Chapter 2 Classical regularity conditions

The results from classical asymptotic theory typically require assumptions of pointwise differentiability of a criterion function with respect an unknown parameter. Taylor expansion about some " true " value in the parameter space then gives a quadratic approximation to the criterion function, within error terms that can be bounded using the remainder from the Taylor expansion. With appropriately small error terms, an estimator defined to minimize the criterion function will lie close to the random variable that minimizes the quadratic, a random variable that typically has a neat closed form representation. The estimator inherits the limiting behaviour of the minimizer of the quadratic. 1. Comparison arguments Suppose G n (θ) = G n (ω, θ) is a random variable for each θ in an index set. Suppose an estimator θ n = θ n (ω) is defined by minimization of G n (·), or at least is required to come close to minimizing G n (·) over , G n (θ n) ≈ inf θ ∈ G n (θ), in a sense soon to be made more precise. What asymptotic properties must θ n have, as a consequence of the minimization? The answer will depend upon how much uniform control we have over G n (·). Arguments based directly on the minimization property of an estimator (as distinct from arguments that locate zeros of a derivative of smooth criterion functions)typically depend on a simple, nearly obvious, principle.