Statistical estimation of quadratic Rényi entropy for a stationary m-dependent sequence

Estimation of quadratic density functionals under m-dependence

“…(ii) Källberg et al (2013) show that the one-sample estimator Q2,0,n attains the rates in Theorem 1 under a dependence condition for {X i } which is nonequivalent to A 2 .…”

“…In view of Theorem 2 and the corresponding result for Q2,0,n in Källberg et al (2013) (see Remark 2(ii)), it is natural to ask whether a dependence condition like A 2 is necessary for an estimator of q k,l to attain the convergence rates in Theorem 1. Next we show that this is not the case by introducing a modified version of Qk,l,n .…”

“…→ denotes convergence in distribution. Because no dependence assumptions except for m-dependence are required, (3) would, in view of the corresponding result for Q2,0,n in Källberg et al (2013), be a significant contribution.…”

“…The functional q 2,0 is important in rank-based statistics where it appears, e.g., in many efficacy expressions (Gibbons and Chakraborti, 1992) and in the asymptotic variance of the Hodges-Lehmann estimator. Källberg et al (2013) study U -statistic estimators of q 2,0 and h 2 based on the number of ǫ-close observations in a sample from a stationary m-dependent sequence, and establish conditions for their consistency and weak convergence. In this paper, we investigate the corresponding U -statistic estimators for a more general class of functionals, including the (two-density) functional q 1,1 , and give alternative conditions for consistency.…”

“…Then it straightforward to combine the results in this paper with the conventional limit theory for plug-in estimators to derive consistency properties of the estimators Dn and ĥn . For applications of such nonparametric divergence and entropy estimators, we refer to, e.g., Leonenko et al (2008), Leonenko and Seleznjev (2010), Källberg et al (2013), and references therein.…”