We propose a method for nonparametric density estimation that exhibits robustness to contamination of the training sample. This method achieves robustness by combining a traditional kernel density estimator (KDE) with ideas from classical $M$-estimation. We interpret the KDE based on a radial, positive semi-definite kernel as a sample mean in the associated reproducing kernel Hilbert space. Since the sample mean is sensitive to outliers, we estimate it robustly via $M$-estimation, yielding a robust kernel density estimator (RKDE).
An RKDE can be computed efficiently via a kernelized iteratively re-weighted least squares (IRWLS) algorithm. Necessary and sufficient conditions are given for kernelized IRWLS to converge to the global minimizer of the $M$-estimator objective function. The robustness of the RKDE is demonstrated with a representer theorem, the influence function, and experimental results for density estimation and anomaly detection.
Hyogo University of Health Sciences（兵库医疗大学）；Chiba University（千叶大学）
This study proposes multivariate kernel density estimation by stagewise minimization algorithm based on $U$-divergence and a simple dictionary. The dictionary consists of an appropriate scalar bandwidth matrix and a part of the original data. The resulting estimator brings us data-adaptive weighting parameters and bandwidth matrices, and realizes a sparse representation of kernel density estimation. We develop the non-asymptotic error bound of estimator obtained via the proposed stagewise minimization algorithm. It is confirmed from simulation studies that the proposed estimator performs compe…
Current meta-learning approaches focus on learning functional representations of relationships between variables, i.e. on estimating conditional expectations in regression. In many applications, however, we are faced with conditional distributions which cannot be meaningfully summarized using expectation only (due to e.g. multimodality). Hence, we consider the problem of conditional density estimation in the meta-learning setting. We introduce a novel technique for meta-learning which combines neural representation and noise-contrastive estimation with the established literature of conditional…
Consistency of the kernel density estimator requires that the kernel bandwidth tends to zero as the sample size grows. In this paper we investigate the question of whether consistency is possible when the bandwidth is fixed, if we consider a more general class of weighted KDEs. To answer this question in the affirmative, we introduce the fixed-bandwidth KDE (fbKDE), obtained by solving a quadratic program, and prove that it consistently estimates any continuous square-integrable density. We also establish rates of convergence for the fbKDE with radial kernels and the box kernel under appropria…
Conditional density estimation is a general framework for solving various problems in machine learning. Among existing methods, non-parametric and/or kernel-based methods are often difficult to use on large datasets, while methods based on neural networks usually make restrictive parametric assumptions on the probability densities. Here, we propose a novel method for estimating the conditional density based on score matching. In contrast to existing methods, we employ scalable neural networks, but do not make explicit parametric assumptions on densities. The key challenge in applying score mat…
We introduce a balloon estimator in a generalized expectation-maximization method for estimating all parameters of a Gaussian mixture model given one data sample per mixture component. Instead of limiting explicitly the model size, this regularization strategy yields low-complexity sparse models where the number of effective mixture components reduces with an increase of a smoothing probability parameter $\mathbf{P>0}$. This semi-parametric method bridges from non-parametric adaptive kernel density estimation (KDE) to parametric ordinary least-squares when $\mathbf{P=1}$. Experiments show that…
Multi-dimensional Hawkes process (MHP) is a class of self and mutually exciting point processes that find wide range of applications -- from prediction of earthquakes to modelling of order books in high frequency trading. This paper makes two major contributions, we first find an unbiased estimator for the log-likelihood estimator of the Hawkes process to enable efficient use of the stochastic gradient descent method for maximum likelihood estimation. The second contribution is, we propose a specific single hidden layered neural network for the non-parametric estimation of the underlying kerne…
Allthough nonparametric kernel density estimation with bias reduce is nowadays a standard technique in explorative data-analysis, there is still a big dispute on how to assess the quality of the estimate and which choice of bandwidth is optimal. This article examines the most important bandwidth selection methods for kernel density estimation with bias reduce, in particular, normal reference, least squares cross-validation, biased crossvalidation and $β$-Divergence loss. Methods are described and expressions are presented. We will compare these various bandwidth selector on simulated data. As …
An offshore wind turbine needs to withstand the environmental loads, which can be expected during its life time. Consequently, designers must define loads based on extreme environmental conditions to verify structural integrity. The environmental contour method is an approach to systematically derive these extreme environmental design conditions. The method needs a probability density function as its input. Here we propose the use of constant bandwidth kernel density estimation to derive the joint probability density function of significant wave height and wind speed. We compare kernel density…
We derive the optimal signed variable in general case kernels for the classical statistic density estimation, which are some generalization of the famous Epanechnikov's ones.
Chia and Nakano (2009) introduced the concept of M-decomposability of probability densities in one-dimension. In this paper, we generalize M-decomposability to any dimension. We prove that all elliptical unimodal densities are M-undecomposable. We also derive an inequality to show that it is better to represent an M-decomposable density via a mixture of unimodal densities. Finally, we demonstrate the application of M-decomposability to clustering and kernel density estimation, using real and simulated data. Our results show that M-decomposability can be used as a non-parametric criterion to lo…
We present a modified version of the non parametric Hawkes kernel estimation procedure studied in arXiv:1401.0903 that is adapted to slowly decreasing kernels. We show on numerical simulations involving a reasonable number of events that this method allows us to estimate faithfully a power-law decreasing kernel over at least 6 decades. We then propose a 8-dimensional Hawkes model for all events associated with the first level of some asset order book. Applying our estimation procedure to this model, allows us to uncover the main properties of the coupled dynamics of trade, limit and cancel ord…
This study proposes a data condensation method for multivariate kernel density estimation by genetic algorithm. First, our proposed algorithm generates multiple subsamples of a given size with replacement from the original sample. The subsamples and their constituting data points are regarded as $\it{chromosome}$ and $\it{gene}$, respectively, in the terminology of genetic algorithm. Second, each pair of subsamples breeds two new subsamples, where each data point faces either $\it{crossover}$, $\it{mutation}$, or $\it{reproduction}$ with a certain probability. The dominant subsamples in terms …
Nonparametric density estimation is of great importance when econometricians want to model the probabilistic or stochastic structure of a data set. This comprehensive review summarizes the most important theoretical aspects of kernel density estimation and provides an extensive description of classical and modern data analytic methods to compute the smoothing parameter. Throughout the text, several references can be found to the most up-to-date and cut point research approaches in this area, while econometric data sets are analyzed as examples. Lastly, we present SIZer, a new approach introduc…
We review recent advances in modal regression studies using kernel density estimation. Modal regression is an alternative approach for investigating relationship between a response variable and its covariates. Specifically, modal regression summarizes the interactions between the response variable and covariates using the conditional mode or local modes. We first describe the underlying model of modal regression and its estimators based on kernel density estimation. We then review the asymptotic properties of the estimators and strategies for choosing the smoothing bandwidth. We also discuss u…
In the this paper, the authors propose to estimate the density of a targeted population with a weighted kernel density estimator (wKDE) based on a weighted sample. Bandwidth selection for wKDE is discussed. Three mean integrated squared error based bandwidth estimators are introduced and their performance is illustrated via Monte Carlo simulation. The least-squares cross-validation method and the adaptive weight kernel density estimator are also studied. The authors also consider the boundary problem for interval bounded data and apply the new method to a real data set subject to informative c…
A nonparametric kernel density estimator for directional-linear data is introduced. The proposal is based on a product kernel accounting for the different nature of both (directional and linear) components of the random vector. Expressions for bias, variance and Mean Integrated Squared Error (MISE) are derived, jointly with an asymptotic normality result for the proposed estimator. For some particular distributions, an explicit formula for the MISE is obtained and compared with its asymptotic version, both for directional and directional-linear kernel density estimators. In this same setting a…
New bandwidth selectors for kernel density estimation with directional data are presented in this work. These selectors are based on asymptotic and exact error expressions for the kernel density estimator combined with mixtures of von Mises distributions. The performance of the proposed selectors is investigated in a simulation study and compared with other existing rules for a large variety of directional scenarios, sample sizes and dimensions. The selector based on the exact error expression turns out to have the best behaviour of the studied selectors for almost all the situations. This sel…
In this note we provide a simple approximation theory motivation for the circular kernel density estimation and further explore the usefulness of the wrapped Cauchy kernel in this context. It is seen that the wrapped Cauchy kernel appears as a natural candidate in connection to orthogonal series density estimation on a unit circle. This adds further weight to the considerable role of the wrapped Cauchy in circular statistics.
This paper considers the partially functional linear model (PFLM) where all predictive features consist of a functional covariate and a high dimensional scalar vector. Over an infinite dimensional reproducing kernel Hilbert space, the proposed estimation for PFLM is a least square approach with two mixed regularizations of a function-norm and an $\ell_1$-norm. Our main task in this paper is to establish the minimax rates for PFLM under high dimensional setting, and the optimal minimax rates of estimation is established by using various techniques in empirical process theory for analyzing kerne…
We investigate the statistical complexity of estimating the parameters of a discrete-state Markov chain kernel from a single long sequence of state observations. In the finite case, we characterize (modulo logarithmic factors) the minimax sample complexity of estimation with respect to the operator infinity norm, while in the countably infinite case, we analyze the problem with respect to a natural entry-wise norm derived from total variation. We show that in both cases, the sample complexity is governed by the mixing properties of the unknown chain, for which, in the finite-state case, there …
A new data-based smoothing parameter for circular kernel density (and its derivatives) estimation is proposed. Following the plug-in ideas, unknown quantities on an optimal smoothing parameter are replaced by suitable estimates. This paper provides a circular version of the well-known Sheather and Jones bandwidths (DOI: 10.1111/j.2517-6161.1991.tb01857.x), with direct and solve-the-equation plug-in rules. Theoretical support for our developments, related to the asymptotic mean squared error of the estimator of the density, its derivatives, and its functionals, for circular data, are provided. …
Kernel Density Estimation is a very popular technique of approximating a density function from samples. The accuracy is generally well-understood and depends, roughly speaking, on the kernel decay and local smoothness of the true density. However concrete statements in the literature are often invoked in very specific settings (simplified or overly conservative assumptions) or miss important but subtle points (e.g. it is common to heuristically apply Taylor's expansion globally without referring to compactness). The contribution of this paper is twofold (a) we demonstrate that, when the bandwi…
This manuscript is a supplemental document providing the omitted material for our paper entitled "Nonparametric kernel estimation of the probability density function of regression errors using estimated residuals" [arXiv:1010.0439]. The paper is submitted to Journal of Nonparametric Statistics.
A hybrid estimator of the log-spectral density of a stationary time series is proposed. First, a multiple taper estimate is performed, followed by kernel smoothing the log-multitaper estimate. This procedure reduces the expected mean square error by $({π^2 \over 4})^{.8}$ over simply smoothing the log tapered periodogram. The optimal number of tapers is $O(N^{8/15})$. A data adaptive implementation of a variable bandwidth kernel smoother is given. When the spectral density is discontinuous, one sided smoothing estimates are used.
In this paper, we introduce a robust nonparametric density estimator combining the popular Kernel Density Estimation method and the Median-of-Means principle (MoM-KDE). This estimator is shown to achieve robustness to any kind of anomalous data, even in the case of adversarial contamination. In particular, while previous works only prove consistency results under known contamination model, this work provides finite-sample high-probability error-bounds without a priori knowledge on the outliers. Finally, when compared with other robust kernel estimators, we show that MoM-KDE achieves competitiv…
We present a new method for estimating the frontier of a multidimensional sample. The estimator is based on a kernel regression on the power-transformed data. We assume that the exponent of the transformation goes to infinity while the bandwidth of the kernel goes to zero. We give conditions on these two parameters to obtain complete convergence and asymptotic normality. The good performance of the estimator is illustrated on some finite sample situations.
We provide new general kernel selection rules thanks to penalized least-squares criteria. We derive optimal oracle inequalities using adequate concentration tools. We also investigate the problem of minimal penalty as described in [BM07].
We focus on the nonparametric density estimation problem with directional data. We propose a new rule for bandwidth selection for kernel density estimation. Our procedure is automatic, fully data-driven and adaptive to the smoothness degree of the density. We obtain an oracle inequality and optimal rates of convergence for the L2 error. Our theoretical results are illustrated with simulations.
In this paper we propose a variable bandwidth kernel regression estimator for $i.i.d.$ observations in $\mathbb{R}^2$ to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of $O(h_n^4)$ under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kerne…
This paper introduces the kernel mixture network, a new method for nonparametric estimation of conditional probability densities using neural networks. We model arbitrarily complex conditional densities as linear combinations of a family of kernel functions centered at a subset of training points. The weights are determined by the outer layer of a deep neural network, trained by minimizing the negative log likelihood. This generalizes the popular quantized softmax approach, which can be seen as a kernel mixture network with square and non-overlapping kernels. We test the performance of our met…
Length-biased data are a particular case of weighted data, which arise in many situations: biomedicine, quality control or epidemiology among others. In this paper we study the theoretical properties of kernel density estimation in the context of length-biased data, proposing two consistent bootstrap methods that we use for bandwidth selection. Apart from the bootstrap bandwidth selectors we suggest a rule-of-thumb. These bandwidth selection proposals are compared with a least-squares cross-validation method. A simulation study is accomplished to understand the behaviour of the procedures in f…
Approximate Shapley value estimation using sampling without replacement and variance estimation via the new Symmetric bootstrap and the Doubled half bootstrap
Renewable energy is essential for energy security and global warming mitigation. However, power generation from renewable energy sources is uncertain due to volatile weather conditions and complex equipment operations. To improve equipment's operation efficiency, it is important to understand and characterize the uncertainty in renewable power generation. In this paper, we proposed a conditional kernel density estimation method to model the distribution of equipment's power output given any weather conditions. It explicitly accounts for the temporal dependence in the data stream and uses an it…
Given a sample $\{X_i\}_{i=1}^n$ from $f_X$, we construct kernel density estimators for $f_Y$, the convolution of $f_X$ with a known error density $f_ε$. This problem is known as density estimation with Berkson error and has applications in epidemiology and astronomy. Little is understood about bandwidth selection for Berkson density estimation. We compare three approaches to selecting the bandwidth both asymptotically, using large sample approximations to the MISE, and at finite samples, using simulations. Our results highlight the relationship between the structure of the error $f_ε$ and the…
The ratio of two probability densities can be used for solving various machine learning tasks such as covariate shift adaptation (importance sampling), outlier detection (likelihood-ratio test), and feature selection (mutual information). Recently, several methods of directly estimating the density ratio have been developed, e.g., kernel mean matching, maximum likelihood density ratio estimation, and least-squares density ratio fitting. In this paper, we consider a kernelized variant of the least-squares method and investigate its theoretical properties from the viewpoint of the condition numb…
Kernel density estimation is a well known method involving a smoothing parameter (the bandwidth) that needs to be tuned by the user. Although this method has been widely used the bandwidth selection remains a challenging issue in terms of balancing algorithmic performance and statistical relevance. The purpose of this paper is to compare a recently developped bandwidth selection method for kernel density estimation to those which are commonly used by now (at least those which are implemented in the R-package). This new method is called Penalized Comparison to Overfitting (PCO). It has been pro…
This tutorial provides a gentle introduction to kernel density estimation (KDE) and recent advances regarding confidence bands and geometric/topological features. We begin with a discussion of basic properties of KDE: the convergence rate under various metrics, density derivative estimation, and bandwidth selection. Then, we introduce common approaches to the construction of confidence intervals/bands, and we discuss how to handle bias. Next, we talk about recent advances in the inference of geometric and topological features of a density function using KDE. Finally, we illustrate how one can …
Recent contributions to kernel smoothing show that the performance of cross-validated bandwidth selectors improve significantly from indirectness. Indirect crossvalidation first estimates the classical cross-validated bandwidth from a more rough and difficult smoothing problem than the original one and then rescales this indirect bandwidth to become a bandwidth of the original problem. The motivation for this approach comes from the observation that classical crossvalidation tends to work better when the smoothing problem is difficult. In this paper we find that the performance of indirect cro…
We consider the problem of estimating the s-th derivative of a density function f by the tilted Kernel estimator introduced in Hall and Doosti (2012). Then we further show this estimator achieves the same convergence rate, in probability, the wavelet estimators achieved as shown in Hall and Patil (1995).
It is a common practice to evaluate probability density function or matter spatial density function from statistical samples. Kernel density estimation is a frequently used method, but to select an optimal bandwidth of kernel estimation, which is completely based on data samples, is a long-term issue that has not been well settled so far. There exist analytic formulae of optimal kernel bandwidth, but they cannot be applied directly to data samples, since they depend on the unknown underlying density functions from which the samples are drawn. In this work, we devise an approach to pick out the…
We consider the problem of estimation of a bivariate density function with support $\Re\times[0,\infty)$, where a classical bivariate kernel estimator causes boundary bias due to the non-negative variable. To overcome this problem, we propose four kernel density estimators and compare their performances in terms of the mean integrated squared error. Simulation study shows that the estimator based on the proposed normal-gamma kernel performs best. Two astronomical data sets are used to demonstrate the applicability of this estimator.
This paper presents new methodology for computationally efficient kernel density estimation. It is shown that a large class of kernels allows for exact evaluation of the density estimates using simple recursions. The same methodology can be used to compute density derivative estimates exactly. Given an ordered sample the computational complexity is linear in the sample size. Combining the proposed methodology with existing approximation methods results in extremely fast density estimation. Extensive experimentation documents the effectiveness and efficiency of this approach compared with the e…
We define a new bandwidth-dependent kernel density estimator that improves existing convergence rates for the bias, and preserves that of the variation, when the error is measured in $L_1$. No additional assumptions are imposed to the extant literature.
This paper deals with the nonparametric density estimation of the regression error term assuming its independence with the covariate. The difference between the feasible estimator which uses the estimated residuals and the unfeasible one using the true residuals is studied. An optimal choice of the bandwidth used to estimate the residuals is given. We also study the asymptotic normality of the feasible kernel estimator and its rate-optimality.
We first revisit the problem of estimating the spot volatility of an Itô semimartingale using a kernel estimator. We prove a Central Limit Theorem with optimal convergence rate for a general two-sided kernel. Next, we introduce a new pre-averaging/kernel estimator for spot volatility to handle the microstructure noise of ultra high-frequency observations. We prove a Central Limit Theorem for the estimation error with an optimal rate and study the optimal selection of the bandwidth and kernel functions. We show that the pre-averaging/kernel estimator's asymptotic variance is minimal for exponen…
In this paper, nonparametric estimation of the conditional Weibull-tail coefficient when the variable of interest is right random censored is addressed. A Weissman-type estimator of conditional extreme quantile is also proposed. In addition, a simulation study is conducted to assess the finite-sample behavior of the proposed estimators and a comparison with alternative strategies is provided. Finally, the practical applicability of the methodology is presented using a real datasets of men suffering from a larynx cancer.
The purpose of this note is to provide an approximation for the generalized bootstrapped empirical process achieving the rate in Kolmos et al. (1975). The proof is based on much the same arguments as in Horvath et al. (2000). As a consequence, we establish an approximation of the bootstrapped kernel-type density estimator
Spectral Equivariance and Geometric Transport in Reproducing Kernel Hilbert Spaces: A Unified Framework for Orthogonal Polynomial and Kernel Estimation
