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Asymptotic normality under contiguity in a dependence case

  • K. L. Mehra
Article

Summary

Let \(X_{vi} = (X_{vi1} ,X_{vi2} ,...,X_{viK_i } ) 1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } n_v\) be a sequence of independent random vectors following the regression model Xvij = α + ΒCvij + σYvij, with −∞<α, Β, C vij <, σ<0, and where \(Y_{vi} = (Y_{vi1} ,...,Y_{viK_i } ), 1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } i\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } n_v\), are independent random vectors with absolutely continuous distributions F (i) (x (i) ) and with densities \(f^{(i)} (x^{(i)} ) (x^{(i)} = (x_1 ,x_2 ,...x_{K_i } ))\). Define \(S_v = \sum\limits_i {\sum\limits_j {d_{vij} } \xi _{vR_{vij} } }\) where \(\{ \xi _{vk} : 1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } k\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } N_v \} (N_v = \sum\limits_{i = 1}^{n_v } {K_i } )\) is a double sequence of real numbers, -∞<d vij <∞ and R vij = rank of X vij in a combined ranking of N v components X vij , 1≦jKi, 1≦jKi, 1≦iKv. Under certain assumptions on the densities f (i) (x (i) ) and the sequences {ξ vij }, {d vij } and {C vij }, the asymptotic normality of the sequence S v , as n v → ∞, is proved. The results extend similar results of Hájek [3] and [4], from independently distributed components to the above pattern of dependence. An extension of the main theorem also covers the case when some of the distributions F (i) (x (i) ) are singular. The connection between the Hájek condition (1.8) of [4] and the present condition (6.1) on the multivariate densities f (i) (x (i) ) is also discussed.

Keywords

Regression Model Real Number Stochastic Process Probability Theory Present Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Feller, W.: An Introduction to Probability Theory and its Applications. Vol. II. New York: John Wiley & Sons 1966.Google Scholar
  2. 2.
    Gumbel, E.J.: Bivariate logistic distributions. J. Amer. statist. Assoc. 56, 355–349 (1961).Google Scholar
  3. 3.
    Hájek, J.: Some extensions of the Wald-Wolfowitz-Noether theorem. Ann. math. Statistics 32, 506–523 (1961).Google Scholar
  4. 4.
    —: Asymptotically most powerful rank-order tests. Ann. math. Statistics 33, 1124–1147 (1962).Google Scholar
  5. 5.
    Hewitt, J., and K. Stromberg: Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer 1965.Google Scholar
  6. 6.
    LeCam, L.: Locally asymptotically normal families of distributions. Univ. California Publ. Statist. 3, 37–98 (1960).Google Scholar
  7. 7.
    Loeve, M.: Probability Theory. New York: Van Nostrand 1955.Google Scholar
  8. 8.
    Mehra, K.L.: Conditional rank-order tests for experimental designs. Technical Report No. 59, Department of Statistics, Stanford University, Stanford, California 1967.Google Scholar
  9. 9.
    Wald, A., and J. Wolfowitz: Statistical tests based on permutations of the observations. Ann. math. Statistics 15, 358–372 (1944).Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • K. L. Mehra
    • 1
  1. 1.Department of MathematicsThe University of AlbertaEdmontonCanada

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