## Abstract

Let {*v* _{n}(*θ*)} be a sequence of statistics such that when*θ* =*θ* _{0},*v* _{n}(*θ* _{0})\(\mathop \to \limits^D \) *N* _{p}(0,*Σ*), where*Σ* is of rank*p* and*θ* ε*R* ^{d}. Suppose that under*θ* =*θ* _{0}, {*Σ* _{n}} is a sequence of consistent estimators of*Σ*. Wald (1943) shows that*v* _{n} ^{T} (*θ* _{0})*Σ* _{ n } ^{−1} *v* _{n}(*θ* _{0})\(\mathop \to \limits^D \) *x* ^{2}(*p*). It often happens that*v* _{n}(*θ* _{0})\(\mathop \to \limits^D \) *N* _{p}(0,*Σ*) holds but*Σ* is singular. Moore (1977) states that under certain assumptions*v* _{n} ^{T} (*θ* _{0})*Σ* _{ n } ^{−} *v* _{n}(*θ* _{0})\(\mathop \to \limits^D \) *x* ^{2}(*k*), where*k* = rank (*Σ*) and*Σ* _{ n } ^{−} is a generalized inverse of*Σ* _{n}. However, Moore’s result as stated is incorrect. It needs the additional assumption that rank (*Σ* _{n}) =*k* for*n* sufficiently large. In this article, we show that Moore’s result (as corrected) holds under somewhat different, but easier to verify, assumptions.

## Key words

Chi-square tests Generalized inverse Limit theorems## Preview

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