Abstract
This paper considers the problem for testing symmetry of a distribution inR m based on the empirical distribution function. Limit theorems which play important roles for investigating asymptotic behavior of such tests are obtained. The limit processes of the theorems are multiparameter Wiener process. Based on the limit theorems, nonparametric tests are proposed whose asymptotic distributions are functionals of a multiparameter standard Wiener process. The tests are compared asymptotically with each other in the sense of Bahadur.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adler, R. J. (1990).An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, IMS Lecture Notes-Monograph Series, Vol. 12, Hayward, California.
Aki, S. (1987). On nonparametric tests for symmetry,Ann. Inst. Statist. Math.,39, 457–472.
Aki, S. (1990). On a test based on a sequential empirical process (unpublished manuscript).
Aki, S. and Kashiwagi, N. (1989). Asymptotic properties of some goodness-of-fit tests based on theL 1-norm,Ann. Inst. Statist. Math.,41, 753–764.
Bahadur, R. R. (1960). Stochastic comparison of tests,Ann. Math. Statist.,31, 276–295.
Bickel, P. J. and Wichura, M. J. (1971). Convergence criteria for multiparameter stochastic process and some applications,Ann. Math. Statist.,42, 1656–1670.
Blum, J. R., Kiefer, J. and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function,Ann. Math. Statist.,32, 485–498.
Butler, C. (1969). A test for symmetry using the sample distribution function,Ann. Math. Statist.,40, 2209–2210.
Cotterill, D. S. and Csörgő, M. (1982). On the limiting distribution of and critical values for the multivariate Cramér-von Mises statistic,Ann. Statist.,10, 233–244.
Cotterill, D. S. and Csörgő, M. (1985). On the limiting distribution of and critical values for the Hoeffding, Blum, Kiefer, Rosenblatt independence criterion,Statist. Decisions,3, 1–48.
Csörgő, S. and Heathcote, C. R. (1987). Testing for symmetry,Biometrika,74, 177–184.
Devroye, L. P. and Wagner, T. J. (1980). The strong uniform consistency of kernel density estimates,Multivariate Analysis, V (ed. P. R. Krishnaiah), 59–77, North-Holland, New York.
Giné, E. and Zinn, J. (1984). Some limit theorems for empirical processes,Ann. Probab.,12, 929–989.
Giné, E. and Zinn, J. (1986). Lectures on central limit theorem for empirical processes (Probability and Banach Spaces),Lecture Notes in Math.,1221, 50–113.
Imhof, J. P. (1961). Computing the distribution of quadratic forms in normal variables,Biometrika,48, 419–426.
Khmaladze, E. V. (1988). An innovation approach to goodness-of-fit tests inR m,Ann. Statist.,16, 1503–1516.
Koning, A. J. (1992). Approximation of stochastic integrals with applications to goodness-of-fit tests,Ann. Statist.,20, 428–454.
Ledoux, P. M. and Talagrand, M. (1986). Conditions d'intégrablité pour les multiplicateurs dans le tlc Banachique,Ann. Probab.,14, 916–921.
Martynov, G. V. (1975). Computation of distribution functions of quadratic forms of normally distributed random variables,Theory Probab. Appl.,20, 782–793.
Pollard, D. (1990).Empirical Processes: Theory and Applications, NSF-CBMS Regional Conference Series in Probability and Statistics,2, IMS, Hayward, California.
Rothman, E. D. and Woodroofe, M. (1972). A Cramér-von Mises type statistic for testing symmetry,Ann. Math. Statist.,43, 2035–2038.
Shorack, G. R. and Wellner, J. A. (1986).Empirical Processes with Applications to Statistics, Wiley, New York.
Wieand, H. S. (1976). A condition under which the Pitman and Bahadur approaches to efficiency coincide,Ann. Statist.,4, 1003–1011.
Zolotarev, V. M. (1961). Concerning a certain probability problem,Theory Probab. Appl.,6, 201–204.
Author information
Authors and Affiliations
About this article
Cite this article
Aki, S. On nonparametric tests for symmetry inR m . Ann Inst Stat Math 45, 787–800 (1993). https://doi.org/10.1007/BF00774788
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00774788