Explicit asymptotically normal estimation of a parameter for some multivariate nonlinear regression problems (Q2713940)

Let a sequence of independent random variables \(X_1,X_2,\dots\) be defined as follows: NEWLINE\[NEWLINE X_i=a_i/\beta_i(\theta)+\sigma_i\xi_i, NEWLINE\]NEWLINE where \(\beta_i(\theta)\) depends on an \(m\)-dimensional parameter \(\theta\) in the following way: NEWLINE\[NEWLINE \beta_i(\theta)=1+\sum_{j=1}^m b_{ji}\theta_j. NEWLINE\]NEWLINE The values \(a_i>0\) and \(b_{ji}\geq 0\) are assumed known, while the values of the parameter \(\theta\) and the variance \(\mathbf{D}X_i\equiv\sigma^2\) are unknown. The random noises \(\xi_1,\dots,\xi_N\) are assumed to be independent, identically distributed random variables with zero mean and unit variance. NEWLINENEWLINENEWLINEThe author considers the problem of estimating the unknown parameter \(\theta\) from the observations \(X_1,\dots,X_N\). An asymptotically normal estimator is found under rather general assumptions. The author makes use of the method developed in a paper of \textit{Yu.Yu. Linke} and \textit{A.I. Sakhanenko} [Sib. Math. J. 41, No. 1, 125-137 (2000); translation from Sib. Mat. Zh. 41, No. 1, 150-163 (2000; Zbl 0943.62025)].