Kybernetika 51 no. 4, 571-587, 2015

Transformations to symmetry based on the probability weighted characteristic function

Simos G. Meintanis and Gilles StupflerDOI: 10.14736/kyb-2015-4-0571

Abstract:

We suggest a nonparametric version of the probability weighted empirical characteristic function (PWECF) introduced by Meintanis {et al.}~\cite{meiswaall2014} and use this PWECF in order to estimate the parameters of arbitrary transformations to symmetry. The almost sure consistency of the resulting estimators is shown. Finite--sample results for i.i.d. data are presented and are subsequently extended to the regression setting. A real data illustration is also included.

Keywords:

characteristic function, empirical characteristic function, probability weighted moments, symmetry transformation

Classification:

62G10, 62G20

paper.pdf

References:

  1. P. J. Bickel: On adaptive estimation. Ann. Statist. 10 (1982), 647-671.   DOI:10.1214/aos/1176345863
  2. P. J. Bickel and K. A. Doksum: An analysis of transformations revisited. J. Amer. Statist. Assoc. 76 (1981), 296-311.   DOI:10.1080/01621459.1981.10477649
  3. G. E. P. Box and D. R. Cox: An analysis of transformations. J. Roy. Statist. Soc. B 26 (1964), 211-243.   CrossRef
  4. J. B. Burbidge, L. Magee and A. L. Robb: Alternative transformations to handle extreme values of the dependent variable. J. Amer. Statist. Assoc. 83 (1988), 123-127.   CrossRef
  5. G. Chen, R. Lockhart and M. A. Stephens: Box-Cox transformations in linear models: large sample theory and tests for normality (with discussion). Canad. J. Statist. 30 (2002), 1-59.   CrossRef
  6. G. González-Rivera and F. C. Drost: Efficiency comparisons of maximum-likelihood-based estimators in GARCH models. J. Econometr. 93 (1999), 93-111.   DOI:10.1016/s0304-4076(99)00005-6
  7. J. L. Horowitz: Semiparametric and Nonparametric Methods in Econometrics. Springer-Verlag, New York 2009.   DOI:10.1007/978-0-387-92870-8
  8. J. A. John and N. R. Draper: An alternative family of transformations. J. Roy. Statist. Soc. C 29 (1980), 190-197.   DOI:10.2307/2986305
  9. B. F. J. Manly: Exponential data transformations. J. Roy. Statist. Soc. D 25 (1976), 37-42.   CrossRef
  10. S. G. Meintanis, J. Swanepoel and J. Allison: The probability weighted characteristic function and goodness-of-fit testing. J. Statist. Plann. Infer. 146 (2014), 122-132.   DOI:10.1016/j.jspi.2013.09.011
  11. W. K. Newey: Adaptive estimation of regression models via moment restrictions. J. Econometr. 38 (1988), 301-339.   DOI:10.1016/0304-4076(88)90048-6
  12. W. K. Newey and D. G. Steigerwald: Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroskedasticity models. Econometrica 65 (1997), 587-599.   DOI:10.2307/2171754
  13. E. Parzen: On estimation of a probability density function and mode. Ann. Math. Statist. 33 (1962), 1065-1076.   DOI:10.1214/aoms/1177704472
  14. G. Pólya and G. Szegő: Problems and Theorems in Analysis, Volume I. Springer-Verlag, Berlin 1998.   CrossRef
  15. M. Rosenblatt: Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (1956), 832-837.   DOI:10.1214/aoms/1177728190
  16. O. Y. Savchuk and A. Schick: Density estimation for power transformations. J. Nonparametr. Statist. 25 (2013), 545-559.   DOI:10.1080/10485252.2013.811788
  17. P. K. Sen: Estimates of the regression coefficient based on Kendall's tau. J. Amer. Statist. Assoc. 63 (1968), 1379-1389.   DOI:10.1080/01621459.1968.10480934
  18. H. Theil: A rank-invariant method of linear and polynomial regression analysis. I, II, III. Nederl. Akad. Wetensch. Proc. 53 (1950), 386-392, 521-525, 1397-1412.   CrossRef
  19. I.-K. Yeo and R. A. Johnson: A new family of power transformations to improve normality or symmetry. Biometrika 87 (2000), 954-959.   CrossRef
  20. I.-K. Yeo and R. A. Johnson: A uniform law of large numbers for $U$-statistics with application to transforming to near symmetry. Statist. Probab. Lett. 51 (2001), 63-69.   CrossRef
  21. I.-K. Yeo and R. A. Johnson: An empirical characteristic function approach to selecting a transformation to symmetry. In: Contemporary Developments in Statistical Theory (S. Lahiri, A. Schick, A. SenGupta and T. Sriram, eds.), Springer International Publishing 2014, pp. 191-202.   DOI:10.1007/978-3-319-02651-0_11
  22. I.-K. Yeo, R. A. Johnson and X. W. Deng: An empirical characteristic function approach to selecting a transformation to normality. Commun. Stat. Appl. Methods 21 (2014), 213-224.   DOI:10.5351/csam.2014.21.3.213