Tracking Interval to Select an Optimal Model Among Non-nested Copula Functions

Abstract

One of the important issues in order to survey multivariate distribution or model dependency structure between interested variables is finding the proper copula function. Extensive studies have been done based on Akaike information criterion (AIC), copula information criterion (CIC), and pseudo-likelihood ratio and fitness test of the copula function. The previous methods of selecting copula functions when the sample size is too small are not satisfactory. Therefore, our method in this paper is based on tracking interval for the parametric copula function which is obtained using expected Kullback–Leibler risk between the two proposed non-nested parametric copula model. It can be find that optimal parametric copula between proposed copula functions in a good level of significance. Finally, efficiency and capability of our method using simulation and applied example have been shown.

Keywords

Copula / Tracking interval / Expected Kullback–Leibler risk

References

[1] Akaike, H.: Information theory and an extension of maximum likelihood principle. In: Second International Symposium on Information Theory, pp. 267–281 (1973)

[2] Bedford, T., Daneshkhah, A., Wilson.: Approximate uncertainty modeling with vine copulas. Eur. J. Oper. Res. (2015)

[3] Breymann W, Dias W, Embrechts P. Dependence structures for multivariate high-frequency data in finance. Quant. Finance. 2003, 3 1-14

[4] Cebrian A, Denuit M, Scaillet O. Testing for concordance ordering. ASTIN Bull.. 2004, 34 151-173

[5] Cherbini U, Luciano E, Vecchiato W. Copula Methods in Finance. Business Economics. 2004 Hoboken: Wiley

[6] Chatrabgoun O, Parham G. Copula density estimation using multiwavelets based on the multiresolution analysis. Commun. Stat. Simul. Comput.. 2016, 45 3350-3372

[7] Chen X, Fan Y. Estimation of copula-based semiparametric time series models. J. Econom.. 2004, 130 307-335

[8] Chen X, Fan Y. Pseudo-likelihood ratio tests for model selection in semiparametric multivariate copula models. Can. J. Stat.. 2005, 33 389-414

[9] Chen X, Fan Y. Estimation and model selection of semiparametric copula-based model selection of multivaritate dynamic models under copula misspecification. J. Econom.. 2006, 135 125-154

[10] Chen X, Fan Y, Pouzo D, Ying Z. Estimation and model selection of semiparametric multivariate survival functions under general censorship. J. Econom.. 2010, 157 129-142

[11] Commanges D, Sayyareh A, Letenneur L, Guedj J, Bar-hen A. Estimating a difference of Kullback–Liebler risks using a normalized difference of AIC. Ann. Appl. Stat.. 2008, 2 1123-1142

[12] Denuit M, Scaillet O. Nonparametric tests for positive quadrant dependence. J. Financ. Econom.. 2004, 2 422-450

[13] Embrechts P, Lindskog F, McNeil A. Rachev ST. Modeling dependence with copulas and applications to risk management. Handbook of Heavy Tailed Distributions in Finance. 2003 Amsterdam: Elsevier. 307-327

[14] Frees EW, Valdez EA. Understanding relationships using copulas. N. Am. Actuar. J.. 1998, 2 1-25

[15] Genest C, Ghoudi K, Rivest L. A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika. 1995, 82 543-552

[16] Genest C, Rivest LP. Statistical inference procedures for bivariate Archimedean copula. J. Am. Stat. Assoc.. 1993, 88 1034-1043

[17] Genest C, Ghoudi K, Rivest LP. Comment on understanding relationships using copulas, by Edward W. Frees and Emiliano A. January 1998. N. Am. Actuar. J.. 1998, 2 143-149

[18] Genest C, Quessy JF, Rmillard B. Goodness-of-fit procedures for copula models based on the integral probability transformation. Scand. J. Stat.. 2006, 33 337-366

[19] Genest C, Masiellob E, Tribouley K. Estimating copula densities through wavelets. Insur. Math. Econ.. 2009, 44 170-181

[20] Grønneberg S, Hiort NL. The copula information criteria. Scand. J. Stat.. 2014, 41 436-459

[21] Gui, W.: Adaptive series estimators for copula densities. Ph.D. thesis, Florida State University College of Arts and Sciences (2009)

[22] Joe H. Multivariate Models and Dependence Concepts. 1997 London: Chapman Hall

[23] Kullback S, Leibler R. A on information and sufficiency. Ann. Math. Stat.. 1951, 22 79-87

[24] Klugman S, Parsa R. Fitting bivariate loss distributions with copulas. Insur. Math. Econ.. 1999, 24 139-148

[25] Linhart H, Zucchini W. Model Selection. 1986 New York: Wiley

[26] Nelsen R. An Introduction to Copulas. 1999 New York: Springer

[27] Oakes D. Multivariate survival distributions. J. Nonparametr. Stat.. 1994, 3 343-354

[28] Sayyareh AR. Tracking interval for selecting between non-nested models: an investigation for Type II right censored data. J. Stat. Plan. Inference. 2012, 142 3201-3208

[29] Shih J, Louis T. Inferences on the association parameter in copula models for bivariate survival data. Biometrics. 1995, 51 1384-1399

[30] Sklar A. Fonctions de r’epartition ’a n dimensions et leurs marges. Publ. lnst. Stat. Univ. Paris. 1959, 8 229-231

[31] Scaillet O. A Kolmogorov–Smirnov type test for positive quadrant dependence. Can. J. Stat.. 2005, 33 415-427

[32] Vuong QH. Likelihood ratio test for model selection and non-nested hypotheses. Econometrica. 1989, 57 307-333

[33] Wang W, Wells M. Model selection and semiparametric inference for bivariate failure-time data. J. Am. Stat. Assoc.. 2000, 95 62-76

[34] White H. Maximum likelihood estimation of misspecified models. Econometrica. 1982, 50 1-25