On assessing vector valued parameters

1. INTRODUCTION We consider the assessment of a vector valued parameter for a regular statistical model with data. The approach involves the Taylor expansion of the log-model, the separation of terms by asymptotic accuracy O(1), O(n −1/2), and O(n −1), and the subsequent recombination of terms in a coordinate-free form. Our particular interest centers on vector valued parameters in the presence of nuisance parameters, with special concern for discrete data. The model f (y; θ) with observed data y 0 leads immediately to the observed likelihood L(θ) = cf (y 0 ; θ) which can often be examined directly. In turn the log-likelihood (θ) = log L(θ) gives the maximum likelihood valuê θ and observed informationˆ θθ = −(∂ 2 /∂θ∂θ)(θ)| ˆ θ recording curvature, a second-derivative value or matrix at the maximum (ˆ θ). The Taylor expansion of the model as demonstrated below gives an approximate Normal distribution (θ; ˆ 