A Class of Renyi Information Estimators for Multidimensional Densities

In a recent paper [1], Leonenko, Pronzato and Savani consider the estimation of the Rényi and Tsallis entropies, respectively, H ∗ q = 1 1−q log ∫ Rm f (x) dx and Hq = 1 q−1(1 − ∫ Rm f (x) dx), q = 1, of an unknown probability measure with density f on R with respect to the Lebesgue measure though kth nearestneighbor distances in a sample X1, . . . ,XN i.i.d. with the density f . The results in [1] about the asymptotic unbiasedness and consistency of the estimator proposed for Iq = E{f q−1(X)} = ∫Rm f (x) dx are correct for q > 1 but, for q < 1, convergence in distribution should be complemented by additional arguments to obtain the required convergence of moments. Following [1], define ÎN,k,q = 1 N ∑N i=1(ζN,i,k)1−q, with ζN,i,k = (N − 1)Ck × Vm(ρ (i) k,N−1), where Vm = πm/2/ (m/2 + 1) is the volume of the unit ball B(0,1) in R, Ck = [ (k) (k+1−q) ]1/(1−q) and ρ k,N−1 denote the kth nearest-neighbor distance from a given Xi to some other Xj in the sample X1, . . . ,XN . Also define rc(f ) = sup{r > 0 : ∫Rm |x|rf (x) dx <∞}, so that E|Xi |r <∞ if r < rc(f ) and E|Xi |r =∞ if r > rc(f ) (see [2]). A correct version of the convergence results of ÎN,k,q to Iq for 0 < q < 1 and f having unbounded support can be obtained by using results on the subadditivity of Euclidean functionals (see [3]) and can be formulated as follows [2]: if Iq < ∞ and rc(f ) > m1−q q , then E[ÎN,k,q] → Iq,N → ∞ (asymptotic unbiasedness; see Theorem 3.1 of [1]); if Iq <∞, q > 2 and rc(f ) > 2m 1−q 2q−1 , then E[ÎN,k,q − Iq]2 → 0,N →∞ (L2 convergence; see Theorem 3.2 of [1]). The situation is simpler when f has bounded support Sf . When f is bounded away from 0 and infinity on Sf and Sf consists of a finite union of convex bounded sets with