Asymptotically normal estimation of a multidimensional parameter in the linear-fractional regression problem (Q2773620)

Let a sequence of random variables \(Z_1,\dots, Z_N\) be defined by: NEWLINE\[NEWLINEZ_i=\alpha_i(\theta)/\beta_i(\theta)+\xi_i,\quad i=1,\dots,N,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\alpha_i(\theta)=\alpha_{0i}+\sum_{j=1}^ma_{ji}\theta_j, \quad \beta_i(\theta)=1+\sum_{j=1}^mb_{ji}\theta_j,NEWLINE\]NEWLINE are linear combinations depending on an unknown \(m\)-dimensional parameter \(\theta\), while \(b_{ji}\geq 0\), \(a_{0i}\), and \(a_{ji}\) are known numbers.NEWLINENEWLINENEWLINEThe authors consider the problem of estimating the unknown vector \(\theta\) from the observations \(Z_1,\dots,Z_N\). They construct an estimator which is asymptotically normal under rather general assumptions on the constants \(\{c_i\}\) and the random errors \(\xi_1,\xi_2,\dots\) . Conditions for optimality of the estimators are also obtained. The article generalizes some previous results of the authors [ibid. 41, No. 1, 125-137 (2000; Zbl 0943.62025)].