Abstract
This paper is concerned with the use of the empirical characteristic function (ecf) in nonparametric testing for independence. Properties of the ecf are briefly reviewed, and a new distributional convergence result (Theorem 2.3) included. Nonparametric testing for independence is discussed briefly, but with particular focus on asymptotic aspects. Some new procedures for testing independence based on the ecf are presented and developed, and a Monte Carlo study carried out. The asymptotic efficiency of the procedure is discussed and suggestions for further work and some open problems noted.
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References
Anderson, T. W. (1958), An Introduction to Multivariate Statistical Analysis. New York: Wiley and Sons.
Bahadur, R. R. (1960), “Stochastic comparison of tests”. Annals of Mathematical Statistics 31, 276–295.
Bahadur, R. R. (1967), “Rates of convergence of estimates and test statistics.” Annals of Mathematical Statistics 38, 303–324.
Blum, J. R., J. Kiefer, and M. Rosenblatt (1961), “Distribution free tests of independence based on the sample distribution function.” Annals of Mathematical Statistics 32, 485–498.
Csorgo, S. (1981a), “Limit behaviour of the empirical characteristic function.” Annals of Probability 9, 130–144.
Csorgo, S. (1981b), “Multivariate empirical characteristic functions.” Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 55, 203–229.
Csorgo, S., and V. Totik (1983), “On how long an interval is the empirical characteristic function uniformly consistent?” Acta Scientiarum Mathematicarum (Szeged) 45, 141–149.
Feuerverger, A., and D. A. S. Fraser (1980), “Categorical information and the singular linear model.” Canadian Journal of Statistics 8, 41–45.
Feuerverger, A., and P. McDunnough (1981a), “On the efficiency of empirical characteristic function procedures.” Journal of the Royal Statistical Society, Series B 43, 20–27.
Feuerverger, A., and P. McDunnough (1981b), “On some Fourier methods for inference.” Journal of the American Statistical Association 76, 379–387.
Feuerverger, A., and R. A. Mureika (1977), “The empirical characteristic function and its applications.” Annals of Statistics 5, 88–97.
Gregory, G. G. (1980), “On efficiency and optimality of quadratic tests.” Annals of Statistics 8, 116–131.
Hoeffding, W. (1948), “A nonparametric test of independence.” Annals of Mathematical Statistics 19, 546–557.
Hollander, M., and D. A. Wolfe (1973), Nonparametric Statistical Methods. New York: Wiley and Sons.
Keller, H. D. (1979), “Einige Untersuchungen zur empirischen charakterischen Funktion und deren Anwendungen.” Dissertation. Universität Dortmund.
Kendall, M. G. (1970), Rank Correlation Methods, 4th edition. London: Griffin.
Konijn, H. S. (1956), “On the power of certain tests for independence in bivariate populations.” Annals of Mathematical Statistics 27, 300–323. Correction 29, (1958) 935-936.
Koziol, J. A., and A. F. Nemec (1979), “On a Cramer-von Mises type statistic for testing bivariate independence.” Canadian Journal of Statistics 7, 43–52.
Lehmann, E. L. (1959), Testing Statistical Hypotheses. New York: Wiley and Sons.
Lehmann, E. L. (1966), “Some concepts of dependence.” Annals of Mathematical Statistics 37, 1137–1153.
Lehmann, E. L. (1975), Nonparametrics: Statistical Methods Based on Ranks. San Francisco: Holden-Day.
Marcus, M. B. (1981), “Weak convergence of the empirical characteristic function.” Annals of Probability 9, 194–201.
Rao, C. R. (1965), Linear Statistical Inference and its Applications. New York: Wiley and Sons.
Rosenblatt, M. (1975), “A quadratic measure of deviation of two-dimensional density estimates and a test of independence.” Annals of Statistics 3, 1–14.
Srivastava, M. S., and G. C. Lee (1984), “On the distribution of the correlation coefficient when sampling from a mixture of two bivariate normal densities: robustness and the effect of outliers.” Canadian Journal of Statistics 12, 119–133.
Stuart, A. (1954), “The asymptotic relative efficiencies of tests and the derivatives of their power functions.” Skandinavisk Aktuarietidskrift 37, 163–169.
Wald, A. (1943), “Tests of statistical hypotheses concerning several parameters when the number of observations is large.” Transactions of the American Mathematical Society 54, 426–483.
Wieand, H. S. (1976), “A condition under which the Pitman and Bahadur approaches to efficiency coincide.” Annals of Statistics 4, 1003–1011.
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© 1987 D. Reidel Publishing Company
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Feuerverger, A. (1987). On Some ECF Procedures for Testing Independence. In: MacNeill, I.B., Umphrey, G.J., Carter, R.A.L., McLeod, A.I., Ullah, A. (eds) Time Series and Econometric Modelling. The University of Western Ontario Series in Philosophy of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-4790-0_14
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DOI: https://doi.org/10.1007/978-94-009-4790-0_14
Publisher Name: Springer, Dordrecht
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