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Fourier methods for model selection

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A test approach to the model selection problem based on characteristic functions (CFs) is proposed. The scheme is close to that proposed by Vuong (Econometrica 57:257–306, 1989), which is based on comparing estimates of the Kullback–Leibler distance between each candidate model and the true population. Other discrepancy measures could be used. This is specially appealing in cases where the likelihood of a model cannot be calculated or even, if it has a closed expression, it is either not easily tractable or not regular enough. In this work, the closeness is measured by means of a distance based on the CFs. As a prerequisite, some asymptotic properties of the minimum integrated squared error estimators are studied. From these properties, consistent tests for model selection based on CFs are given for separate, overlapping and nested models. Several examples illustrate the application of the proposed methods.

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  • Athreya, K. B., Lahiri, S. N. (2006). Measure theory and probability theory. New York: Springer.

  • Broniatowski, M., Keziou, A. (2009). Parametric estimation and testing through divergences and the duality technique. Journal of Multivariate Analysis, 100, 16–36.

  • Castaño-Martínez, A., López-Blázquez, F. (2005). Distribution od a sum of weighted central chi-square variables. Communications in Statistics-Theory and Methods, 34, 515–524.

  • Cox, D. R. (1961). Tests of separate families of hypothesis. Proceedings of the forth Berkeley symposium in mathematical statistics and probability (pp. 105–123). Berkeley: University of California Press.

  • Cox, D. R. (1962). Further results on tests of separate families of hypothesis. Journal of the Royal Statistical Society B, 24, 406–424.

  • Csörgő, S. (1981). The empirical characteristic process when parameters are estimated. In J. Gani, V. K. Rohatgi (Eds.), Contributions to probability: A collection of papers dedicated to Eugene Lukacs (pp. 708–723). New York: Academic Press.

  • Epps, T. W., Singleton, K. J., Pulley, L. B. (1982). A test of separate families of distributions based on the empirical moment generating function. Biometrika, 69, 391–399.

  • Feigin, P. D., Heathcote, C. R. (1976). The empirical characteristic function and the Cramér–von Mises statistic. Sankhya, 38, 309–325.

  • Feller, W. (1971). An introduction to probability theory and its applications (Vol. 2). New York: Wiley.

  • Heathcote, C. R. (1977). The integrated squared error estimation of parameters. Biometrika, 64, 64–255.

  • Jiménez-Gamero, M. D., Alba-Fernández, V., Muñoz-García, J., Chalco-Cano, Y. (2009). Goodness-of-fit tests based on empirical characteristic functions. Computational Statistics & Data Analysis, 53, 3957–3971.

  • Jiménez-Gamero, M. D., Pino-Mejías, R., Alba-Fernández, V., Moreno-Rebollo, J. L. (2011). Minimum \(\phi \)-divergence estimation in misspecified multinomial models. Computational Statistics & Data Analysis, 55, 3365–3378.

  • Jiménez-Gamero, M. D., Pino-Mejías, R., Rufián-Lizana, A. (2014). Minimum \(K_{\phi }\)-divergence estimators for multinomial models and applications. Computational Statistics, 29, 363–401.

  • Kishino, H., Hasegawa, M. (1989). Evaluation of the maximum likelihood estimate of the evolutionary tree topologies from DNA sequence data, and the branching order in Hominoidea. Journal of Molecular Evolution, 29, 170–179.

  • Kotz, S., Johnson, N. L., Boyd, D. W. (1967). Series representations of quadratic forms in normal variables. I. Central case. The Annals of Mathematical Statistics, 38, 823–837.

  • Lindsay, B. G. (1994). Efficiency versus robustness: the case for minimum Hellinger distance and related methods. The Annals of Statistics, 22, 1081–1114.

  • Linhart, H. (1988). A test whether two AIC’s differ significantly. South African Statistical Journal, 22, 153–161.

  • Matsui, M., Takemura, A. (2005). Empirical characteristic function approach to goodness-of-fit tests for the Cauchy distribution with parameters estimated by MLE or EISE. Annals of the Institute of Mathematical Statistics, 57, 183–199.

  • Matsui, M., Takemura, A. (2008). Goodness-of-fit tests for symmetric stable distributions-Empirical characteristics function approach. Test, 17(3), 546–566.

  • Meintanis, S. G. (2005). Consistent tests for symmetric stability with finite mean based on the empirical characteristic function. Journal of Statistical Planning and Inference, 128(2), 373–380.

  • Pardo, L. (2006). Statistical inference based on divergence measures. Boca Raton: Chapman & Hall.

  • Randles, R. H. (1982). On the asymptotic normality of statistics with estimated parameters. The Annals of Statistics, 10, 462–474.

  • Serfling, R. J. (1980). Approximation theorems of mathematical statistics. New York: Wiley.

  • Shimodaira, H. (1998). An application of multiple comparison techniques to model selection. Annals of the Institute of Mathematical Statistics, 50, 1–13.

  • Vuong, Q. H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57, 257–306.

  • Vuong, Q. H., Wang, W. (1993). Minimum chi-square estimation and tests for model selection. Journal of Econometrics, 56, 141–168.

  • White, H. (1981). Misspecified nonlinear regression models. Journal of the American Statistical Society, 76, 419–433.

  • White, H. (1982a). Maximum likelihood estimation of misspecified models. Econometrica, 50, 1–25.

  • White, H. (1982b). Regularity conditions for Cox’s test of non-nested hypothesis. Journal of Econometrics, 19, 301–315.

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The authors thank the anonymous referees for their constructive comments and suggestions which helped to improve the presentation. M.D. Jiménez-Gamero and M.V. Alba-Fernández have been partially supported by the research Project UJA2013/08/01 (University of Jaén and Caja Rural Provincial of Jaén).

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Correspondence to M. V. Alba-Fernández.

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Jiménez-Gamero, M.D., Batsidis, A. & Alba-Fernández, M.V. Fourier methods for model selection. Ann Inst Stat Math 68, 105–133 (2016).

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