ABSTRACT
Most modern statistical questions involve large data sets, the modeling of whose stochastic
structure involves complex models governed by several, often many, real parameters and
frequently even more semi-or nonparametric models. In this final chapter of Volume I
we develop the analogues of the asymptotic analyses of the behaviors of estimates, tests,
and confidence regions in regular one-dimensional parametric models for d-dimensional models {Pθ : θ ∈ Θ}, Θ ⊂ Rd. We have presented several such models already, for instance, the multinomial (Examples 1.6.7, 2.3.3), multiple regression models (Examples 1.1.4, 1.4.3, 2.1.1) and more generally have studied the theory of multiparameter exponential families (Sections 1.6.2, 2.2, 2.3). However, with the exception of Theorems 5.2.2 and 5.3.5, in which we looked at asymptotic theory for the MLE in multiparameter exponential families, we have not considered asymptotic inference, testing, confidence regions, and
prediction in such situations. We begin our study with a thorough analysis of the Gaus-
sian linear model with known variance in which exact calculations are possible. We shall
show how the exact behavior of likelihood procedures in this model correspond to limit-
ing behavior of such procedures in the unknown variance case and more generally in large
samples from regular d-dimensional parametric models and shall illustrate our results with a number of important examples.
