Journal of Statistical Theory and Practice

, Volume 3, Issue 3, pp 705–733 | Cite as

Informative Statistical Analyses Using Smooth Goodness of Fit Tests

  • O. ThasEmail author
  • J. C. W. Rayner
  • D. J. Best
  • B. De Boeck


We propose a methodology for informative goodness of fit testing that combines the merits of both hypothesis testing and nonparametric density estimation. In particular, we construct a data-driven smooth test that selects the model using a weighted integrated squared error (WISE) loss function. When the null hypothesis is rejected, we suggest plotting the estimate of the selected model. This estimate is optimal in the sense that it minimises the WISE loss function. This procedure may be particularly helpful when the components of the smooth test are not diagnostic for detecting moment deviations. Although this approach relies mostly on existing theory of (generalised) smooth tests and nonparametric density estimation, there are a few issues that need to be resolved so as to make the procedure applicable to a large class of distributions. In particular, we will need an estimator of the variance of the smooth test components that is consistent in a large class of distributions for which the nuisance parameters are estimated by method of moments. This estimator may also be used to construct diagnostic component tests.

The properties of the new variance estimator, the new diagnostic components and the proposed informative testing procedure are evaluated in several simulation studies. We demonstrate the new methods on testing for the logistic and extreme value distributions.


Generalised score test Method of moment estimator Nonparametric density estimation Orthonormal polynomials 

AMS Subject Classification

62G07 62G10 


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  1. Akaike, H., 1973. Information theory and an extension of the maximum likelihood principle. In Second International Symposium on Inference Theory, Petrov, B., Csàki, F. (editors), 267–281. Akadémiai Kiadó, Budapest.Google Scholar
  2. Akaike, H., 1974. A new look at statistical model identification. I.E.E.E. Trans. Auto. Control, 19, 716–723.MathSciNetCrossRefGoogle Scholar
  3. Anderson, G., de Figueiredo, R., 1980. An adaptive orthogonal-series estimator for probability density functions. Annals of Statistics, 8, 347–376.MathSciNetCrossRefGoogle Scholar
  4. Bain, L., Easterman, J., Engelhardt, M., 1973. A study of life-testing models and statistical analyses for the logistic distribution. Technical Report ARL-73-0009, Aerospace Research Laboratories, Wright Patterson AFB.Google Scholar
  5. Baringhaus, L., Henze, N., 1992. Limit distributions for Mardia measure of multivariate skewness. Annals of Statistics, 20, 1889–1902.MathSciNetCrossRefGoogle Scholar
  6. Barton D., 1953. On Neyman’s smooth test of goodness of fit and its power with respect to a particular system of alternatives. Skandinavisk Aktuarietidskrift, 36, 24–63.MathSciNetzbMATHGoogle Scholar
  7. Bickel, P., Ritov, Y., Stoker, T., 2006. Tailor-made tests of goodness of fit to semiparametric hypotheses. Annals of Statistics, 34, 721–741.MathSciNetCrossRefGoogle Scholar
  8. Boos, D., 1992. On generalized score tests. The American Statistician, 46, 327–333.Google Scholar
  9. Buckland, S., 1992. Fitting density functions with polynomials. Applied Statistics, 41, 63–76.MathSciNetCrossRefGoogle Scholar
  10. Cencov, N., 1962. Evaluation of an unknown distribution density from observations. Soviet. Math., 3, 1559–1562.Google Scholar
  11. Claeskens, G., Hjort, N., 2004. Goodness of fit via non-parametric likelihood ratios. Scandinavian Journal of Statistics, 31, 487–513.MathSciNetCrossRefGoogle Scholar
  12. Clutton-Brock, M., 1990. Density estimation using exponentials of orthogonal series. Journal of the American Statistical Association, 85, 760–764.MathSciNetCrossRefGoogle Scholar
  13. Diggle, P., Hall, P., 1986. The selection of terms in an orthogonal series density estimator. Journal of the American Statistical Association, 81, 230–233.MathSciNetCrossRefGoogle Scholar
  14. Efron, B., Tibshirani, R., 1996. Using specially designed exponential families for density estimation. Annals of Statistics, 24, 2431–2461.MathSciNetCrossRefGoogle Scholar
  15. Emerson, P., 1968. Numerical construction of orthogonal polynomials from a general recurrence formula. Biometrics, 24, 695–701.CrossRefGoogle Scholar
  16. Engelhardt, M., 1975. Simple linear estimation of the parameters of the logistic distribution from a complete or censored sample. Journal of the American Statistical Association, 70, 899–902.CrossRefGoogle Scholar
  17. Eubank, R., LaRiccia, V., Rosenstein, R., 1987. Test statistics derived as components of Pearson’s phi-squared distance measure. Journal of the American Statistical Association, 82, 816–825.MathSciNetCrossRefGoogle Scholar
  18. Gajek, G., 1986. On improving density estimators which are not bona fide functions. Annals of Statistics, 14, 1612–1618.MathSciNetCrossRefGoogle Scholar
  19. Glad, I., Hjort, N., Ushakov, N., 2003. Correction of density estimators that are not densities. Scandinavian Journal of Statistics, 30, 415–427.MathSciNetCrossRefGoogle Scholar
  20. Hall, W., Mathiason, D., 1990. On large-sample estimation and testing in parametric models. International Statistical Review, 58, 77–97.CrossRefGoogle Scholar
  21. Henze, N., 1997. Do components of smooth tests of fit have diagnostic properties? Metrika, 45, 121–130.MathSciNetCrossRefGoogle Scholar
  22. Henze, N., Klar, B., 1996. Properly rescaled components of smooth tests of fit are diagnostic. Australian Journal of Statistics, 38, 61–74.MathSciNetCrossRefGoogle Scholar
  23. Hjort, N., Glad, I., 1995. Nonparametric density estimation with a parametric start. Annals of Statistics, 23, 882–904.MathSciNetCrossRefGoogle Scholar
  24. Kallenberg, W., Ledwina, T., 1995. Consistency and Monte Carlo simulation of a data driven version of smooth goodness-of-fit tests. Annals of Statistics, 23, 1594–1608.MathSciNetCrossRefGoogle Scholar
  25. Kallenberg, W., Ledwina, T., 1997. Data-driven smooth tests when the hypothesis is composite. Journal of the American Statistical Association, 92, 1094–1104.MathSciNetCrossRefGoogle Scholar
  26. Kallenberg, W., Ledwina, T., Rafajlowicz, E., 1997. Testing bivariate independence and normality. Sankhyā, Series A, 59,42-59.Google Scholar
  27. Klar, B., 2000. Diagnostic smooth tests of fit. Metrika, 52, 237–252.MathSciNetCrossRefGoogle Scholar
  28. Ledwina, T., 1994. Data-driven version of Neyman’s smooth test of fit. Journal of the American Statistical Association, 89, 1000–1005.MathSciNetCrossRefGoogle Scholar
  29. Lehmann, E., 1999. Elements of Large-Sample Theory. Springer, New York.CrossRefGoogle Scholar
  30. Mardia, K., Kent, J., 1991. Rao score tests for goodness-of-fit and independence. Biometrika, 78, 355–363.MathSciNetCrossRefGoogle Scholar
  31. Rayner J., Best D., 1989. Smooth Tests of Goodness-of-Fit. Oxford University Press, New York.zbMATHGoogle Scholar
  32. Rayner, J., Best, D., Mathews, K., 1995. Interpreting the skewness coefficient. Communications in Statistics — Theory and Methods, 24, 593–600.CrossRefGoogle Scholar
  33. Rayner, J., Best, D., Thas, O., 2009a. Generalised smooth tests of goodness of fit. Journal of Statistical Theory and Practice, 3(3), 665–679. Accompanying paper.MathSciNetCrossRefGoogle Scholar
  34. Rayner, J., Thas, O., Best, D., 2009b. Smooth Tests of Goodness of Fit. Wiley, New York, USA.CrossRefGoogle Scholar
  35. Rayner, J., Thas, O., De Boeck, B., 2008. A generalised Emerson recurrence relation. Australian and New Zealand Journal of Statistics, 50, 235–240.MathSciNetCrossRefGoogle Scholar
  36. Schwarz, G., 1978. Estimating the dimension of a model. Annals of Statistics, 6, 461–464.MathSciNetCrossRefGoogle Scholar
  37. Stuart, A., Ord, J., 1994. Kendall’s Advanced Theory of Statistics. Arnold / Halsted, London.zbMATHGoogle Scholar
  38. Tarter, M., 1976. An introduction to the implementation and theory of nonparametric density estimation. The American Statistician, 30, 105–112.zbMATHGoogle Scholar
  39. van der Vaart, A., 1998. Asymptotic Statistics. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
  40. Wasserman, L., 2005. All of Nonparametric Statistics. Springer.zbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2009

Authors and Affiliations

  • O. Thas
    • 1
    Email author
  • J. C. W. Rayner
    • 2
  • D. J. Best
    • 2
  • B. De Boeck
    • 1
  1. 1.Department of Applied Mathematics, Biometrics and Process ControlGhent UniversityGentBelgium
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleAustralia

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