On Fisher information inequalities in the presence of nuisance parameters

The existence of a generalized Fisher information matrix for a vector parameter of interest is established for the case where nuisance parameters are present under general conditions. A matrix inequality is established for the information in an estimating function for the vector parameter of interest. It is shown that this inequality leads to a sharper lower bound for the variance matrix of unbiased estimators, for any set of functionally independent functions of parameters of interest, than the lower bound provided by the Cramér-Rao inequality in terms of the full parameter.