Journal of Statistical Theory and Practice

, Volume 3, Issue 3, pp 645–664 | Cite as

Data-Driven Smooth Tests for a Location-Scale Family Revisited

  • A. JanicEmail author
  • T. Ledwina


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.


Efficient score statistic Goodness-of-fit test Model selection Schwarz’s rule Test for extreme value distribution Test for normality 

AMS Subject Classification

62G10 62G20 


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  1. Aerts, M., Claeskens, G., Hart, J., 1999. Testing the fit of a parametric function. J. Amer. Statist. Assoc., 94, 869–879.MathSciNetzbMATHGoogle Scholar
  2. Aerts, M., Claeskens, G., Hart, J., 2000. Testing lack of fit in multiple regression. Biometrika, 87, 405–424.MathSciNetzbMATHGoogle Scholar
  3. Bai, Z., Chen, L., 2003. Weighted W test for normality and asymptotics a revisit of Chen-Shapiro test for normality. J. Statist. Plann. Inference, 113, 485–503.MathSciNetzbMATHGoogle Scholar
  4. Baklanov, E., Borisov, I., 2003. Probability inequalities and limit theorems for generalized L-statistics. Lith. Math. J., 43, 125–140.MathSciNetzbMATHGoogle Scholar
  5. Bera, A., Bilias, Y., 2001a. 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.MathSciNetzbMATHGoogle Scholar
  6. Bera, A., Bilias, Y., 2001b. On optimality properties of Fisher-Rao score function in testing and estimation. Commun. Statist.-Theory Meth., 30, 1533–1559.MathSciNetzbMATHGoogle Scholar
  7. Bickel, P., Ritov, Y., Stoker, T., 2006. Tailor-made tests for goodness of fit to semiparametric hypotheses. Ann. Statist., 34, 721–741.MathSciNetzbMATHGoogle Scholar
  8. Bogdan, M., Ledwina, T., 1996. Testing uniformity via log-spline modeling. Statistics, 28, 131–157.MathSciNetzbMATHGoogle Scholar
  9. Bühler, W., Puri, P., 1966. On optimal asymptotic tests of composite hypotheses with several constraints. Z. Wahrsch. verw. Geb., 5, 71–88.MathSciNetzbMATHGoogle Scholar
  10. Chen, L., Shapiro, S., 1995. An alternative test for normality based on normalized spacings. J. Statist. Comput. Simulation, 53, 269–288.zbMATHGoogle Scholar
  11. Claeskens, G., Hjort, N., 2004. Goodness of fit via non-parametric likelihood ratios. Scand. J. Statist., 31, 487–513.MathSciNetzbMATHGoogle Scholar
  12. D’Agostino, R., Stephens, M., 1986. Goodness-of-Fit Techniques. Dekker, New York.zbMATHGoogle Scholar
  13. Downton, F., 1966. Linear estimates with polynomial coefficients. Biometrika, 53, 129–141.MathSciNetzbMATHGoogle Scholar
  14. Fromont, M., Laurent, B., 2006. Adaptive goodness-of-fit tests in a density model. Ann. Statist., 34, 680–720.MathSciNetzbMATHGoogle Scholar
  15. Greenwood, J., Landwehr, J., Matalas, N., 1979. Probability weighted moments: definition and relation to parameters of several distributions expressible in inverse form. Water Resour. Res., 15, 1049–1054.Google Scholar
  16. Hosking, J., Wallis, J., Wood, E., 1985. Estimation of the generalized extreme-value distribution by the method of probability-weighted moments. Technometrics, 27, 251–261.MathSciNetGoogle Scholar
  17. Huber, P., 1981. Robust Statistics. Wiley, New York.zbMATHGoogle Scholar
  18. Inglot, T., Kallenberg, W., 2003. Moderate deviations of minimum contrast estimators under contamination. Ann. Statist., 31, 852–879.MathSciNetzbMATHGoogle Scholar
  19. Inglot, T., Kallenberg, W., Ledwina, T., 1994. On selection rules with an application to goodness-of-fit for composite hypotheses. Memorandum 1242. University of Twente, Faculty of Applied Mathematics.Google Scholar
  20. Inglot, T., Kallenberg, W., Ledwina, T., 1997. Data driven smooth tests for composite hypotheses. Ann. Statist., 25, 1222–1250.MathSciNetzbMATHGoogle Scholar
  21. Inglot, T., Ledwina, T., 1993. Moderately large deviations and expansions of large deviations for some functionals of weighted empirical process. Ann. Probab., 21, 1691–1705.MathSciNetzbMATHGoogle Scholar
  22. Inglot, T., Ledwina, T., 2001. Asymptotic optimality of data driven smooth tests for location-scale family. Sankhyā Ser. A, 60,41-71.Google Scholar
  23. Inglot, T., Ledwina, T., 2006a. Data-driven score tests for homoscedastic linear regression model: asymptotic results. Probab. Math. Statist., 26, 41–61.MathSciNetzbMATHGoogle Scholar
  24. Inglot, T., Ledwina, T., 2006b. Data-driven score tests for homoscedastic linear regression model: the construction and simulations. In M. Hušková, M. Janžura (eds.), Prague Stochastics 2006. Proceedings, 124–137. Matfyzpress, Prague.Google Scholar
  25. Inglot, T., Ledwina, T., 2006c. Towards data driven selection of a penalty function for data driven Neyman tests. Linear Algebra and its Appl., 417, 579–590.MathSciNetzbMATHGoogle Scholar
  26. Janic-Wróblewska, A., 2004a. Data-driven smooth test for a location-scale family. Statistics, 38, 337–355.MathSciNetzbMATHGoogle Scholar
  27. Janic-Wróblewska, A., 2004b. Data-driven smooth test for extreme value distribution. Statistics, 38, 413–426.MathSciNetzbMATHGoogle Scholar
  28. Javitz, H., 1975. Generalized smooth tests of goodness of fit, independence and equality of distributions. Ph.D. thesis, University of California., Berkeley, USA.Google Scholar
  29. Kallenberg, W., Ledwina, T., 1995. On data driven Neyman’s tests. Probab. Math. Statist., 15, 409–426.MathSciNetzbMATHGoogle Scholar
  30. Kallenberg, W., Ledwina, T., 1997a. Data driven smooth tests for composite hypotheses: Comparison of powers. J. Statist. Comput. Simul., 59, 101–121.MathSciNetzbMATHGoogle Scholar
  31. Kallenberg, W., Ledwina, T., 1997b. Data driven smooth tests when the hypothesis is composite. J. Amer. Statist. Assoc., 92, 1094–1104.MathSciNetzbMATHGoogle Scholar
  32. Kallenberg, W., Ledwina, T., 1999. Data driven rank tests for independence. J. Amer. Statist. Assoc., 94, 285–301.MathSciNetzbMATHGoogle Scholar
  33. Landwehr, J., Matalas, N., 1979. Probability weighted moments compared with some traditional techniques in estimating Gumbel parameters and quantiles. Water Resour. Res., 15, 1055–1064.Google Scholar
  34. LaRiccia, V., 1991. Smooth goodness-of-fit tests: a quantile function approach. J. Amer. Statist. Assoc., 86, 427–431.MathSciNetGoogle Scholar
  35. LaRiccia, V., Mason, D., 1985. 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.MathSciNetzbMATHGoogle Scholar
  36. Le Cam, L., Lehmann, E., 1974. J. Neyman — on the occasion of his 80th birthday. Ann. Statist., 2, vii-xiii.Google Scholar
  37. Ledwina, T., 1994. Data-driven version of Neyman’s smooth test of fit. J. Amer. Statist. Assoc., 89, 1000–1005.MathSciNetzbMATHGoogle Scholar
  38. Meintanis, S., Swanepoel, J., 2007. Bootstrap goodness-of-fit tests with estimated parameters based on empirical transforms. Statist. Probab. Lett., 77, 1004–1013.MathSciNetzbMATHGoogle Scholar
  39. Morales, D., Pardo, L., Vajda, I., 1997. Some new statistics for testing hypotheses in parametric models. J. Multivariate Anal., 62, 137–168.MathSciNetzbMATHGoogle Scholar
  40. Neyman, J., 1954. Sur une famille de tests asymptotiques des hypothèses statistiques composées. Trabajos de Estadistica, 5, 161–168.zbMATHGoogle Scholar
  41. Neyman, J., 1959. Optimal asymptotic tests of composite statistical hypotheses. In U. Grenander (ed.), Probability and Statistics, Harald Cramér Volume, 212–234. Wiley, New York.Google Scholar
  42. Pearson, E., D’Agostino, R., Bowman, K., 1977. Tests for departure from normality: comparison of powers. Biometrika, 64, 231–246.zbMATHGoogle Scholar
  43. Rayner, J., Best, D., 1989. Smooth Tests of Goodness of Fit. Oxford University Press, New York.zbMATHGoogle Scholar
  44. Rubin, H., Sethuraman, J., 1965. Probabilities of moderate deviations. Sankhyā Ser. A, 27, 325–346.MathSciNetzbMATHGoogle Scholar
  45. Shorack, G., Wellner, J., 1986. Empirical Processes with Applications to Statistics. Wiley, New York.zbMATHGoogle Scholar
  46. Stute, W., Manteiga, W., Quindimil, M., 1993. Bootstrap based goodness-of-fit tests. Metrika, 40, 243–256.MathSciNetzbMATHGoogle Scholar
  47. Thomas, D., Pierce, D., 1979. Neyman’s smooth goodness-of-fit test when the hypothesis is composite. J. Amer. Statist. Assoc., 74, 441–445.MathSciNetGoogle Scholar

Copyright information

© Grace Scientific Publishing 2009

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsWrocław University of TechnologyWrocławPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWrocławPoland

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