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

TY - JOUR

T1 - The Le Cam distance between density estimation, Poisson processes and Gaussian white noise

AU - Ray, Kolyan Michael

AU - Schmidt-Hieber, Johannes

PY - 2018/9/10

Y1 - 2018/9/10

N2 - It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities have Holder 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.

AB - It is well-known that density estimation on the unit interval is asymptotically equivalent to a Gaussian white noise experiment, provided the densities have Holder 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.

KW - Asymptotic equivalence

KW - Le Cam distance

KW - Density estimation

KW - Poisson intensity estimation

KW - Gaussian shift experiments

U2 - 10.4171/MSL/1-2-1 Published online: 2018-09-05

DO - 10.4171/MSL/1-2-1 Published online: 2018-09-05

M3 - Article

SN - 2520-2324

VL - 1

SP - 101

EP - 170

JO - Mathematical Statistics and Learning

JF - Mathematical Statistics and Learning

IS - 2

ER -