Approximate estimation of non-identifiable parameters in a convolution

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 Mv converging in distribution to the distribution of the convolution but the parameters are identifiable in each term of the sequence Mv. 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, [phi]2) by a sequence of t-distributions with v degrees of freedom and [phi] as a scale parameter. We show that it is possible to construct consistent estimators of the parameters of signal Y which is N([theta], [sigma]2) whereas in the original model [sigma]2 is not identifiable.