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Informative Statistical Analyses Using Smooth Goodness of Fit Tests

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Abstract

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.

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References

  • 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 

  • Akaike, H., 1974. A new look at statistical model identification. I.E.E.E. Trans. Auto. Control, 19, 716–723.

    Article  MathSciNet  Google Scholar 

  • Anderson, G., de Figueiredo, R., 1980. An adaptive orthogonal-series estimator for probability density functions. Annals of Statistics, 8, 347–376.

    Article  MathSciNet  Google Scholar 

  • 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 

  • Baringhaus, L., Henze, N., 1992. Limit distributions for Mardia measure of multivariate skewness. Annals of Statistics, 20, 1889–1902.

    Article  MathSciNet  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Bickel, P., Ritov, Y., Stoker, T., 2006. Tailor-made tests of goodness of fit to semiparametric hypotheses. Annals of Statistics, 34, 721–741.

    Article  MathSciNet  Google Scholar 

  • Boos, D., 1992. On generalized score tests. The American Statistician, 46, 327–333.

    Google Scholar 

  • Buckland, S., 1992. Fitting density functions with polynomials. Applied Statistics, 41, 63–76.

    Article  MathSciNet  Google Scholar 

  • Cencov, N., 1962. Evaluation of an unknown distribution density from observations. Soviet. Math., 3, 1559–1562.

    Google Scholar 

  • Claeskens, G., Hjort, N., 2004. Goodness of fit via non-parametric likelihood ratios. Scandinavian Journal of Statistics, 31, 487–513.

    Article  MathSciNet  Google Scholar 

  • Clutton-Brock, M., 1990. Density estimation using exponentials of orthogonal series. Journal of the American Statistical Association, 85, 760–764.

    Article  MathSciNet  Google Scholar 

  • Diggle, P., Hall, P., 1986. The selection of terms in an orthogonal series density estimator. Journal of the American Statistical Association, 81, 230–233.

    Article  MathSciNet  Google Scholar 

  • Efron, B., Tibshirani, R., 1996. Using specially designed exponential families for density estimation. Annals of Statistics, 24, 2431–2461.

    Article  MathSciNet  Google Scholar 

  • Emerson, P., 1968. Numerical construction of orthogonal polynomials from a general recurrence formula. Biometrics, 24, 695–701.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Gajek, G., 1986. On improving density estimators which are not bona fide functions. Annals of Statistics, 14, 1612–1618.

    Article  MathSciNet  Google Scholar 

  • Glad, I., Hjort, N., Ushakov, N., 2003. Correction of density estimators that are not densities. Scandinavian Journal of Statistics, 30, 415–427.

    Article  MathSciNet  Google Scholar 

  • Hall, W., Mathiason, D., 1990. On large-sample estimation and testing in parametric models. International Statistical Review, 58, 77–97.

    Article  Google Scholar 

  • Henze, N., 1997. Do components of smooth tests of fit have diagnostic properties? Metrika, 45, 121–130.

    Article  MathSciNet  Google Scholar 

  • Henze, N., Klar, B., 1996. Properly rescaled components of smooth tests of fit are diagnostic. Australian Journal of Statistics, 38, 61–74.

    Article  MathSciNet  Google Scholar 

  • Hjort, N., Glad, I., 1995. Nonparametric density estimation with a parametric start. Annals of Statistics, 23, 882–904.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Kallenberg, W., Ledwina, T., 1997. Data-driven smooth tests when the hypothesis is composite. Journal of the American Statistical Association, 92, 1094–1104.

    Article  MathSciNet  Google Scholar 

  • Kallenberg, W., Ledwina, T., Rafajlowicz, E., 1997. Testing bivariate independence and normality. Sankhyā, Series A, 59,42-59.

    Google Scholar 

  • Klar, B., 2000. Diagnostic smooth tests of fit. Metrika, 52, 237–252.

    Article  MathSciNet  Google Scholar 

  • Ledwina, T., 1994. Data-driven version of Neyman’s smooth test of fit. Journal of the American Statistical Association, 89, 1000–1005.

    Article  MathSciNet  Google Scholar 

  • Lehmann, E., 1999. Elements of Large-Sample Theory. Springer, New York.

    Book  Google Scholar 

  • Mardia, K., Kent, J., 1991. Rao score tests for goodness-of-fit and independence. Biometrika, 78, 355–363.

    Article  MathSciNet  Google Scholar 

  • Rayner J., Best D., 1989. Smooth Tests of Goodness-of-Fit. Oxford University Press, New York.

    MATH  Google Scholar 

  • Rayner, J., Best, D., Mathews, K., 1995. Interpreting the skewness coefficient. Communications in Statistics — Theory and Methods, 24, 593–600.

    Article  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Rayner, J., Thas, O., Best, D., 2009b. Smooth Tests of Goodness of Fit. Wiley, New York, USA.

    Book  Google Scholar 

  • Rayner, J., Thas, O., De Boeck, B., 2008. A generalised Emerson recurrence relation. Australian and New Zealand Journal of Statistics, 50, 235–240.

    Article  MathSciNet  Google Scholar 

  • Schwarz, G., 1978. Estimating the dimension of a model. Annals of Statistics, 6, 461–464.

    Article  MathSciNet  Google Scholar 

  • Stuart, A., Ord, J., 1994. Kendall’s Advanced Theory of Statistics. Arnold / Halsted, London.

    MATH  Google Scholar 

  • Tarter, M., 1976. An introduction to the implementation and theory of nonparametric density estimation. The American Statistician, 30, 105–112.

    MATH  Google Scholar 

  • van der Vaart, A., 1998. Asymptotic Statistics. Cambridge University Press, Cambridge.

    Book  Google Scholar 

  • Wasserman, L., 2005. All of Nonparametric Statistics. Springer.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Biometrics and Process Control, Ghent University, 9000, Gent, Belgium

    O. Thas & B. De Boeck

  2. School of Mathematical and Physical Sciences, University of Newcastle, NSW, 2308, Australia

    J. C. W. Rayner & D. J. Best

Authors
  1. O. Thas
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  2. J. C. W. Rayner
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  3. D. J. Best
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  4. B. De Boeck
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Correspondence to O. Thas.

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Thas, O., Rayner, J.C.W., Best, D.J. et al. Informative Statistical Analyses Using Smooth Goodness of Fit Tests. J Stat Theory Pract 3, 705–733 (2009). https://doi.org/10.1080/15598608.2009.10411955

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  • DOI: https://doi.org/10.1080/15598608.2009.10411955

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