Asymptotic normality under contiguity in a dependence case

  • K. L. Mehra


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.


Regression Model Real Number Stochastic Process Probability Theory Present Condition 
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Copyright information

© Springer-Verlag 1969

Authors and Affiliations

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

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