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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  and , 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  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
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## Copyright information

© Springer-Verlag 1969

## Authors and Affiliations

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