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 X vij = α + ΒC vij + σY vij, 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≦j≦K i, 1≦j≦K i, 1≦i≦K v. 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Feller, W.: An Introduction to Probability Theory and its Applications. Vol. II. New York: John Wiley & Sons 1966.
Gumbel, E.J.: Bivariate logistic distributions. J. Amer. statist. Assoc. 56, 355–349 (1961).
Hájek, J.: Some extensions of the Wald-Wolfowitz-Noether theorem. Ann. math. Statistics 32, 506–523 (1961).
—: Asymptotically most powerful rank-order tests. Ann. math. Statistics 33, 1124–1147 (1962).
Hewitt, J., and K. Stromberg: Real and Abstract Analysis. Berlin-Heidelberg-New York: Springer 1965.
LeCam, L.: Locally asymptotically normal families of distributions. Univ. California Publ. Statist. 3, 37–98 (1960).
Loeve, M.: Probability Theory. New York: Van Nostrand 1955.
Mehra, K.L.: Conditional rank-order tests for experimental designs. Technical Report No. 59, Department of Statistics, Stanford University, Stanford, California 1967.
Wald, A., and J. Wolfowitz: Statistical tests based on permutations of the observations. Ann. math. Statistics 15, 358–372 (1944).
Author information
Authors and Affiliations
Additional information
Prepared with the partial support of the National Research Council of Canada Grant A-3061.
Rights and permissions
About this article
Cite this article
Mehra, K.L. Asymptotic normality under contiguity in a dependence case. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 173–184 (1969). https://doi.org/10.1007/BF00534839
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00534839