The Le Cam distance between density estimation, Poisson processes and Gaussian white noise

Abstract

It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities have Hölder smoothness larger than 1/2 and are uniformly bounded away from zero. We derive matching lower and constructive upper bounds for the Le Cam deficiencies between these experiments, with explicit dependence on both the sample size and the size of the densities in the parameter space. As a consequence, we derive sharp conditions on how small the densities can be for asymptotic equivalence to hold. The related case of Poisson intensity estimation is also treated.

Minimax theory for a class of nonlinear statistical inverse problems

“…[30,8]. However the situation is in fact more subtle, with asymptotic equivalence only holding in the second regime of (3.5) and under further conditions (see [33] for more details).…”

A Statistical Model of Microscope Resolution

“…In one dimension, provided that t √ n log 8 n, we can apply recent results from (Ray and Schmidt-Hieber, 2018) on asymptotic equivalence in the Le Cam sense, to prove that the Poisson model is asymptotically equivalent to the VSG model. Combining asymptotic equivalence with the CLT, we see that (2.15) holds true asymptotically for any coupling between t and n.…”

“…2 by some absolute constants c 1 , c 2 > 0 and it is Hölder with exponent β > 1/2. This result was extended in Theorem 4 of (Ray and Schmidt-Hieber, 2018) to include functions f which are not bounded away from zero: functions f = f n that may depend on n ∈ N, satisfy…”

“…and f n are Hölder with 1/2 < β ≤ 1. Thus, we only need to extend our f n 's to functions on [0, 1] and prove that they satisfy the assumptions of Theorem 4 of (Ray and Schmidt-Hieber, 2018) to complete the proof. As a first step, extend the image function g to a function on (5.49) for some δ > 0.…”

“…• Can the results of (Ray and Schmidt-Hieber, 2018) on the 1D Poisson intensity estimation be extended to higher dimensional Poisson experiments? Would these results be valid at least in the whole complement of the CLT validity domain, thereby proving (2.15) for any relationship between t and n, just like in 1D?…”