Kernel-Type Estimators of Divergence Measures and Its Strong Uniform Consistency

Abstract

Nonparametric density estimation, based on kernel-type estimators, is a very popular method in statistical research, especially when we want to model the probabilistic or stochastic structure of a data set. In this paper, we investigate the asymptotic confidence bands for the distribution with kernel-estimators for some types of divergence measures (Rényi-α and Tsallis-α divergence). Our aim is to use the method based on empirical process techniques, in order to derive some asymptotic results. Under different assumptions, we establish a variety of fundamental and theoretical properties, such as the strong consistency of an uniform-in-bandwidth of the divergence estimators. We further apply the previous results in simulated examples, including the kernel-type estimator for Hellinger, Bhattacharyya and Kullback-Leibler divergence, to illustrate this approach, and we show that that the method performs competitively.

Nonparametric \(\phi\)-Divergence Estimation and Test for Model Selection

“…We choose two values of π which are π = 0.5 and π = 0.75. Note that the value π = 0.5 is the value for Dhaker et al (2016) showed that the Kullback-Leibler and Hellinger divergences estimators based on the kernel density estimator are strongly consistent estimators. We consider the Kullback-Leibler and the Hellinger divergences based on the bias reduced kernel density estimator ( D KL1 and D H1 ) and we further consider the case of Kullback-Leibler and Hellinger divergences based on the kernel density estimator ( D KL2 and D H2 ).…”

“…Einmahl and Mason (2005) proved the uniform in bandwidth consistency of kernel-type function estimators. Dhaker et al (2016) proposed a strong uniformly consistent kernel-type estimator of divergence measures. Rudemo (1982) and Bowman et al (1984) introduced a convenient method for the choice of optimal bandwidth in practice for kernel density estimator using cross-validation.…”

Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes I

“…Instead, the literature presented us many kinds of results on almost-sure efficiency of the estimation, with rates of convergences and laws of the iterated logarithm, L p (p = 1, 2) convergence, etc. To be precise, [Dhakher et al(2016)] used recent techniques based on functional empirical process to provide a series of interesting rates of convergence of the estimators in the case of one-sided approach for the class de Renyi, Tsallis, Kullback-Leibler to cite a few. Unfortunately, the authors did not address the problem of integrability, taking for granted that the divergence measures are finite.…”