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A class of CUSUM tests using empirical distributions for tail changes in weakly dependent processes

  • Kim, JunHyeong (Department of Industrial Engineering, Hanyang University) ;
  • Hwang, Eunju (Department of Applied Statistics, Gachon University)
  • Received : 2019.07.08
  • Accepted : 2020.01.06
  • Published : 2020.03.31

Abstract

We consider a wide class of general weakly-dependent processes, called ψ-weak dependence, which unify almost all weak dependence structures of interest found in statistics under natural conditions on process parameters, such as mixing, association, Bernoulli shifts, and Markovian sequences. For detecting the tail behavior of the weakly dependent processes, change point tests are developed by means of cumulative sum (CUSUM) statistics with the empirical distribution functions of sample extremes. The null limiting distribution is established as a Brownian bridge. Its proof is based on the ψ-weak dependence structure and the existence of the phantom distribution function of stationary weakly-dependent processes. A Monte-Carlo study is conducted to see the performance of sizes and powers of the CUSUM tests in GARCH(1, 1) models; in addition, real data applications are given with log-returns of financial data such as the Korean stock price index.

Acknowledgement

Supported by : Gachon University

References

  1. Ango Nze P, Buhlmann P, and Doukhan P (2002). Nonparametric regression estimation under weak dependence beyond mixing and association, Annals of Statistics, 30, 397-430. https://doi.org/10.1214/aos/1021379859
  2. Ango Nze PA and Doukhan P (2004). Weak dependence: models and applications to econometrics, Econometric Theory, 20, 995-1045.
  3. Coulon-Prieur C and Doukhan P (2000). A triangular central limit theorem under a new weak dependent condition, Statistics and Probability Letters, 47, 61-68. https://doi.org/10.1016/S0167-7152(99)00138-8
  4. Dedecker J, Doukhan P, Lang G, and Leon JR, Louhichi S, and Prieur C (2007) Weak Dependence: With Examples and Applications. In Lecture Notes in Statistics (Vol 190), Springer, New York.
  5. Doukhan P, Jakubowski A, and Lang G (2015). Phantom distribution functions for some stationary sequences, Extremes, 18, 697-725. https://doi.org/10.1007/s10687-015-0228-y
  6. Doukhan P and Louhichi S (1999). A new weak dependence condition and applications to moment inequalities, Stochastic Processes and Their Applications, 84, 313-342. https://doi.org/10.1016/S0304-4149(99)00055-1
  7. Doukhan P and Louhichi S (2001). Functional estimation of a density under a new weak dependence condition, Scandinavian Journal of Statistics, 28, 325-341. https://doi.org/10.1111/1467-9469.00240
  8. Doukhan P and Neumann MH (2007). Probability and moment inequalities for sums of weakly dependent random variables with applications, Stochastic Processes and Their Applications, 117, 878-903. https://doi.org/10.1016/j.spa.2006.10.011
  9. Hill B (1975). A simple general approach to inference about the tail of a distribution, The Annals of Statistics, 3, 1163-1174. https://doi.org/10.1214/aos/1176343247
  10. Hoga Y (2017). Change point tests for the tail index of ${\beta}$-mixing random variables, Econometric Theory, 33, 915-954. https://doi.org/10.1017/S0266466616000189
  11. Hwang E and Shin DW (2011). Semiparametric estimation for partially linear models with ${\psi}$-weak dependent errors, Journal of the Korean Statistical Society, 40, 411-424. https://doi.org/10.1016/j.jkss.2011.01.002
  12. Hwang E and Shin DW (2012a). Stationary bootstrap for kernel density estimators under ${\psi}$-weak dependence, Computational Statistics and Data Analysis, 56, 1581-1593. https://doi.org/10.1016/j.csda.2011.10.001
  13. Hwang E and Shin DW (2012b). Random central limit theorems for linear processes with weakly dependent innovations, Journal of the Korean Statistical Society, 41, 313-322. https://doi.org/10.1016/j.jkss.2011.10.004
  14. Hwang E and Shin DW (2013). A study on moment inequalities under a weak dependence, Journal of the Korean Statistical Society, 42, 133-141. https://doi.org/10.1016/j.jkss.2012.06.003
  15. Hwang E and Shin DW (2016a). Maximal inequalities and an application under a weak dependence, Journal of Korean Mathematical Society, 53, 57-72. https://doi.org/10.4134/JKMS.2016.53.1.057
  16. Hwang E and Shin DW (2016b). Kernel estimators of mode under ${\psi}$-weak dependence, Annals of the Institute of Statistical Mathematics, 68, 301-327. https://doi.org/10.1007/s10463-014-0489-2
  17. Jakubowski A (1991). Relative extremal index of two stationary processes, Stochastic Processes and their Applications, 37, 281-297. https://doi.org/10.1016/0304-4149(91)90048-H
  18. Jureckova J, Koul HL, and Picek J (2009). Testing the tail index in autoregressive models, Annals of the Institute of Statistical Mathematics, 61, 579-598. https://doi.org/10.1007/s10463-007-0155-z
  19. Jureckova J and Picek J (2001). A class of tests on the tail index, Extremes, 4, 165-183. https://doi.org/10.1023/A:1013925226836
  20. KimMand Lee S (2009). Test for tail index change in stationary time series with Pareto-type marginal distribution, Bernoulli, 15, 325-356. https://doi.org/10.3150/08-BEJ157
  21. Kim M and Lee S (2011). Change point test for tail index for dependent data, Metrika, 74, 297-311. https://doi.org/10.1007/s00184-010-0304-x
  22. Kim M and Lee S (2012). Change point test of tail index for autoregressive processes, Journal of the Korean Statistical Society, 41, 305-312. https://doi.org/10.1016/j.jkss.2011.10.003
  23. Kim M and Lee S (2016). On the tail index inference for heavy-tailed GARCH-type innovations, Annals of the Institute of Statistical Mathematics, 68, 237-267. https://doi.org/10.1007/s10463-014-0495-4
  24. Lo AW (1991). Long-term memory in stock market prices, Econometrica, 59, 1279-1313. https://doi.org/10.2307/2938368
  25. O'Brien GL (1987). Extreme values for stationary and Markov sequences, Annals of Probability, 15, 281-291. https://doi.org/10.1214/aop/1176992270
  26. Quintos C, Fan Z, and Phillips PCB (2001). Structural change tests in tail behaviour and the Asian crisis, The Review of Economics Studies, 68, 633-663. https://doi.org/10.1111/1467-937X.00184