The Asymptotic Effect of Substituting Estimators for Parameters in Certain Types of Statistics

In a variety of statistical problems, one is interested in the limiting distribution of statistics T̂ₙ = Tₙ(y₁, y₂, ⋯, yₙ; λ̂ₙ), where λ̂ₙ is an estimator of a parameter in the distribution of the yᵢ and where the limiting distribution of Tₙ = Tₙ(y₁, y₂, ⋯, yₙ; λ) is relatively easy to find. For cases in which the limiting distribution of Tₙ is normal with mean independent of λ, a useful method is given for finding the limiting distribution of T̂ₙ. A simple application to testing normality in regression models is given.