Approximate estimation of non-identifiable parameters in a convolution

Abstract

In this paper we introduce a general method for the estimation of parameters in a convolution when these are non-identifiable or confounded as in the case of Gaussian signal plus Gaussian noise model. The method proposed is to replace the original convolution by a sequence of convolutions M v converging in distribution to the distribution of the convolution but the parameters are identifiable in each term of the sequence M v . For example in Gaussian signal plus noise models, X = Y + Z, where Y is the signal and Z is the noise, we approximate the error distribution N(0, φ 2) by a sequence of t-distributions with v degrees of freedom and φ as a scale parameter. We show that it is possible to construct consistent estimators of the parameters of signal Y which is N( θ, σ 2) whereas in the original model σ 2 is not identifiable.