Maximum likelihood estimation of functional relationships

Consider the following measurement model relating n observation p-vectors \(x_i = (x_{i1}, \cdots, x_{ip})^T\) to their underlying true values \( xi_i = (\xi _{i1} ,....,\xi _{ip} )^T \) through, $$ x_i = \xi _i + \varepsilon _i \;\;(i = 1,..n) $$ (2.2) with \( \varepsilon _i = (\varepsilon _{i1} , \cdots ,\varepsilon _{ip} )T \) a vector of measurement errors, which we shall assume to have a zero mean i.e. \( E\varepsilon _i = 0 \). This means that ξi. is measured without systematic errors. Unless stated otherwise we shall assume that the distribution of e is not dependent on ξi.