Data-driven smooth tests for a location-scale family revisited. (English) Zbl 1211.62073

Janic, A.; Ledwina, T.

J. Stat. Theory Pract. 3, No. 3, 645-664 (2009).

Summary: A new data-driven, smooth goodness of test for a location-scale family is proposed and studied. The new test statistic is a combination of an efficient score statistic and an appropriate selection rule. Some examples are presented and by using extensive simulations the test is shown to have desirable properties.

MSC:

62G10	Nonparametric hypothesis testing
62G20	Asymptotic properties of nonparametric inference
65C60	Computational problems in statistics (MSC2010)

Keywords:

efficient score statistic; goodness-of-fit test; model selection; Schwarz’s rule; test for extreme value distribution; test for normality

References:

[1]	Aerts, M.; Claeskens, G.; Hart, J., Testing the fit of a parametric function, J. Amer. Statist. Assoc., 94, 869-879 (1999) · Zbl 0996.62044
[2]	Aerts, M.; Claeskens, G.; Hart, J., Testing lack of fit in multiple regression, Biometrika, 87, 405-424 (2000) · Zbl 0948.62028
[3]	Bai, Z.; Chen, L., Weighted W test for normality and asymptotics a revisit of Chen-Shapiro test for normality, J. Statist. Plann. Inference, 113, 485-503 (2003) · Zbl 1026.62042
[4]	Baklanov, E.; Borisov, I., Probability inequalities and limit theorems for generalized L-statistics, Lith. Math. J., 43, 125-140 (2003) · Zbl 1052.60015
[5]	Bera, A.; Bilias, Y., Rao’s score, Neyman’s C(α) and Silvey’s LM tests: an essay on historical developments and some new results, J. Statist. Plann. Inference, 97, 9-44 (2001) · Zbl 1053.62023
[6]	Bera, A.; Bilias, Y., On optimality properties of Fisher-Rao score function in testing and estimation, Commun. Statist.-Theory Meth., 30, 1533-1559 (2001) · Zbl 1008.62545
[7]	Bickel, P.; Ritov, Y.; Stoker, T., Tailor-made tests for goodness of fit to semiparametric hypotheses, Ann. Statist., 34, 721-741 (2006) · Zbl 1092.62050
[8]	Bogdan, M.; Ledwina, T., Testing uniformity via log-spline modeling, Statistics, 28, 131-157 (1996) · Zbl 1077.62515
[9]	Bühler, W.; Puri, P., On optimal asymptotic tests of composite hypotheses with several constraints, Z. Wahrsch. verw. Geb., 5, 71-88 (1966) · Zbl 0203.21006
[10]	Chen, L.; Shapiro, S., An alternative test for normality based on normalized spacings, J. Statist. Comput. Simulation, 53, 269-288 (1995) · Zbl 0873.62045
[11]	Claeskens, G.; Hjort, N., Goodness of fit via non-parametric likelihood ratios, Scand. J. Statist., 31, 487-513 (2004) · Zbl 1065.62056
[12]	D’Agostino, R., Stephens, M., 1986. Goodness-of-Fit Techniques. Dekker, New York. · Zbl 0597.62030
[13]	Downton, F., Linear estimates with polynomial coefficients, Biometrika, 53, 129-141 (1966) · Zbl 0173.46403
[14]	Fromont, M.; Laurent, B., Adaptive goodness-of-fit tests in a density model, Ann. Statist., 34, 680-720 (2006) · Zbl 1096.62040
[15]	Greenwood, J.; Landwehr, J.; Matalas, N., Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form, Water Resour. Res., 15, 1049-1054 (1979)
[16]	Hosking, J.; Wallis, J.; Wood, E., Estimation of the generalized extreme-value distribution by the method of probability-weighted moments, Technometrics, 27, 251-261 (1985)
[17]	Huber, P., 1981. Robust Statistics. Wiley, New York. · Zbl 0536.62025
[18]	Inglot, T.; Kallenberg, W., Moderate deviations of minimum contrast estimators under contamination, Ann. Statist., 31, 852-879 (2003) · Zbl 1028.62012
[19]	Inglot, T.; Kallenberg, W.; Ledwina, T., On selection rules with an application to goodness-of-fit for composite hypotheses (1994)
[20]	Inglot, T.; Kallenberg, W.; Ledwina, T., Data driven smooth tests for composite hypotheses, Ann. Statist., 25, 1222-1250 (1997) · Zbl 0904.62055
[21]	Inglot, T.; Ledwina, T., Moderately large deviations and expansions of large deviations for some functionals of weighted empirical process, Ann. Probab., 21, 1691-1705 (1993) · Zbl 0786.60027
[22]	Inglot, T.; Ledwina, T., Asymptotic optimality of data driven smooth tests for location-scale family (2001) · Zbl 0999.62034
[23]	Inglot, T.; Ledwina, T., Data-driven score tests for homoscedastic linear regression model: asymptotic results, Probab. Math. Statist., 26, 41-61 (2006) · Zbl 1104.62051
[24]	Inglot, T.; Ledwina, T.; Hušková, M. (ed.); Janžura, M. (ed.), Data-driven score tests for homoscedastic linear regression model: the construction and simulations, 124-137 (2006), Prague
[25]	Inglot, T.; Ledwina, T., Towards data driven selection of a penalty function for data driven Neyman tests, Linear Algebra and its Appl., 417, 579-590 (2006) · Zbl 1095.62023
[26]	Janic-Wróblewska, A., Data-driven smooth test for a location-scale family, Statistics, 38, 337-355 (2004) · Zbl 1106.62052
[27]	Janic-Wróblewska, A., Data-driven smooth test for extreme value distribution, Statistics, 38, 413-426 (2004) · Zbl 1085.62049
[28]	Javitz, H., 1975. Generalized smooth tests of goodness of fit, independence and equality of distributions. Ph.D. thesis, University of California., Berkeley, USA.
[29]	Kallenberg, W.; Ledwina, T., On data driven Neyman’s tests, Probab. Math. Statist., 15, 409-426 (1995) · Zbl 0847.62034
[30]	Kallenberg, W.; Ledwina, T., Data driven smooth tests for composite hypotheses: Comparison of powers, J. Statist. Comput. Simul., 59, 101-121 (1997) · Zbl 1067.62535
[31]	Kallenberg, W.; Ledwina, T., Data driven smooth tests when the hypothesis is composite, J. Amer. Statist. Assoc., 92, 1094-1104 (1997) · Zbl 1067.62534
[32]	Kallenberg, W.; Ledwina, T., Data driven rank tests for independence, J. Amer. Statist. Assoc., 94, 285-301 (1999) · Zbl 1072.62574
[33]	Landwehr, J.; Matalas, N., Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles, Water Resour. Res., 15, 1055-1064 (1979)
[34]	LaRiccia, V., Smooth goodness-of-fit tests: a quantile function approach, J. Amer. Statist. Assoc., 86, 427-431 (1991)
[35]	LaRiccia, V.; Mason, D., Optimal goodness-of-fit tests for location/scale families of distributions based on the sum of squares of L-statistics, Ann. Statist., 13, 315-330 (1985) · Zbl 0576.62031
[36]	Cam, L.; Lehmann, E., J. Neyman — on the occasion of his 80th birthday (1974) · Zbl 0278.01020
[37]	Ledwina, T., Data-driven version of Neyman’s smooth test of fit, J. Amer. Statist. Assoc., 89, 1000-1005 (1994) · Zbl 0805.62022
[38]	Meintanis, S.; Swanepoel, J., Bootstrap goodness-of-fit tests with estimated parameters based on empirical transforms, Statist. Probab. Lett., 77, 1004-1013 (2007) · Zbl 1116.62052
[39]	Morales, D.; Pardo, L.; Vajda, I., Some new statistics for testing hypotheses in parametric models, J. Multivariate Anal., 62, 137-168 (1997) · Zbl 0877.62020
[40]	Neyman, J., Sur une famille de tests asymptotiques des hypothèses statistiques composées, Trabajos de Estadistica, 5, 161-168 (1954) · Zbl 0059.13202
[41]	Neyman, J.; Grenander, U. (ed.), Optimal asymptotic tests of composite statistical hypotheses, 212-234 (1959), New York · Zbl 0104.12602
[42]	Pearson, E.; D’Agostino, R.; Bowman, K., Tests for departure from normality: comparison of powers, Biometrika, 64, 231-246 (1977) · Zbl 0371.62102
[43]	Rayner, J., Best, D., 1989. Smooth Tests of Goodness of Fit. Oxford University Press, New York. · Zbl 0731.62064
[44]	Rubin, H.; Sethuraman, J., Probabilities of moderate deviations, Sankhyā Ser. A, 27, 325-346 (1965) · Zbl 0178.53802
[45]	Shorack, G., Wellner, J., 1986. Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[46]	Stute, W.; Manteiga, W.; Quindimil, M., Bootstrap based goodness-of-fit tests, Metrika, 40, 243-256 (1993) · Zbl 0770.62016
[47]	Thomas, D.; Pierce, D., Neyman’s smooth goodness-of-fit test when the hypothesis is composite, J. Amer. Statist. Assoc., 74, 441-445 (1979)