Incidental Versus Random Nuisance Parameters

Let \P,: (, ) H\, with R and H arbitrary, be a family of mutually absolutely continuous probability measures on a measurable space (X, A). The problem is to estimate, based on a sample (x₁, , xₙ) from ⁿ₁ P, _. If (₁, , ₙ) are independently distributed according to some unknown prior distribution, then the distribution of n^1/2 (^ (n) -) under Pⁿ, (P, being the -mixture of P, , H) cannot be more concentrated asymptotically than a certain normal distribution with mean 0, say N (₀, ℂ䃐 (, ) ). Folklore says that such a bound is also valid if (₁, , ₙ) are just unknown values of the nuisance parameter: In this case, the distribution cannot be more concentrated asymptotically than N (₀, ℂ䃐 (, ₄^ (₍) (䃑, , 䂸) ) ), where E^ (n) (䃑, , 䂸) is the empirical distribution of (₁, , ₙ). The purpose of the present paper is to discuss to which extent this conjecture is true. The results are summarized at the end of Sections 1 and 3.