Bootstrapping the mode

The problem of constructing bootstrap confidence intervals for the mode of a density is considered. Estimates of the mode are derived from kernel density estimates based on fixed and data-dependent bandwidths. The asymptotic validity of bootstrap techniques to estimate the sampling distribution of the estimates is investigated. In summary, the results are negative in the sense that a straightforward application of a naive bootstrap yields invalid inferences. In particular, the bootstrap fails if resampling is done from the kernel density estimate. On the other hand, if one resamples from a smoother kernel density estimate (which is necessarily different from the one which yields the original estimate of the mode), the bootstrap is consistent. The bootstrap also fails if resampling is done from the empirical distribution, unless the choice of bandwidth is suboptimal. Similar results hold when applying bootstrap techniques to other functionals of a density.     

方向数据回归函数核估计平均偏差的指数界

本文讨论方向数据回归函数核估计的平均偏差，在关于应变量、核函数及窗宽的温和假定下得到了 这类估计的平均偏差的指数界．     

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In the situation of \rho-mixing dependent sequences, this paper studied the mean square error and the optimal bandwidth of distribution kernel estimator nu_{p,h} of VaR. And the optimal bandwidth minimized the mean square error. The density function of Laplace distribution is used in the calculation of bandwidth and we adopt the method of interpolation to compute specific value of bandwidth in this paper. According to the numerical simulations, the distribution kernel estimator is more accurate by comparing the performance of VaR distribution kernel estimation with a common order statistic. Finally, Shangzheng A-share index and Shenzheng B-share index are chosen for an empirical research, which concludes that the risk of the latter is significantly higher than that of the former.     

Learning mixtures of truncated basis functions from data

In this paper we investigate methods for learning hybrid Bayesian networks from data. First we utilize a kernel density estimate of the data in order to translate the data into a mixture of truncated basis functions (MoTBF) representation using a convex optimization technique. When utilizing a kernel density representation of the data, the estimation method relies on the specification of a kernel bandwidth. We show that in most cases the method is robust wrt. the choice of bandwidth, but for certain data sets the bandwidth has a strong impact on the result. Based on this observation, we propose an alternative learning method that relies on the cumulative distribution function of the data.Empirical results demonstrate the usefulness of the approaches: Even though the methods produce estimators that are slightly poorer than the state of the art (in terms of log-likelihood), they are significantly faster, and therefore indicate that the MoTBF framework can be used for inference and learning in reasonably sized domains. Furthermore, we show how a particular sub-class of MoTBF potentials (learnable by the proposed methods) can be exploited to significantly reduce complexity during inference.     

A bias reducing technique in kernel distribution function estimation

In this paper we suggest a bias reducing technique in kerneldistribution function estimation. In fact, it uses a convex combination of three kernel estimators, and it turned out that the bias has been reduced to the fourth power of the bandwidth, while the bias of the kernel distribution function estimator has the second power of the bandwidth. Also, the variance of the proposed estimator remains at the same order as the kernel distribution function estimator. Numerical results based on simulation studies show this phenomenon, too.     

Boundary kernels for adaptive density estimators on regions with irregular boundaries

In some applications of kernel density estimation the data may have a highly non-uniform distribution and be confined to a compact region. Standard fixed bandwidth density estimates can struggle to cope with the spatially variable smoothing requirements, and will be subject to excessive bias at the boundary of the region. While adaptive kernel estimators can address the first of these issues, the study of boundary kernel methods has been restricted to the fixed bandwidth context. We propose a new linear boundary kernel which reduces the asymptotic order of the bias of an adaptive density estimator at the boundary, and is simple to implement even on an irregular boundary. The properties of this adaptive boundary kernel are examined theoretically. In particular, we demonstrate that the asymptotic performance of the density estimator is maintained when the adaptive bandwidth is defined in terms of a pilot estimate rather than the true underlying density. We examine the performance for finite sample sizes numerically through analysis of simulated and real data sets.     

Nonparametric Density Estimation for a Long-Range Dependent Linear Process

We estimate the marginal density function of a long-range dependent linear process by the kernel estimator. We assume the innovations are i.i.d. Then it is known that the term of the sample mean is dominant in the MISE of the kernel density estimator when the dependence is beyond some level which depends on the bandwidth and that the MISE has asymptotically the same form as for i.i.d. observations when the dependence is below the level. We call the latter the case where the dependence is not very strong and focus on it in this paper. We show that the asymptotic distribution of the kernel density estimator is the same as for i.i.d. observations and the effect of long-range dependence does not appear. In addition we describe some results for weakly dependent linear processes.     

Fourier methods for smooth distribution function estimation

The limit behavior of the optimal bandwidth sequence for the kernel distribution function estimator is analyzed, in its greatest generality, by using Fourier transform methods. We show a class of distributions for which the kernel estimator achieves a first-order improvement in efficiency over the empirical estimator.     

Strong consistency of automatic kernel regression estimates

Regression function estimation from independent and identically distributed bounded data is considered. TheL ₂ error with integration with respect to the design measure is used as an error criterion. It is shown that the kernel regression estimate with an arbitrary random bandwidth is weakly and strongly consistent forall distributions whenever the random bandwidth is chosen from some deterministic interval whose upper and lower bounds satisfy the usual conditions used to prove consistency of the kernel estimate for deterministic bandwidths. Choosing discrete bandwidths by cross-validation allows to weaken the conditions on the bandwidths. Research supported by DAAD, NSERC and Alexander von Humboldt Foundation. The research of the second author was completed during his stay at the Technical University of Szczecin, Poland.     

Rate of convergence of a convolution-type estimator of the marginal density of a MA(1) process

In this paper moving-average processes with no parametric assumption on the error distribution are considered. A new convolution-type estimator of the marginal density of a MA(1) is presented. This estimator is closely related to some previous ones used to estimate the integrated squared density and has a structure similar to the ordinary kernel density estimator. For second-order kernels, the rate of convergence of this new estimator is investigated and the rate of the optimal bandwidth obtained. Under limit conditions on the smoothing parameter the convolution-type estimator is proved to be -consistent, which contrasts with the asymptotic behavior of the ordinary kernel density estimator, that is only -consistent.     

Maximum Kernel Likelihood Estimation

We introduce an estimator for the population mean based on maximizing likelihoods formed by parameterizing a kernel density estimate. Due to these origins, we have dubbed the estimator the maximum kernel likelihood estimate (MKLE). A speedy computational method to compute the MKLE based on binning is implemented in a simulation study which shows that the MKLE at an optimal bandwidth is decidedly superior in terms of efficiency to the sample mean and other measures of location for heavy tailed symmetric distributions. An empirical rule and a computational method to estimate this optimal bandwidth are developed and used to construct bootstrap confidence intervals for the population mean. We show that the intervals have approximately nominal coverage and have significantly smaller average width than the standard t and z intervals. Finally, we develop some mathematical properties for a very close approximation to the MKLE called the kernel mean. In particular, we demonstrate that the kernel mean is indeed unbiased for the population mean for symmetric distributions.     

Real-Time Density and Mode Estimation With Application to Time-Dynamic Mode Tracking

We introduce a nonparametric time-dynamic kernel type density estimate for the situation where an underlying multivariate distribution evolves with time. Based on this time-dynamic density estimate, we propose nonparametric estimates for the time-dynamic mode of the underlying distribution. Our estimators involve boundary kernels for the time dimension so that the estimator is always centered at current time, and multivariate kernels for the spatial dimension of the time-evolving distribution. Under certain mild conditions, the asymptotic behavior of density and mode estimators, especially their uniform convergence in both time and space, is derived. A time-dynamic algorithm for mode tracking is proposed, including automatic bandwidth choices, and is implemented via a mean update algorithm. Simulation studies and real data illustrations demonstrate that the proposed methods work well in practice.     

Bayesian bandwidth estimation for a semi-functional partial linear regression model with unknown error density

In the context of semi-functional partial linear regression model, we study the problem of error density estimation. The unknown error density is approximated by a mixture of Gaussian densities with means being the individual residuals, and variance a constant parameter. This mixture error density has a form of a kernel density estimator of residuals, where the regression function, consisting of parametric and nonparametric components, is estimated by the ordinary least squares and functional Nadaraya–Watson estimators. The estimation accuracy of the ordinary least squares and functional Nadaraya–Watson estimators jointly depends on the same bandwidth parameter. A Bayesian approach is proposed to simultaneously estimate the bandwidths in the kernel-form error density and in the regression function. Under the kernel-form error density, we derive a kernel likelihood and posterior for the bandwidth parameters. For estimating the regression function and error density, a series of simulation studies show that the Bayesian approach yields better accuracy than the benchmark functional cross validation. Illustrated by a spectroscopy data set, we found that the Bayesian approach gives better point forecast accuracy of the regression function than the functional cross validation, and it is capable of producing prediction intervals nonparametrically.     

The asymptotic distributions of kernel estimators of the mode

Summary In a decreasing sequence of intervals centered on the true mode the normalized kernel estimate of the density converges weakly to a nonstationary Gaussian random process. The expected value of this process is a parabola through the origin. The covariance function of this process depends on the smoothness of the kernel. When the kernel is mean-square differentiable the location of the maximum of this process has a normal distribution. When the kernel is discontinuous the location of the maximum has a distribution related to a solution of the heat equation.Research supported in part by the National Science Foundation under grant MCS-78-02422 and MCS-80-05115 to Carnegie-Mellon University     

Optimal kernel estimation of spot volatility of stochastic differential equations

A unified framework to optimally select the bandwidth and kernel function of spot volatility kernel estimators is put forward. The proposed models include not only classical Brownian motion driven dynamics but also volatility processes that are driven by long-memory fractional Brownian motions or other Gaussian processes. We characterize the leading order terms of the mean squared error, which in turn enables us to determine an explicit formula for the leading term of the optimal bandwidth. Central limit theorems for the estimation error are also obtained. A feasible plug-in type bandwidth selection procedure is then proposed, for which, as a sub-problem, a new estimator of the volatility of volatility is developed. The optimal selection of the kernel function is also investigated. For Brownian Motion type volatilities, the optimal kernel turns out to be an exponential function, while, for fractional Brownian motion type volatilities, easily implementable numerical results to compute the optimal kernels are devised. Simulation studies further confirm the good performance of the proposed methods.     

Statistical estimation in varying coefficient models with surrogate data and validation sampling

Varying coefficient error-in-covariables models are considered with surrogate data and validation sampling. Without specifying any error structure equation, two estimators for the coefficient function vector are suggested by using the local linear kernel smoothing technique. The proposed estimators are proved to be asymptotically normal. A bootstrap procedure is suggested to estimate the asymptotic variances. The data-driven bandwidth selection method is discussed. A simulation study is conducted to evaluate the proposed estimating methods.     

Assessing log-concavity of multivariate densities

We develop a test for log-concavity of multivariate densities. The method uses kernel density estimation, where the test statistic is the smallest bandwidth for which the estimate is log-concave. We examine the properties of this technique through numerical studies.     

Estimating a density under pointwise constraints on the derivatives

Suppose we want to estimate a density at a point where we know the values of its first or higher order derivatives. In this case a given kernel estimator of the density can be modified by adding appropriately weighted kernel estimators of these derivatives. We give conditions under which the modified estimators are asymptotically normal. We also determine the optimal weights. When the highest derivative is known to vanish at a point, then the bias is asymptotically negligible at that point and the asymptotic variance of the kernel estimator can be made arbitrarily small by choosing a large bandwidth.     

Local linear hazard rate estimation and bandwidth selection

A new kernel based local linear estimate of the hazard rate, under the random right censorship model is proposed in this article. We study its finite sample and asymptotic properties and prove its asymptotic normality. Then we bring in three popular methods for bandwidth selection to the hazard setting as potential bandwidth choice rules for the estimate. We discuss their practical implementation and through Monte Carlo simulations we use four distributions with different hazard rate shapes to compare their performance over various sample sizes and levels of censoring.     

On the estimation of a monotone conditional variance in nonparametric regression

A monotone estimate of the conditional variance function in a heteroscedastic, nonparametric regression model is proposed. The method is based on the application of a kernel density estimate to an unconstrained estimate of the variance function and yields an estimate of the inverse variance function. The final monotone estimate of the variance function is obtained by an inversion of this function. The method is applicable to a broad class of nonparametric estimates of the conditional variance and particularly attractive to users of conventional kernel methods, because it does not require constrained optimization techniques. The approach is also illustrated by means of a simulation study.