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Metrika

, Volume 37, Issue 1, pp 309–315 | Cite as

A note on generalized wald’s method

  • A. S. Hadi
  • M. T. Wells
Publications

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 n T (θ 0)Σ n −1 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 n T (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.

Key words

Chi-square tests Generalized inverse Limit theorems 

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Copyright information

© Physica-Verlag Ges.m.b.H 1990

Authors and Affiliations

  • A. S. Hadi
    • 1
  • M. T. Wells
    • 1
  1. 1.Department of Economic and Social StatisticsCornell UniversityIthacaUSA

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