A note on generalized wald’s method

Abstract

Let {v _n(θ)} be a sequence of statistics such that whenθ =θ ₀,v _n(θ ₀)$\mathop \to \limits^D$ N _p(0,Σ), whereΣ is of rankp andθ εR ^d. Suppose that underθ =θ ₀, {Σ _n} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatv ^T_n (θ ₀)Σ ⁻¹ _n v _n(θ ₀)$\mathop \to \limits^D$ x ²(p). It often happens thatv _n(θ ₀)$\mathop \to \limits^D$ N _p(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsv ^T_n (θ ₀)Σ ⁻ _n v _n(θ ₀)$\mathop \to \limits^D$ x ²(k), wherek = rank (Σ) andΣ ⁻ _n is a generalized inverse ofΣ _n. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σ _n) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions.