Estimation of regression parameters from quantum measurements (Q1824967)

The analysis of grouped data with the purpose of estimating a parameter of a distribution of measurements is the subject of numerous investigations. In particular, the consistency and asymptotic normality of maximum likelihood (ML) and Bayesian estimates were established. We study methods of actual computation of, say, ML estimates by means of algorithms for which computer programs are available. We investigate the statistical behavior of a Gauss-Newton type iterative procedure for the calculation of ML estimates in the asymptotic situation in which the number n of cells of the grid and the number N of measurements tend simultaneously to infinity. The basic facts of classical estimation theory for nonnecessarily linear regression models are generalized by the second author in Izv. Vyssh. Uchebn. Zaved., Math. 1983, No.11(258), 19-41 (1983; Zbl 0556.62058), English translation in Sov. Math. 27, No.11, 21-47 (1983), and extended to the so-called F-models, in which not only the mean values \(\eta (x_ i,\theta)\) of the independent measurements \(y_ i\in {\mathbb{R}}^ p\), but also the covariance matrices \(V(x_ i,\theta)\), \(i=1,...,N\), depend on the unknown parameter \(\theta\), and \(\theta\) is identified through \([\eta (x_ 1,\theta),...,\eta (x_ N,\theta)].\) In the present paper these investigations are carried over to the case of a degenerate matrix V(x,\(\theta)\) (which occurs in describing grouped measurements). This permits to extend the domain of applicability of a method of solving the estimation problem which is effective and computationally simple.