Nonnegative minimum biased quadratic estimation in the linear regression models
Let us consider a linear model \(M\{y, X\beta, \sigma^2 I\}\), where \(y\) is an \(n \times 1\) normally distributed vector of observations, with the expectation vector \(E(y) = X \beta\) and the covariance matrix \(\text{Cov} (y) = \sigma^2 I\). The nonstochastic \(n \times p\) matrix \(X\) is known and of rank \(p - q\), \(0 \leq q < p < n\), \(\beta\) is a \(p \times 1\) vector of unknown parameters, while \(\sigma^2 > 0\) is the unknown variance of the disturbances. We consider the problem of nonnegative estimation of \(\beta' H\beta + h\sigma^2\). Here \(H\) is a nonnegative definite matrix, while \(h\) is a nonnegative scalar. The problem arises, for instance, if we want to estimate accuracy of linear estimators \(Ly\) of \(\beta\) by means of the total mean squared error (TMSE), which has the structure \(\beta' H\beta + h \sigma^2\) and it is given by \[ \text{TMSE} (Ly) = E[(Ly - \beta)' (Ly - \beta)] = \beta' (X' L' - I) (X' L' - I)' \beta + \sigma^2 \text{tr} \{LL'\}. \] An iterative procedure for the nonnegative minimum biased quadratic estimator is described. Moreover, in the case that \(H\) and \(X' X\) commute, an explicit formula for this estimator is given. Admissibility of the estimator is proved. The results are applied to nonnegative estimation of the total mean squared error of a linear biased estimator.
nonnegative minimum biased quadratic estimator
scientific article; zbMATH DE number 797144