Nonnegative piecewise linear histograms *

We introduce a modified version ?_nof the piecewiss linear hisiugrimi uf Beirlant et al. (1998) which is a true probability density, i.e., ?_n[d] 0 and [d]?_n=1. We prove that ?_nestimates the underlying densitv ? strongly consistently in the L₁mmn, derive large deviation inequalities for the t\ error \?_n- f\ and prove that £||/"-/|| tends to zero with the rate n ^-1\3, We also show that the derivative lf'n estimates consistently in ine expected Lx error the derivative/ of sufficiently smooth density and evaluate the rate of convergence n-i/5 for Epf'n -f'% The estimator/" thus enables to approximate/in the Besov space with a guaranteed rate of convergence. Optimization of the smoothing parameter is also studied. The theoretical or experimentally approximated values of the expected errors E\\?_n- f\\ and E||2?'_n-?' are compared with tiie errors aCiiieveu u-y t"e histogram of Beirlant et ah, and other nonparametric methods.     

The exact solution to the mle of a decreasing density based on arbitrarily right censored data

Nonparametric maximum likelihood estimation of decreasing and unimodal density functions, based on observations subject to arbitrary right censorship, was considered by McNichols and Padgett(1982). In order to compute their estimators, however, nonlinear equations with linear constraints had to be maximized using numerical techniques. The exact solution to this problem can now be found. An example illustrates the simplicity of the method.     

Kernel density estimation with binned data

Continuous data are often measured or used in binned or rounded form. In this paper we follow up on Hall's work analyzing the effect of using equally-spaced binned data in a kernel density estimator. It is shown that a surprisingly large amount of binning does not adversely affect the integrated mean squared error of a kernel estimate.     

Interpreting the Principal Component Analysis of Multivariate Density Functions

Functional principal component analysis (FPCA) as a reduction data technique of a finite number T of functions can be used to identify the dominant modes of variation of numeric three-way data.

We carry out the FPCA on multidimensional probability density functions, relate this method to other standard methods and define its centered or standardized versions. Grounded on the relationship between FPCA of densities, FPCA of their corresponding characteristic functions, PCA of the MacLaurin expansions of these characteristic functions and dual STATIS method applied to their variance matrices, we propose a method for interpreting the results of the FPCA of densities. This method is based on the investigations of the relationships between the scores of the FPCA and the moments associated to the densities.

The method is illustrated using known Gaussian densities. In practice, FPCA of densities deals with observations of multidimensional variables on T occasions. These observations can be used to estimate the T associated densities (i) by estimating the parameters of these densities, assuming that they are Gaussian, or (ii) by using the Gaussian kernel method and choosing the matrix bandwidth by the normal reference rule. Thereafter, FPCA estimate is derived from these estimates and the interpretation method is carried out to explore the dominant modes of variation of the types of three-way data encountered in sensory analysis and archaeology.     

Dot Plots

Dot plots represent individual observations in a batch of data with symbols, usually circular dots. They have been used for more than 100 years to depict distributions in detail. Hand-drawn examples show their authors' efforts to arrange symbols so that they are as near as possible to their proper locations on a scale without overlapping enough to obscure each other. Recent computer programs that attempt to reproduce these historical plots have unfortunately resorted to simple histogram binning instead of using methods that follow the rules for the hand-drawn examples. This article introduces an algorithm that more accurately represents the dot plots cited in the literature.     

Semiparametric ARCH Models

This article introduces a semiparametric autoregressive conditional heteroscedasticity (ARCH) model that has conditional first and second moments given by autoregressive moving average and ARCH parametric formulations but a conditional density that is assumed only to be sufficiently smooth to be approximated by a nonparametric density estimator. For several particular conditional densities, the relative efficiency of the quasi-maximum likelihood estimator is compared with maximum likelihood under correct specification. These potential efficiency gains for a fully adaptive procedure are compared in a Monte Carlo experiment with the observed gains from using the proposed semiparametric procedure, and it is found that the estimator captures a substantial proportion of the potential. The estimator is applied to daily stock returns from small firms that are found to exhibit conditional skewness and kurtosis and to the British pound to dollar exchange rate.     

NONPARAMETRIC ESTIMATION OF ANGULAR DISTRIBUTIONS FROM CARTESIAN OBSERVATIONS WITH ERROR

If angular data are obtained from Cartesian observations, then any measurement error in these observations will produce a particular error structure in the angular data. The paper shows how non-parametric density estimation by orthogonal series may be performed in this case.     

An optimal variable cell histogram

A simple procedure for specifying a histogram with variable cell sizes is proposed. The procedure chooses a set of cutpoints that maximizes a criterion function based on the sample spacings:Under some conditions, this estimated set of cutpoints is shown to converge in probability to the theoretical set of cutpoints for the histogram estimate that minimizes the Hellingerdistance to the underlying density. An algorithm for finding the set of cutpoints that numerically maximizes the criterion function is presented along with an example. Performance for finite sample sizes is evaluated by simulations.     

Bayesian optimal sequential design for nonparametric regression via inhomogeneous evolutionary MCMC

We develop a novel computational methodology for Bayesian optimal sequential design for nonparametric regression. This computational methodology, that we call inhomogeneous evolutionary Markov chain Monte Carlo, combines ideas of simulated annealing, genetic or evolutionary algorithms, and Markov chain Monte Carlo. Our framework allows optimality criteria with general utility functions and general classes of priors for the underlying regression function. We illustrate the usefulness of our novel methodology with applications to experimental design for nonparametric function estimation using Gaussian process priors and free-knot cubic splines priors.     

Fourier series methods in nonparametric estimation

Two different approaches to the design of optimal observations networks are compared. One approach is based on the traditional experimental design theory, the other essentially uses the covariance analysis methodology of observed fields, It is found that for random fields generated by regression models with random parameters both approaches lead to similar solutions     

Shape constrained kernel density estimation

In this paper, a method for estimating monotone, convex and log-concave densities is proposed. The estimation procedure consists of an unconstrained kernel estimator which is modified in a second step with respect to the desired shape constraint by using monotone rearrangements. It is shown that the resulting estimate is a density itself and shares the asymptotic properties of the unconstrained estimate. A short simulation study shows the finite sample behavior.     

Bewertung von sheätzungen bei einem linearen modell auf der grundlage des fiduzialkonzepts

Usingf fiducial inference the quality of estimations is investigated for a linear model under the conditionj that information about the vector to be estimated is derived from a small sample. By application of that model to the deduction of atmospheric temperature profiles the obtained results are illustrated numerically.     

A Note on Asymptotic Behavior of the Nonparametric Density Estimators in Multivariate Mixtures

The asymptotic behavior of the nonparametric density estimator has been given for a multivariate mixture model. It has been observed that the estimator is asymptotically normally distributed with bias of size h ² and variance of size (nh)^?1.     

Posterior consistency of logistic Gaussian process priors in density estimation

We establish weak and strong posterior consistency of Gaussian process priors studied by Lenk [1988. The logistic normal distribution for Bayesian, nonparametric, predictive densities. J. Amer. Statist. Assoc. 83 (402), 509–516] for density estimation. Weak consistency is related to the support of a Gaussian process in the sup-norm topology which is explicitly identified for many covariance kernels. In fact we show that this support is the space of all continuous functions when the usual covariance kernels are chosen and an appropriate prior is used on the smoothing parameters of the covariance kernel. We then show that a large class of Gaussian process priors achieve weak as well as strong posterior consistency (under some regularity conditions) at true densities that are either continuous or piecewise continuous.     

Nonparametric estimation of the mode of a distribution of random curves

Motivated by the need to develop meaningful empirical approximations to a 'typical' data value, we introduce methods for density and mode estimation when data are in the form of random curves. Our approach is based on finite dimensional approximations via generalized Fourier expansions on an empirically chosen basis. The mode estimation problem is reduced to a problem of kernel-type multivariate estimation from vector data and is solved using a new recursive algorithm for finding the empirical mode. The algorithm may be used as an aid to the identification of clusters in a set of data curves. Bootstrap methods are employed to select the bandwidth.     

Nonparametric estimation of a regression function from backward recurrence times in a cross-sectional sampling

This study considers the nonparametric estimation of a regression function when the response variable is the waiting time between two consecutive events of a stationary renewal process, and where this variable is not completely observed. In these circumstances, our data are the recurrence times from the occurrence of the last event up to a pre-established time, along with the corresponding values of a certain set of covariates. Estimation of the error density function and some of its characteristics are also considered. For the proposed estimators, we first analyze their asymptotic behavior and, thereafter, carry out a simulation study to highlight their behavior in finite samples. Finally, we apply this methodology to an illustrative example with biomedical data.     

Clustering via nonparametric density estimation

Although Hartigan (1975) had already put forward the idea of connecting identification of subpopulations with regions with high density of the underlying probability distribution, the actual development of methods for cluster analysis has largely shifted towards other directions, for computational convenience. Current computational resources allow us to reconsider this formulation and to develop clustering techniques directly in order to identify local modes of the density. Given a set of observations, a nonparametric estimate of the underlying density function is constructed, and subsets of points with high density are formed through suitable manipulation of the associated Delaunay triangulation. The method is illustrated with some numerical examples.     

Deconvolution of supersmooth densities with smooth noise

The author considers the estimation of the common probability density of independent and identically distributed random variables observed with added white noise. She assumes that the unknown density belongs to some class of supersmooth functions, and that the error distribution is ordinarily smooth, meaning that its characteristic function decays polynomially asymptotically. In this context, the author evaluates the minimax rate of convergence of the pointwise risk and describes a kernel estimator having this rate. She computes upper bounds for the L₂ risk of this estimator.     

Estimating the Cost of a Warranty

The cost of certain types of warranties is closely related to functions that arise in renewal theory. The problem of estimating the warranty cost for a random sample of size n can be reduced to estimating these functions. In an earlier paper, I gave several methods of estimating the expected number of renewals, called the renewal function. This answered an important accounting question of how to arrive at a good approximation of the expected warranty cost. In this article, estimation of the renewal function is reviewed and several extensions are given. In particular, a resampling estimator of the renewal function is introduced. Further, I argue that managers may wish to examine other summary measures of the warranty cost, in particular the variability. To estimate this variability, I introduce estimators, both parametric and nonparametric, of the variance associated with the number of renewals. Several numerical examples are provided.