Minimax Optimal Additive Functional Estimation with Discrete Distribution: Slow Divergence Speed Case

TL;DR

This paper extends the understanding of the minimax optimal rates for additive functional estimation of discrete distributions, especially in the slow divergence speed case, providing precise rates across different parameter ranges.

Contribution

It generalizes the minimax rate characterization for a broader divergence speed range, including slow divergence cases, and derives explicit rates for different parameter regimes.

Findings

Minimax rate for lpha  (1,3/2): rac{1}{n}+rac{k^2}{(n\u2217ln n)^{2\u03b1}}

Minimax rate for lpha  [3/2,2]: rac{1}{n}

Extended the divergence speed analysis to more general cases.

Abstract

This paper addresses an estimation problem of an additive functional of $\phi$, which is defined as $\theta(P;\phi)=\sum_{i=1}^k\phi(p_i)$, given $n$ i.i.d. random samples drawn from a discrete distribution $P=(p_1,...,p_k)$ with alphabet size $k$. We have revealed in the previous paper that the minimax optimal rate of this problem is characterized by the divergence speed of the fourth derivative of $\phi$ in a range of fast divergence speed. In this paper, we prove this fact for a more general range of the divergence speed. As a result, we show the minimax optimal rate of the additive functional estimation for each range of the parameter $\alpha$ of the divergence speed. For $\alpha \in (1,3/2)$, we show that the minimax rate is $\frac{1}{n}+\frac{k^2}{(n\ln n)^{2\alpha}}$. Besides, we show that the minimax rate is $\frac{1}{n}$ for $\alpha \in [3/2,2]$.