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 Tn (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 Tn (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.
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Research partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell University.
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Hadi, A.S., Wells, M.T. A note on generalized wald’s method. Metrika 37, 309–315 (1990). https://doi.org/10.1007/BF02613538
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DOI: https://doi.org/10.1007/BF02613538