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

A note on generalized wald’s method

  • A. S. Hadi
  • M. T. Wells


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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  1. Andrews DWK (1987) Asymptotic Results for Generalized Wald Tests. Econometric Theory 3:348–358CrossRefGoogle Scholar
  2. Akritas MG (1988) Pearson-Type Goodness-of-Fit Tests: The Univariate Case. J Amer Statist Assoc 83:222–230zbMATHCrossRefMathSciNetGoogle Scholar
  3. Chamberlain G (1984) In: Guliches Z, Intriligater MD (eds) Panel Data. Handbook of Econometrics, vol 2. North Holland, New YorkGoogle Scholar
  4. Eaton M (1983) Multivariate Statistics. John Wiley, New YorkzbMATHGoogle Scholar
  5. Hadi AS, Wells MT (1990) Minimum Distance Method of Estimation and Testing When Statistics Have Limiting Singular Multivariate Normal Distribution. Sankhya (in press)Google Scholar
  6. Koutrouvelis IA, Kellermeier J (1981) A Goodness-of-fit Test Based on the Empirical Characteristic Function When Parameters Must be Estimated. J R Statist Soc B, 43:173–176zbMATHMathSciNetGoogle Scholar
  7. LeCam L, Mahau C, Singh A (1983) An Extension of a Theorem of H. Chernoff and E. L., Lehmann. In: Rizvi MH, Rustagi JS, Siegmund D (eds) Recent Advances in Statistics: Papers in Honor of Herman Chernoff. Academic Press, New York, pp 303–337Google Scholar
  8. Mihalko DP, Moore DS (1980) Chi-Square Tests of Fit for Type II Censored Data. Ann Statist 8(3):625–644zbMATHMathSciNetGoogle Scholar
  9. Moore DS (1977) Generalized Inverses, Wald’s Method, and the Construction of Chi-Squared Tests of Fit. J Amer Statist Assoc 72:131–137zbMATHCrossRefMathSciNetGoogle Scholar
  10. Rao CR, Mitra SK (1971) Generalized Inverses of Matrices and Its Applications. John Wiley, New YorkGoogle Scholar
  11. Steward GW (1969) On the Continuity of the Generalized Inverse. SIAM J Appl Math 17:33–45CrossRefMathSciNetGoogle Scholar
  12. Wald, Abraham (1943) Tests of Statistical Hypothesis When the Number of Observations is large. Transaction of the A.M.S. 54:426–482zbMATHCrossRefMathSciNetGoogle Scholar

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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