Summary
It is proved that the martingale term of the empirical distribution function converges weakly to a Gaussian process inD[0, 1]. Some statistics for goodness-of-fit tests based on the martingale term of the empirical distribution function are proposed. Asymptotic distributions of these statistics under the null hypothesis are given. The approximate Bahadur efficiencies of the statistics to the Kolmogorov-Smirnov statistic and to the Cramér-von Mises statistic are also calculated.
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References
Aalen, O. (1978). Nonparametric inference for a family of counting processes,Ann. Statist.,6, 701–726.
Al-Hussaini, A. and Elliott, R. J. (1984). The single jump process with some statistical applications,Theory Prob. Appl.,29, 607–613.
Bahadur, R. R. (1960). Stochastic comparison of tests,Ann. Math. Statist.,31, 276–295.
Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York.
Butler, C. (1969). A test for symmetry using the sample distribution function,Ann. Math. Statist.,40, 2209–2210.
David, H. A. (1970).Order Statistic, Wiley, New York.
Feller, W. (1966).An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York.
Gregory, G. G. (1977). Large sample theory ofU-statistics and tests of fit,Ann. Statist.,5, 110–123.
Gregory, G. G. (1980). On efficiency and optimality of quadratic tests,Ann. Statist.,8, 116–131.
Hida, T. (1980).Brownian Motion, Springer, New York.
Jacobsen, M. (1982).Statistical Analysis of Counting Processes, Lecture Notes in Statistics, Vol. 12, Springer, New York.
Khmaladze, E. V. (1981). Martingale approach in the theory of goodness-of-fit tests,Theory. Prob. Appl.,26, 240–257.
Lehmann, E. L. (1959).Testing Statistical Hypotheses, Wiley, New York.
Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function spaceD[0, ∞),J. Appl. Prob.,10, 109–121.
Liptser, R. S. and Shiryaev, A. N. (1981). On a problem of necessary and sufficient conditions in the functional central limit theorem for local martingales,Z. Wahrsh. Verw. Gebiete,59, 311–318.
MacNeill, I. B. (1974). Tests for change of parameter at unknown times and distributions of some related functionals on Brownian motion,Ann. Statist.,2, 950–962.
Neyman, J. (1937). ‘Smooth’ test for goodness of fit,Skandinavisk Aktuarietidskrift,20, 149–199.
Rebolledo, R. (1978). Sur les applications de la théorie des martingales à l'étude statistique d'une famille de processus ponctuels, Lecture Notes in Mathematics, Vol. 636, 27–70, Springer, New York.
Rebolledo, R. (1980). Central limit theorems for local martingales,Z. Wahrsh. Verw. Gebiete,51, 269–286.
Rothman, E. D. and Woodroofe, M. (1972). A Cramér-von Mises type statistic for testing symmetry,Ann. Math. Statist.,43, 2035–2038.
Wieand, H. S. (1976). A condition under which the Pitman and Bahadur approaches to efficiency coinside,Ann. Statist.,4, 1003–1011.
Zolotarev, V. M. (1961). Concerning a certain probability problem,Theory Prob. Appl. 6, 201–204.
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Aki, S. Some test statistics based on the martingale term of the empirical distribution function. Ann Inst Stat Math 38, 1–21 (1986). https://doi.org/10.1007/BF02482496
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DOI: https://doi.org/10.1007/BF02482496