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Goodness-of-fit tests for semiparametric and parametric hypotheses based on the probability weighted empirical characteristic function

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Abstract

We investigate the finite-sample properties of certain procedures which employ the novel notion of the probability weighted empirical characteristic function. The procedures considered are: (1) Testing for symmetry in regression, (2) Testing for multivariate normality with independent observations, and (3) Testing for multivariate normality of random effects in mixed models. Along with the new tests alternative methods based on the ordinary empirical characteristic function as well as other more well known procedures are implemented for the purpose of comparison.

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References

  • Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726

    Article  MathSciNet  MATH  Google Scholar 

  • Baringhaus L (1996) Fibonacci numbers, Lucas numbers and integrals of certain Gaussian processes. Proc Am Math Soc 124:3875–3884

    Article  MathSciNet  MATH  Google Scholar 

  • Baringhaus L, Henze N (1988) A consistent test for multivariate normality based on the empirical characteristic function. Metrika 35:339–348

    Article  MathSciNet  MATH  Google Scholar 

  • Bates D (2011) Computational methods for mixed models. URL http://cran.r-project.org/web/packages/lme4/vignettes/Theory.pdf

  • Caeiro F, Gomes MI (2011) Semi-parametric tail inference through probability-weighted moments. J Stat Plan Inference 141:935–950

    Article  MathSciNet  MATH  Google Scholar 

  • Caeiro F, Gomes MI, Vanderwalle B (2014) Semi-parametric probability-weighted moments estimation revisited. Methodol Comput Appl Probab 16:1–29

    Article  MathSciNet  MATH  Google Scholar 

  • Claeskens G, Hart JD (2009) Goodness-of-fit in mixed models (with discussion and rejoinder). Test 18:213–270

    Article  MathSciNet  MATH  Google Scholar 

  • Datta S, Bandyopadhyay D, Satten GA (2010) Inverse probability of censoring weighted \(u\)-statistics for right-censored data with an application to testing hypotheses. Scand J Stat 37:680

    Article  MathSciNet  MATH  Google Scholar 

  • de Wet T, Goegebeur Y, Guillou A (2012) Weighted moment estimators for the second order parameter. Methodol Comput Appl Probab 14:753–783

    Article  MathSciNet  MATH  Google Scholar 

  • Feuerverger A, Mureika R (1977) The empirical characteristic function and its applications. Ann Stat 5:88–97

    Article  MathSciNet  MATH  Google Scholar 

  • Ghosh S, Ruymgaart F (1992) Application of empirical characteristic functions in some multivariate problems. Can J Stat 20:429–440

    Article  MathSciNet  MATH  Google Scholar 

  • Gupta AK, Aziz MA (2012) Estimation of parameters of the unified skew normal distribution using the method of weighted moments. J Stat Theory Pract 6:402

    Article  MathSciNet  Google Scholar 

  • Gurka MJ, Edwards LJ (2008) Mixed models. In: Rao CR et al (eds) Handbook of statistics, vol 27. North-Holland, Amsterdam, pp 253–280

    Google Scholar 

  • Henze N (2002) Invariant tests for multivariate normality: a critical review. Stat Pap 43:467–506

    Article  MathSciNet  MATH  Google Scholar 

  • Henze N, Wagner T (1997) A new approach to the BHEP tests for multivariate normality. J Multivar Anal 62:1–23

    Article  MathSciNet  MATH  Google Scholar 

  • Hettmansperger TP, McKean JW, Sheather SJ (2002) Finite sample performance of tests for symmetry of the errors in a linear model. J Stat Comput Simul 72:863–879

    Article  MathSciNet  MATH  Google Scholar 

  • Hušková M, Meintanis SG (2012) Tests for symmetric error distribution in linear and nonparametric regression models. Commun Stat-Simul Comput 41:833–851

    Article  MathSciNet  MATH  Google Scholar 

  • Janssen A (2000) Global power of goodness-of-fit tests. Ann Stat 28:239–253

    Article  MathSciNet  MATH  Google Scholar 

  • Jiménez-Gamero MD (2014) On the empirical characteristic function process of the residuals in GARCH models and applications. Test 23:409–432

    Article  MathSciNet  MATH  Google Scholar 

  • Koutrouvelis IA (1985) Distribution-free procedures for location and symmetry inference problems based on the empirical characteristic function. Scand J Stat 12:257–269

    MathSciNet  MATH  Google Scholar 

  • Meintanis SG, Stupfler G (2015) Transformations to symmetry based on the probability weighted characteristic function. Kybernetika 51:571–587

    MathSciNet  MATH  Google Scholar 

  • Meintanis SG, Ushakov N (2016) Nonparametric probability weighted empirical characteristic function and applications. Stat Probab Lett 108:52–61

    Article  MathSciNet  MATH  Google Scholar 

  • Meintanis SG, Swanepoel J, Allison J (2014) The probability weighted characteristic function and goodness-of-fit testing. J Stat Plan Inference 146:122–132

    Article  MathSciNet  MATH  Google Scholar 

  • Meintanis SG, Allison J, Santana L (2016) Diagnostic tests for the distribution of random effects in multivariate mixed effects models. Commun Stat-Theory Methods 45:201–215

  • Neumeyer N, Dette H (2007) Testing for symmetric error distribution in nonparametric regression models. Stat Sin 17:775–795

    MathSciNet  MATH  Google Scholar 

  • Neumeyer N, Dette H, Nagel ER (2005) A note on testing symmetry of the error distribution in linear regression models. J Nonparametr Stat 17:697–715

    Article  MathSciNet  MATH  Google Scholar 

  • Ngatchou-Wandji J (2009) Testing for symmetry in multivariate distributions. Stat Methodol 6:230–250

    Article  MathSciNet  MATH  Google Scholar 

  • Raftery AE (1984) A continuous multivariate exponential distribution. Commun Stat-Theory Methods 13:947–965

    Article  MathSciNet  MATH  Google Scholar 

  • Székely GJ, Rizzo ML (2005) A new test for multivariate normality. J Multivar Anal 93:58–80

    Article  MathSciNet  MATH  Google Scholar 

  • Taufer E, Leonenko N (2009) Characteristic function estimation of non-Gaussian Ornstein-Uhlenbeck processes. J Stat Plan Inference 139:3050–3063

    Article  MathSciNet  MATH  Google Scholar 

  • Tenreiro C (2009) On the choice of the smoothing parameter for the BHEP goodness-of-fit test. Comput Stat Data Anal 53:1039–1053

    Article  MathSciNet  MATH  Google Scholar 

  • Ushakov N (1999) Selected topics in characteristic functions. VSP, Utrecht

    Book  MATH  Google Scholar 

  • Villaseñor Alva JA, González Estrada E (2009) A generalization of Shapiro-Wilk’s test for multivariate normality. Commun Stat-Theory Methods 38:1870–1883

    Article  MATH  Google Scholar 

  • Witkovský V, Wimmer G (2015) Exact statistical inference by using numerical FFT inversion of the characteristic function. J Palacký Univ Olomouc (to appear)

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Acknowledgments

S. Meintanis acknowledges support by the Special Account for Research Grants (ELKE) (Research Grant 11699) of the National & Kapodistrian University of Athens. J. Allison thanks the National Research Foundation of South Africa for financial support.

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Correspondence to Simos G. Meintanis.

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Simos G. Meintanis—On sabbatical leave from the University of Athens.

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Meintanis, S.G., Allison, J. & Santana, L. Goodness-of-fit tests for semiparametric and parametric hypotheses based on the probability weighted empirical characteristic function. Stat Papers 57, 957–976 (2016). https://doi.org/10.1007/s00362-016-0760-0

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  • DOI: https://doi.org/10.1007/s00362-016-0760-0

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