Bounds on generalized linear predictors with incomplete outcome data
The generalized linear predictors for a response \(Y\) are considered of the form \(\widehat Y=G(X\vartheta)\), where \(X\) is the explanatory variables vector, \(G\) is some pre-assigned strictly increasing function, \(\vartheta\) is a vector of coefficients (to be determined). The best \(\vartheta\) (for the quadratic loss function) is \(\vartheta=(\mathbf{E}X'X)^{-1}\text\textbf{E}X'G^{-1}(X)\). The paper is devoted to the bounds for \(c'\vartheta\) (\(c\) being a fixed vector) in the case when lower and upper bounds \(\overline F\) and \(\underline{F}\) for the conditional CDF \(F_{Y| X}(y,x)\) are known, i.e., \(\underline{F}(y,x)\leq F_{Y| X}(y,x)\leq \overline F(y,x)\) for all \(x\) and \(y\). The obtained bounds are used for estimation of \(\vartheta\) in the case of missing values of \(Y\) in the sample. A real data example is presented.
scientific article; zbMATH DE number 5145086