## Electronic Journal of Statistics

### Estimation of a delta-contaminated density of a random intensity of Poisson data

#### Abstract

In the present paper, we constructed an estimator of a delta contaminated mixing density function $g(\lambda)$ 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 [8] 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.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 683-705.

Dates
First available in Project Euclid: 16 March 2016

https://projecteuclid.org/euclid.ejs/1458133058

Digital Object Identifier
doi:10.1214/16-EJS1118

Mathematical Reviews number (MathSciNet)
MR3474842

Zentralblatt MATH identifier
1333.62109

#### Citation

De Canditiis, Daniela; Pensky, Marianna. Estimation of a delta-contaminated density of a random intensity of Poisson data. Electron. J. Statist. 10 (2016), no. 1, 683--705. doi:10.1214/16-EJS1118. https://projecteuclid.org/euclid.ejs/1458133058

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