Let {vn(θ)} be a sequence of statistics such that whenθ =θ0,vn(θ0)
$$\mathop \to \limits^D $$
Np(0,Σ), whereΣ is of rankp andθ εRd. Suppose that underθ =θ0, {Σn} is a sequence of consistent estimators ofΣ. Wald (1943) shows thatvnT
(θ0)Σn−1vn(θ0)
$$\mathop \to \limits^D $$
x2(p). It often happens thatvn(θ0)
$$\mathop \to \limits^D $$
Np(0,Σ) holds butΣ is singular. Moore (1977) states that under certain assumptionsvnT
(θ0)Σn−vn(θ0)
$$\mathop \to \limits^D $$
x2(k), wherek = rank (Σ) andΣn−
is a generalized inverse ofΣn. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (Σn) =k forn sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions.