The Annals of Statistics

Transformed empirical processes and modified Kolmogorov-Smirnov tests for multivariate distributions

A. Cabaña and E. M. Cabaña

Full-text: Open access


A general way of constructing classes of goodness-of-fit tests for multivariate samples is presented. These tests are based on a random signed measure that plays the same role as the empirical process in the construction of the classical Kolmogorov-Smirnov tests. The resulting tests are consistent against any fixed alternative, and, for each sequence of contiguous alternatives, a test in each class can be chosen so as to optimize the discrimination of those alternatives.

Article information

Ann. Statist., Volume 25, Number 6 (1997), 2388-2409.

First available in Project Euclid: 30 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G10: Hypothesis testing 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions 60G15: Gaussian processes

Goodness-of-fit power improvement TEPs


Cabaña, A.; Cabaña, E. M. Transformed empirical processes and modified Kolmogorov-Smirnov tests for multivariate distributions. Ann. Statist. 25 (1997), no. 6, 2388--2409. doi:10.1214/aos/1030741078.

Export citation


  • Caba na, A. (1993). Transformations du pont brownien empirique et tests de ty pe Kolmogorov- Smirnov. C.R. Acad. Sci. Paris S´er. I. Math. 317 315-318.
  • Caba na, A. (1996). Transformations of the empirical process and Kolmogorov-Smirnov tests. Ann. Statist. 24 2020-2035.
  • Caba na, A. and Caba na, E. M. (1996). Bridge-to-bridge transformations and Kolmogorov- Smirnov tests. Comm. Statist. Theory Methods 25 227-234.
  • Donsker, M. D. (1952). Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23 277-281.
  • Efron, B. and Johnston, I. (1990). Fisher's information in terms of hazard rate. Ann. Statist. 18 38-62.
  • Groeneboom, P. and Wellner, J. A. (1992). Information bounds and nonparametric maximum likelihood estimation. DMV Sem. 19. Birkh¨auser, Boston.
  • H´ajek, J. and Sid´ak, Z. (1967). Theory of Rank Tests. Academic Press, New York.
  • Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • Khmaladze, E. V. (1981). Martingale approach in the theory of goodness-of-fit. Theory Probab. Appl. 26 240-257.
  • Khmaladze, E. V. (1993). Goodness-of-fit problem and scanning innovation martingales. Ann. Statist. 21 798-829.
  • Le Cam L. and Yang, G. L. (1990). Asy mptotics in Statistics. Some Basic Concepts. Springer, New York.
  • Mor´an, M. and Urbina, W. (1996). Some properties of the Gaussian-Hilbert transform. Unpublished manuscript.
  • Oosterhoff, J. and van Zwet, W. R. (1979). A note on contiguity and Hellinger distance. In Contributions to Statistics. Jaroslav H´ajek Memorial Volume (J. Jore ckov´a, ed.) 157- 166. Reidel, Dordrecht.
  • Ossiander, M. (1987). A central limit theorem under metric entropy with L2 bracketing. Ann. Probab. 15 897-919.
  • Ritov, J. and Wellner, J. A. (1988). Censoring, martingales and the Cox model. Contemp. Math. 80 191-220.
  • Sansone, G. (1959). Orthogonal Functions (R. Courant, L. Bers and J. J. Stoker, eds.) Pure and Applied Mathematics 9. Interscience, New York.