Density estimation with Haar series

1994

The LIand L2-errors of the histogram estimate of a density f from a sample X l r X 2 ,. .. , X , using a cubic partition are shown to be asymptotically normal without any unnecessary conditions imposed on the density f. The asymptotic variances are shown to depend on f only through the corresponding norm off. From this follows the asymptotic null distribution of a goodness-of-fit test based on the total variation distance, introduced by Gyorfi and van der Meulen (1991). This note uses the idea of partial inversion for obtaining characteristic functions of conditional distributions, which goes back at least to Bartlett (1938). RESUME On s'interesse aux erreurs. suivant les normes L , et L2, inherentes a I'estimateur de type histogramme pour une fonction de densite f. ce dernier &ant obtenu i partir d'un Cchantillon X I , X 2 ,. .. ,X,, en utilisant une partition cubique. I1 est dtmontre que ces erreurs poss&dent une distribution asymptotique gaussienne, sans avoir a imposer des conditions superflues sur la fonction de densite f. Les variances asymptotiques ne dependent de f que par l'intermkdiaire de sa norme correspondante. Ceci nous permet d'obtenir la distribution asymptotique, sous I'hypothese nulle, d'une statistique de validitt de I'ajustement fondee sur la variation totale, introduite par Gyorfi et van der Meulen (1991). Cet article fait appel la notion d'inversion partielle, remontant au moins a Bartlett (1 938), afin d'obtenir les fonctions caracttristiques de distributions conditionnelles.

Given a random sample from some unknown density f 0 : R → [0, ∞) we devise Haar wavelet estimators for f 0 with variable resolution levels constructed from localised test procedures (as in Lepski, Mammen, and Spokoiny (1997, Ann. Statist.)). We show that these estimators adapt to spatially heterogeneous smoothness of f 0 , simultaneously for every point x in a fixed interval, in sup-norm loss. The thresholding constants involved in the test procedures can be chosen in practice under the idealised assumption that the true density is locally constant in a neighborhood of the point x of estimation, and an information theoretic justification of this practice is given.

The problem of estimating an unknown probability density function (pdf) is of fundamental importance in statistics and required for many statistical applications. In recent years, efficient nonparametric estimation has had greater focus on the problem of nonparametric regression, while the more challenging problem of density estimation has been given much less attention. In this thesis, we consider a class of kernel-type density estimators with Fejér-type kernels and theoretical smoothing parameters h n = (2γθ n)/ log n, where the parameter γ > 0 describes the class of underlying pdfs and 0 ≤ θ n < 1. In theory, the estimator under consideration dominates in L p , 1 ≤ p < ∞, all other known estimators from the literature in the locally asymptotic minimax (LAM) sense. We demonstrate via simulations that the estimator in question is good by comparing its performance to other fixed kernel estimators. The kernel-type estimator is also studied under empirical bandwidth selection methods such as the common cross-validation and the less-known method based on the Fourier analysis of kernel density estimators. The common L 2-risk is used to assess the quality of estimation. The estimator of interest is then tried to real financial data for a risk measure that is widely used in many applications. The simulation results testify that, for a good choice of γ, the theoretical estimator under study provides very good finite sample performance compared to the other kernel estimators. The study also suggests that the bandwidth obtained by using the Fourier analysis techniques performs better than the one from cross-validation in most settings.

Statistica Neerlandica, 1999

We study piecewise linear density estimators from the L 1 point of view: the frequency polygons investigated by SCOTT (1985) and JONES et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L 1 strongly consistent. We derive large deviation inequalities. For twice dierentiable densities with compact support their expected L 1 error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.

2002

The problem of prior elicitation often arises when one is interested in doing inference in a Bayesian setting. Common solutions to this problem consist of hi- erarchical modelling, noninformative priors, dierent empirical Bayes techniques, etc. An alternative to these solutions might be to estimate the unknown,prior directly from the available observations. In the case of the prior distribution of

2016

Functional Analytic Perspectives on Nonparametric Density Estimation by Robert A. Vandermeulen Chair: Clayton Scott Nonparametric density estimation is a classic problem in statistics. In the standard estimation setting, when one has access to iid samples from an unknown distribution, there exist several established and well-studied nonparametric density estimators. Yet there remains interesting alternative settings which are less well-studied. This work considers two such settings. First we consider the case where the data contains some contamination, i.e. a portion of the data is not distributed according to the density we would like to estimate. In this setting one would like an estimator which is robust to the contaminating data. An approach to this was suggested in Kim and Scott (2012). The estimator in that paper was analytically and experimentally shown to be robust, but no consistency result was presented. In Chapter II it is demonstrated that this estimator is indeed consis...

1975

2016

A histogram estimate of the Radon-Nikodym derivative of a probability measure with respect to a dominating measure is developed for binary sequences in {0, 1} N. A necessary and sufficient condition for the consistency of the estimate in the mean-square sense is given. It is noted that the product topology on {0, 1} N and the corresponding dominating product measure pose considerable restrictions on the rate of sampling required for the requisite convergence.

Statistics & Probability Letters, 2013

A histogram estimate of the Radon-Nikodym derivative of a probability measure with respect to a dominating measure is developed for binary sequences in {0, 1} N. A necessary and sufficient condition for the consistency of the estimate in the mean-square sense is given. It is noted that the product topology on {0, 1} N and the corresponding dominating product measure pose considerable restrictions on the rate of sampling required for the requisite convergence.

International Journal of Mathematical Education in Science and Technology, 1978

The empirical density function, a simple modification and improvement of the usual histogram, is defined and its properties are studied. An analysis is presented which enables the interval width to be chosen. The estimators are modified for the important practical case of bounded random variables. Finally, the problems of writing a programme to compute the functions are considered along with some Monte Carlo examples and a practical example from the National Uranium Resource Evaluation study conducted by the United States Energy Research and Development Administration. It is recommended that these techniques be introduced at all levels of statistical courses so that they will become more widely utilized.