuracy of the resulting kernel estimators is studied, i.e. we derive mean squared error results for the closeness of these estimators to both the true function and the kernel estimator based on the original data set. Our results show the influence of the choice of the auxiliary density on the binned kernel estimators and they reveal that non-uniform binning can be worthwhile.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.     

Adaptive Nonparametric Comparison of Regression Curves

In this article, we develop a new and novel kernel density estimator for a sum of weighted averages from a single population based on utilizing the well defined kernel density estimator in conjunction with classic inversion theory. This idea is further developed for a kernel density estimator for the difference of weighed averages from two independent populations. The resulting estimator is “bootstrap-like” in terms of its properties with respect to the derivation of approximate confidence intervals via a “plug-in” approach. This new approach is distinct from the bootstrap methodology in that it is analytically and computationally feasible to provide an exact estimate of the distribution function through direct calculation. Thus, our approach eliminates the error due to Monte Carlo resampling that arises within the context of simulation based approaches that are oftentimes necessary in order to derive bootstrap-based confidence intervals for statistics involving weighted averages of i.i.d. random variables. We provide several examples and carry forth a simulation study to show that our kernel density estimator performs better than the standard central limit theorem based approximation in term of coverage probability.     

Large deviations for the L1-distance in kernel density estimation

We establish a large deviation limit theorem of Chernoff type for the L₁-distance between the nonparametric kernel density estimator and the underlying density. The estimation is based on a sequence of independent and identically distributed random variables. The rate function is well identified, distribution-free and independent of the choice of the kernel.     

Root n consistent and optimal density estimators for moving average processes

Abstract.  The marginal density of a first order moving average process can be written as a convolution of two innovation densities. Saavedra & Cao [Can. J. Statist. (2000), 28, 799] propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the variance of their estimator decreases at the rate 1/ n . Their estimator can be interpreted as a specific U -statistic. We suggest a slightly simplified U -statistic as estimator of the marginal density, prove that it is asymptotically normal at the same rate, and describe the asymptotic variance explicitly. We show that the estimator is asymptotically efficient if no structural assumptions are made on the innovation density. For innovation densities known to have mean zero or to be symmetric, we describe improvements of our estimator which are again asymptotically efficient.     

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.     

On the asymptotic normality of the kernel estimators of the density function and its derivatives under censoring

In this paper, we study asymptotic normality of the kernel estimators of the density function and its derivatives as well as the mode in the randomly right censorship model. The mode estimator is defined as the random variable that maximizes the kernel density estimator. Our results are stated under some suitable conditions upon the kernel function, the smoothing parameter and both distributions functions that appear in this model. Here, the Kaplan–Meier estimator of the distribution function is used to build the estimates. We carry out a simulation study which shows how good the normality works.     

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.     

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.     

Weak and strong uniform consistency of a kernel error density estimator in nonparametric regression

This paper considers the problem of estimating the kernel error density function in nonparametric regression models. Sufficient conditions are given under which the kernel error density estimator based on nonparametric residuals is uniformly weakly and strongly consistent.     

A simple,automatic and adaptive bivariate density estimator based on conditional densities

The standard approach to non-parametric bivariate density estimation is to use a kernel density estimator. Practical performance of this estimator is hindered by the fact that the estimator is not adaptive (in the sense that the level of smoothing is not sensitive to local properties of the density). In this paper a simple, automatic and adaptive bivariate density estimator is proposed based on the estimation of marginal and conditional densities. Asymptotic properties of the estimator are examined, and guidance to practical application of the method is given. Application to two examples illustrates the usefulness of the estimator as an exploratory tool, particularly in situations where the local behaviour of the density varies widely. The proposed estimator is also appropriate for use as a pilot estimate for an adaptive kernel estimate, since it is relatively inexpensive to calculate.     

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.     

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.     

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.     

A Semiparametric Model for Line Transect Sampling

This paper introduces an appealing semiparametric model for estimating wildlife abundance based on line transect data. The proposed method requires the existence of a parametric model and then improves the estimator using a kernel method. Properties of the resultant estimator are derived and an expression for the asymptotic mean square error (AMSE) of the estimator is given. Minimization of the AMSE leads to an explicit formula for an optimal choice of the smoothing parameter. Small-sample properties of the proposed estimator using the parametric half-normal model are investigated and compared with the classical kernel estimator using both simulations and real data. Numerical results show that improvements over the classical kernel estimator often can be realized even when the true density is far from the half-normal model.     

Smooth estimation of a monotone density

We investigate the interplay of smoothness and monotonicity assumptions when estimating a density from a sample of observations. The nonparametric maximum likelihood estimator of a decreasing density on the positive half line attains a rate of convergence of [Formula: See Text] at a fixed point t if the density has a negative derivative at t. The same rate is obtained by a kernel estimator of bandwidth [Formula: See Text], but the limit distributions are different. If the density is both differentiable at t and known to be monotone, then a third estimator is obtained by isotonization of a kernel estimator. We show that this again attains the rate of convergence [Formula: See Text], and compare the limit distributions of the three types of estimators. It is shown that both isotonization and smoothing lead to a more concentrated limit distribution and we study the dependence on the proportionality constant in the bandwidth. We also show that isotonization does not change the limit behaviour of a kernel estimator with a bandwidth larger than [Formula: See Text], in the case that the density is known to have more than one derivative.     

Estimation de densités unimodales

This paper proposes a new nonparametric unimodal estimator of a unimodal probability density function, in the case where the mode is known. The classical solution to this problem is the maximum-likelihood estimator under monotonicity constraint, considered by Grenander (1956). Our approach is based on a unimodal rearrangement of the kernel estimator of the density. Asymptotic properties of this estimator are studied, and its small-sample behaviour is examined through simulations.     

Asymptotic Distributions of Innovation Density Estimators in Linear Processes

In this article, we demonstrate that at a fixed point, the asymptotic distribution of the innovation density estimator is normal for stationary linear process. Also, we show that the asymptotic distribution of the global measure of the deviation of the density estimator from the expectation of the kernel innovation density (based on the true innovations) is the same as that in the case when we can observe the true innovations.