Abstract
We establish a data-driven version of Neyman’s smooth goodness-of-fit test for the marginal distribution of observations generated by an α-mixing discrete time stochastic process \({(X_t)_{t \in \mathbb {Z}}}\) . This is a simple extension of the test for independent data introduced by Ledwina (J Am Stat Assoc 89:1000–1005, 1994). Our method only requires additional estimation of the cumulative autocovariance. Consistency of the test will be shown at essentially any alternative. A brief simulation study shows that the test performs reasonable especially for the case of positive dependence. Finally, we illustrate our approach by analyzing the validity of a forecasting method (“historical simulation”) for the implied volatilities of traded options.
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References
Abramowitz, M., Stegun I. A. (1972). Legendre functions and orthogonal polynomials. Ch. 22 in Chs. 8 and 22. In Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing (pp. 331–339 and 771–802). New York: Dover.
Artzner P., Delbaen F., Eber J.M., Heath D. (1999) Coherent measures of risk. Mathematical Finance 9: 203–228
Barone-Adesi G., Giannopoulos K., Vosper L. (1999) VaR without correlations for non-linear portfolios. Journal of Futures Markets 19: 583–602
Bosq D. (1989) Test du χ2 généralisés. Comparaison avec le test du χ2 classique. Revue de Statistique Appliquée XXXVII: 43–52
Bosq, D. (1998). Nonparametric statistics for stochastic processes (Vol. 110). Lecture Notes in Statistics. Heidelberg: Springer.
Bosq, D. (2002). Functional tests of fit. Goodness-of-fit tests and model validity. In Statistics for industry and technology (pp. 341–356). Boston: Birkhäuser.
Bradley, R. C. (1986). Basic properties of strong mixing conditions. In E. Eberlein, M. S. Taqqu (Eds.), Dependence in probability and statistics, Progress in probability and statistics (Vol. 11, pp. 165–192). Boston: Birkhäuser.
Bradley, R. C. (2002). Introduction to strong mixing conditions. Technical Report, 1, Bloomington: Indiana University.
Brockwell J.P., Davis A.R. (1991) Time series: Theory and methods (2nd ed). Springer, New York
Chanda, K. C. (1981). Chi-square goodness of fit tests based on dependent observations. In C. Taillie, G. P. Patil, B. A. Baldessari (Eds.), Statistical distributions in scientific work (Vol. 5, pp. 35–49). Dordrecht: D. Reidel Publishing Company.
Chanda, K. C. (1999). Chi-squared tests of goodness-of-fit for dependent observations. Asymptotics, non-parametrics and time series. Statistics: Textbooks and monographs, 158 (pp. 743–756). New York: Dekker.
Doukhan P., Massart P., Rio E. (1994) The functional central limit theorem for strongly mixing processes. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 30: 63–82
Ducharme G.R., Micheaux P.L. (2004) Goodness-of-fit tests of normality for the innovations in ARMA models. Journal of Time Series Analysis 25: 373–395
Duffie D., Pan J. (1997) An overview of value at risk. Journal of Derivatives 7: 7–49
Eaton M.L., Tyler D.E. (1991) On Wielandt’s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix. The Annals of Statistics 19: 260–271
Fan J., Gijbels I. (1996) Local polynomial modelling and its applications. Chapman and Hall, London
Fan J., Yao Q. (1998) Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85: 645–660
Gasser T. (1975) Goodness-of-fit tests for correlated data. Biometrika 62: 563–570
Gleser L.J., Moore D.S. (1983) The effect of dependence on chi-squared and empiric distribution tests of fit. The Annals of Statistics 11: 1100–1108
Hannan E.J. (1970) Multiple time series. Wiley, New York
Hull J.C. (2002) Options, futures, and other derivatives (5th ed). Pearson Education, New York
Ignaccolo R. (2004) Goodness-of-fit tests for dependent data. Journal of Nonparametric Statistics 16: 19–38
Inglot T., Ledwina T. (1996) Asymptotic optimality of data driven Neyman’s tests for uniformity. The Annals of Statistics 24: 1982–2019
Inglot T., Kallenberg W.C.M., Ledwina T. (1997) Data driven smooth tests for composite hypotheses. The Annals of Statistics 25: 1222–1250
Janic-Wroblewska A., Ledwina T. (2000) Data driven test for two-sample problem. Scandinavian Journal of Statistics 27: 281–298
Kallenberg W.C.M. (2002) The penalty in data driven Neyman’s tests. Mathematical Methods of Statistics 11: 323–340
Kallenberg W.C.M., Ledwina T. (1995a) Consistency and Monte Carlo simulation of a data driven version of smooth goodness-of-fit tests. The Annals of Statistics 23: 1594–1608
Kallenberg W.C.M., Ledwina T. (1995b) On data driven Neyman’s tests. Probability and Mathematical Statistics 15: 409–426
Kallenberg W.C.M., Ledwina T. (1997a) Data driven smooth tests when the hypothesis is composite. Journal of The American Statistical Association 92: 1095–1105
Kallenberg W.C.M., Ledwina T. (1997b) Data driven smooth tests for composite hypotheses: Comparison of powers. Journal of Statistical Computation and Simulation 59: 101–121
Ledwina T. (1994) Data driven version of Neyman’s smooth test of fit. Journal of the American Statistical Association 89: 1000–1005
Mokkadem A. (1990) Propriétés de mélange des processus autorégressifs polynomiaux. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 26: 219–260
Moore D.S. (1982) The effect of dependence on chi-squared tests of fit. The Annals of Statistics 10: 1163–1171
Neumann M.H., Kreiss J.P. (1998) Regression type inference in nonparametric autoregression. The Annals of Statistics 26: 1570–1613
Neumann M.H., Paparoditis E. (2000) On bootstrapping L 2-type statistics in density testing. Statistics and Probabability Letters 50: 137–147
Neyman J. (1937) Smooth test for goodness of fit. Skandinavian Aktuarietidskrift 20: 150–199
Pritsker, M. (2001). The hidden risks of historical simulation. Federal Reserve Board, Washington. Available at http://www.gloriamundi.org.
Rayner J., Best D. (1989) Smooth tests of goodness-of-fit. Oxford University Press, Oxford
Rio E. (1993) Covariance inequalities for strongly mixing processes. Annales de l’institut Henri Poincaré (B) Probabilités et Statistiques 29: 587–597
Rosenblatt D. (1952) Remarks on a multivariate transformation. Annals of Mathematical Statistics 23: 470–472
Schwarz G. (1978) Estimating the dimension of a model. The Annals of Statistics 6: 461–464
Song, W.-M. T., Hsiao, L. C. (1993). Generation of autocorrelated random variables with a specified marginal distribution. In G. W. Evans, M. Mollaghasemi, E. C. Russell, W. E. Biles (Eds.), Proceedings of the 1993 winter simulation conference (pp. 374–377).
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Munk, A., Stockis, JP., Valeinis, J. et al. Neyman smooth goodness-of-fit tests for the marginal distribution of dependent data. Ann Inst Stat Math 63, 939–959 (2011). https://doi.org/10.1007/s10463-009-0260-2
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DOI: https://doi.org/10.1007/s10463-009-0260-2