Abstract
In the present paper, we constructed an estimator of a delta contaminated mixing density function $g(\lam)$
of an intensity $\lambda$ of the Poisson distribution.
The estimator is based on an expansion of the continuous portion $g_0(\lambda)$ of the unknown pdf over an overcomplete dictionary
with the recovery of the coefficients obtained as the solution of an optimization problem with Lasso penalty. In order to
apply Lasso technique in the, so called, prediction setting where it requires virtually no assumptions on the dictionary
and, moreover, to ensure fast convergence of Lasso estimator, we use a novel formulation of the optimization problem
based on the inversion of the dictionary elements.
We formulate conditions on the dictionary and the unknown mixing density that yield a sharp oracle inequality
for the norm of the difference between $g_0 (\lambda)$ and its estimator and, thus, obtain a smaller error than
in a minimax setting. Numerical simulations and comparisons with the Laguerre functions based estimator
recently constructed by \cite{Comte} also show advantages of our procedure.
At last, we apply the technique developed in the paper to estimation of
a delta contaminated mixing density of the Poisson intensity of the Saturn's rings data.
Anno
2016
Autori IAC
Tipo pubblicazione
Altri Autori
Daniela De Canditiis and Marianna pensky
Editore
Institute of Mathematical Statistics
Rivista
Electronic journal of statistics