Andrea Ridolfi

Penalized Maximum Likelihood Estimator for Normal Mixtures

The estimation of the parameters of a mixture of Gaussian densities is considered, within the framework of maximum likelihood. Due to unboundedness of the likelihood function, the maximum likelihood estimator fails to exist. We adopt a solution to likelihood function degeneracy which consists in penalizing the likelihood function. The resulting penalized likelihood function is then bounded over the parameter space and the existence of the penalized maximum likelihood estimator is granted. As original contribution we provide asymptotic properties, and in particular a consistency proof, for the penalized maximum likelihood estimator. Numerical examples are provided in the finite data case, showing the performances of the penalized estimator compared to the standard one.

Ultrawide bandwidth signals as shot noise: a unifying approach

We present a shot noise based model for a large family of ultrawide bandwidth (UWB) signals. These include time-hopping and direct-sequence signaling with pulse position, interval, and amplitude modulations. Each specific signal is constructed by adding features to a basic model in a modular, simple, and tractable way. Our work unifies the contributions scattered in the literature and provides a general approach that allows various extensions of previous works as well as new results. The exact power spectrum is then evaluated using shot noise spectral theory, which provides a simpler, systematic, and rigorous approach to the spectra evaluation of complicated UWB signals. The strength of our methodology is that different features of the signal model contribute clearly and separately to the resulting spectral expressions.

Power spectra of random spike fields and related processes

In this article, we review known results and present new ones concerning the power spectra of large classes of signals and random fields driven by an underlying point process, such as spatial shot noises (with random impulse response and arbitrary basic stationary point processes described by their Bartlett spectra) and signals or fields sampled at random times or points (where the sampling point process is again quite general). We also obtain the Bartlett spectrum for the general linear Hawkes spatial branching point process (with random fertility rate and general immigrant process described by its Bartlett spectrum). We then obtain the Bochner spectra of general spatial linear birth and death processes. Finally, we address the issues of random sampling and linear reconstruction of a signal from its random samples, reviewing and extending former results.

Penalized Maximum Likelihood Estimation for univariate normal mixture distributions

Due to singularities of the likelihood function, the maximum likelihood approach for the estimation of the parameters of normal mixture models is an acknowledged ill posed optimization problem. Ill posedness is solved by penalizing the likelihood function. In the Bayesian framework, it amounts to incorporating an inverted gamma prior in the likelihood function. A penalized version of the EM algorithm is derived, which is still explicit and which intrinsically assures that the estimates are not singular. Numerical evidence of the latter property is put forward with a test.

Power spectral measure and reconstruction error of randomly sampled signals

We say that a signal is randomly sampled when the samples are taken at random instants of time. The study of random sampling and randomly sampled signals is motivated both by practical and theoretical interests. The first one includes spectral analysis (estimation of spectra from a finite number of samples) and quality of service (signal reconstruction), and the second one includes statistical analysis of reconstruction methods. The present paper focuses on the computation of the (theoretical) spectrum of randomly sampled signals and on the computation of the reconstruction error. Using a point process approach, we obtain general formulas for spatial random sampling, providing powerful tools for the analysis and the processing of randomly sampled signals.