Comments on “entropies of degree β and lower bounds for the average error rate”

Consider a two-dimensional random variable (X, θ) with θ ∊ {θ1 …, θm) and X ∊ R^d, and suppose that the distribution of (X, θ) is given by a) P(θ = θi) = pi, 1 ≤ i ≤ m, and b) P{X < x/θ = θi} has a probability density p(x/θi)1 ≤ i ≤ m. We call θ the state of the observation X. Let R^∗ denote the Bayes´ rate, viz., R^∗ = E{r^∗(x)}, and r^∗(x) = 1 − maxi p(θi/x). Then for any β > 1, a lower bound on R^∗ due to Devijver¹ is the inequality