The Le Cam distance between density estimation and the Gaussian white noise model in the case of small signals

Consider nonparametric density estimation where we observe $n$ i.i.d. copies of a random variable with density $f$ on the unit interval. It is well-known that estimation of the density $f$ is asymptotically equivalent to a Gaussian white noise experiment with drift $2\sqrt{f},$ provided that $f$ lies in a H\"older ball with smoothness index larger than $1/2$ and is uniformly bounded away from zero. We study the case when the latter assumption does not hold and the density is possibly small. We derive matching lower and constructive upper bounds for the Le Cam deficiency in terms of the sample size and parameter space $\Theta$. The closely related case of Poisson intensity estimation is also considered.