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A comparison of uniformity tests

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2005-08
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Taylor & Francis Group Ltd
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Problems of goodness-of-fit to a given distribution can usually be reduced to test uniformity. The uniform distribution appears due to natural random events or due to the application of methods for transforming samples from any other distribution to the samples with values uniformily distributed in the interval (0,1), Thus, one can solve the problem of testing if a sample comes from a given distribution by testing whether its transformed sample is distributed according to the uniform distribution. For this reason, the methods of testing for goodness-of-fit to a uniform distribution have been widely investigated. In this paper, a comparative power analysis of a selected set of statistics is performed in order to give suggestions on which one to use for testing uniformity against the families of alternatives proposed by Stephens [Stephens, M.A., 1974, EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730-737.]. Definition and some relevant features of the considered test statistics are given in section 1. Implemented numerical processes to calculate percentage points of every considered statistic are described in section 2. Finally, a Monte Carlo simulation experiment has been carried out to fulfill the mentioned target of this paper.
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This work was supported by the grants BMF2003-00892, BMF2003-04820 and GV04B/670. The authors thank the referees for their comments and proposals leading to a better and more complete revised version.
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Kolmogorov, A.N., 1933, Sulla determinazione empirica di una legge di distibuziane. Giornale dell’Istituta Italiano degli Attuari, 4, 83–91. Kuiper, N.H., 1960, Tests concerning random points on a circle. Proceedings of the Koninklijke Nederlondse Akademie van Wetenschappen Series A, 63, 38–47. Stephens, M.A., 1965, The goodness-of-fit statistic VN: distribution and significance points. Biometrika, 52, 309–321. Anderson, T.W. and Darling, D.A., 1954, A test of goodness-of-fit. Journal of the American Statistical Association, 49, 765–769. Lewis, P.A.W., 1961, Distribution of the Anderson–Darling statistic. Annals of Mathematical Statistics, 32, 1118–1124. Watson, G.S., 1961, Goodness-of-fit on a circle.Biometrika, 48, 109–114. Durbin, J., 1969, Test for serial correlation in regression analysis based on the periodogram of least-squares residuals. Biometrika, 56, 1–16. Brunk, H.D., 1962, On the range of the difference between hypothetical distribution function and Pyke’s modified empirical distribution function. Annals of Mathematical Statistics, 33, 525–532. Stephens, M.A., 1969, Results from the relation between two statistics of the Kolgomorov–Smirnov type. Annals of Mathematical Statistics, 40, 1833–1837. Hegazy, Y.A.S. and Green, J.R., 1975, Some new goodness-of-fit tests using order statistics. Applied Statistics, 24, 299–308. D-Agostino, R.B. and Stephens, M.A., 1986, Goodness–of–Fit Techniques, Statistics, Textbook and Monographs, Vol. 68 (New York: Marcel Dekker, Inc.). Greenwood, M., 1946, The statistical study of infectious diseases. Journal of Royal Statistical Society Series A, 109, 85–110. Moran, P.A.P., 1947, The random division of an interval–Part I. Journal of Royal Statistical Society Series B, 9, 92–98. Moran, P.A.P., 1951, The random division of an interval–Part II. Journal of Royal Statistical Society Series B, 13, 147–150. Burrows, P.M., 1979, Selected percentage points of Greenwood’s statistic. Journal of Royal Statistical Society, 142, 256–258. Hill, I.D., 1979, Approximating the distribution of Greenwood’s statistic with Johnson distributions.Journal of Royal Statistical Society Series A,142,378–380;(Corrigendum (1981). Journal of Royal Statistical Society Series A, 144, 388). Currie, I.D.,1981,Further percentage points for Greenwood’s Statistic. Journal of Royal Statistical Society Series A, 144, 360–363. Stephens, M.A., 1981, Further percentage points for Greenwood’s statistic. Journal of Royal Statistical Society Series A, 144, 364–366. Quesenberry, C.P. and Miller, F.L. Jr., 1977, Power studies of some tests for uniformity. Journal of Statistical Computation and Simulation, 5, 169–191. Read, R.C. and Cressie, N.A.C., 1988, Goodness-of-fit Statistics for Discrete Multivariate Data (New York: Springer Verlag). Darling, D.A., 1953, On a class of problems related to the random division of an interval. Annals of Mathematical Statistics, 24, 239–253. Gebert, J.R. and Kale, B.K., 1969, Goodness of fit tests based on discriminatory information. Statistische Hefte, 10, 192–200. Del Pino, G.E., 1979, On the asymptotic distribution of k-spacing with applications to goodness-of-fit tests. Annals of Statistics, 7, 1058–1065. Hartley, H.O. and Pfaffenberger, R.C., 1972, Quadratic forms in order statistics used as goodness-of-fit criteria. Biometrika, 59, 605–611. Cressie,N.,1976, On the logarithms of high order spacings. Biometrika, 63, 343–355. Cressie, N., 1977, The minimum of higher order gaps. Australian Journal of Statistics, 19, 132–143. Cressie, N., 1978, Power results for tests based on high order gaps. Biometrika, 65, 214–218. Cressie, N., 1979, An optimal statistic based on higher order gaps. Biometrika, 66, 619–627. Deken, J.G., 1980, Exact distributions for gaps and stretches. Technical Report, Department of Statistics, Stanford University. McLaren, C.G. and Stephens, M.A., 1985, Percentage points and power for spacings statistics for testing uniformity. Technical Report, Department of Mathematics and Statistics, Simon Fraser University. Vasicek,O.,1976,Atest for normality based on sample entropy. Journal of the Royal Statistical Society Series B, 38, 54–59. Dudewicz, E.J. and van der Meulen, E.C., 1981, Entropy-based tests of uniformity. Journal of the American Statistical Association, 76, 967–974. Swartz, T., 1992, Goodness-of-fit tests using Kullback–Leibler information. Communications in Statistics. Theory and Methods, 21, 711–729. Stephens, M.A., 1974, EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association, 69, 730–737. Morales, D., Pardo, L., Pardo, M.C. and Vajda, I., 2003, Limit laws for disparities of spacings. Journal of Nonparametric Statistics, 15(3), 325–342. Pardo, M.C., 2003, A test for uniformity based on informational energy. Statistical Papers, 44,521–534. Cressie, N. and Read, T.R.C., 1984, Multinomial goodness-of-fit tests. Journal of the Royal Statistic Society Series B, 46, 440–464. Marhuenda, M.A., Marhuenda, Y. and Morales, D., 2005a, Uniformity tests under quantile categorization. Kybernetes (in press Vol. 34, number 5/6). Neyman, J., 1937, Smooth test for goodness of fit. Skandinavisk Aktuarietidskrift, 20, 150–199. Ledwina, T., 1994, Data-driven version of Neyman’s smooth test of fit. Journal of the American Statistical Association, 89, 1000–1005. Schwarz, G., 1978, Estimating the dimension of a model. Annals of Statistics, 6, 461–464. Zhang, J., 2002, Powerful goodness-of-fit tests based on the likelihood ratio. Journal of Royal Statistical Society Series B, 64, 281–294. Wichmann, B.A. and Hill, I.D., 1982, Algorithm AS 183: An efficient and portable pseudorandom number generator. Applied Statistics, 31, 188–190; (Corrections (1984). Applied Statistics,33, 123). Marhuenda, Y., 2002, Contrastes de uniformidad. PhD thesis, Miguel Hernández University, Spain. Marhuenda, Y., Morales, D. and Pardo, M.C., 2005, Power results of tests for the uniform distribution. Technical Report (I-2005–09), Operation Research Center, Miguel Hernández University of Elche (Spain). Stephens, M.A., 1970, Use of the Kolgomorov–Smirnov, Cramer-von Mises and related statistics without extensive tables. Journal of Royal Statistical Society Series B, 32, 115–122.
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