Quasi-maximum likelihood estimation of parameters in a multivariate Poisson process (Q1915118)

Consider a family of statistical experiments \[ {\mathcal E}_L= \{{\mathcal X}_L, {\mathcal A}_L, P^L_\theta,\;\theta\in \Theta\}, \qquad L>0, \] where \({\mathcal X}_L= N^M\), \({\mathcal A}_L= 2^{\mathcal X}\), \(m\) is a positive integer, the density of \(P^L_\theta\) w.r.t. the counting measure on \(N^m\) is \(p^L_\theta (\overline {n})= \prod^m_{i=1} (Lg_i )^{n_i} (n_i !)^{-1} e^{-Lg_i}\), \(\overline {n}= (n_1, \dots, n_m)\), \(g_i= g_i (\theta)\), and \(\Theta \subset R^k\). Such models, with linear functions \(g_i (\theta)\), typically apply when histogrammed data are produced over some period of time with a known intensity and measurements are subject to limited resolution and limited acceptance. The functions \(g_i\) then describe the data distortion mechanism and the number of outcomes is a Poisson random variable. The experiments \({\mathcal E}_L\) yield `folded' or `convoluted' data and estimation of true bin probabilities is referred to as unfolding or deconvolution. Problems of that kind have extensively been analysed in the high energy physics context. The main problem is that such models are typically ill-conditioned and a kind of regularization is necessary. In effect, the existing solutions lack a rigid asymptotic treatment. One possible approach could be to put the regularized solutions into the framework of quasi-maximum likelihood estimators. It is the aim of this paper to provide tools needed for accomplishment of that approach.