Abstract
The paper reviews the topic of extremal time series. The literature documenting the presence of extremes in time series data is first reviewed, followed by a discussion of various probabilistic measures, along with the associated statistical inference problems. The impact of extremes upon statistical analyses is discussed, and the connection to extremal latent components is emphasized. Two data sets illustrate the methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aguilar, M., Hill, J.B.: Robust score and portmanteau tests of volatility spillover. J. Econ. 184, 37–61 (2015)
Anderson, P.L., Meerschaert, M.M.: Periodic moving averages of random variables with regularly varying tails. Ann. Stat. 25, 771–785 (1997)
Anderson, P.L., Meerschaert, M.M.: Modeling river flows with heavy tails. Water Resour. Res. 34, 2271–2280 (1998)
Basrak, B., Davis, R.A., Mikosch, T.: Regular variation of GARCH processes. Stoch. Proc. Appl. 99, 95–115 (2002)
Basrak, B., Krizmanić, D., Segers, J.: A functional limit theorem for dependent sequences with infinite variance stable limits. Ann. Probab. 40, 2008–2033 (2012)
Beirlant, J., Goegebeur, Y., Teugels, J., Segers, J., de Wall, D., Ferro, C.: Statistics of Extremes: Theory and Applications. Wiley, New York (2004)
Beran, J., Sherman, R., Taqqu, M., Willinger, W.: Long-range dependence in Variable-bit rate video traffic. IEEE Trans. Comm. 43, 1566–1579 (1995)
Berger, J.M., Mandelbrot, B.B.: A new model for error clustering in telephone circuits. IBM J. Res. Develop. 7, 224–236 (1963)
Berkes, I., Horváth, L., Kokoszka, P.: Estimation of the maximal moment exponent of a GARCH(1,1) sequence. Econ. Theory 19, 565–586 (2003)
Bollerslev, T., Todorov, V., Li, S.Z.: Jump tails, extreme dependencies, and the distribution of stock returns. J. Econ. 172, 307–324 (2013)
Brito, M., Freitas, A.C.M.: Consistent estimation of the tail index for dependent data. Stat. Probab. Lett. 80, 1835–1843 (2010)
Cappe, O., Moulines, E., Pesquet, J., Petropulu, A., Xueshi, Y.: Long-range dependence and heavy-tail modeling for teletraffic data. Signal Process. Mag. IEEE 19, 14–27 (2002)
Castillo, E.: Extreme Value Theory in Engineering. Academic Press, San Diego (1988)
Chernozhukov, V., Fernández-Val, I.: Inference for extremal conditional quantile models, with an application to market and birthweight risks. Rev. Econ. Stud. 78, 559–589 (2011)
Clark, P.K.: A subordinate stochastic process model with finite variance for speculative prices. Econometrica 41, 135–155 (1973)
Coles, S.: An Introducton to Statistical Modeling of Extreme Values. Springer, London (2001)
Crovella, M., Taqqu, M., Bestavros, A.: Heavy-tailed probability distributions in the World Wide Web. In: Adler, R.J., Feldman, R.E., Taqqu, M.S. (eds.) A Practical Guide to Heavy Tails. Birkhäuser, Boston (1998)
Csȯrgȯ, M., Yu, H.: Weak approximations for quantile processes of stationary sequences. Canad. J. Stat. 24, 403–430 (1996)
Danielsson, J, de Haan, l., Peng, L., de Vries, C.G: Using a bootstrap method to choose the sample fraction in tail index estimation. J. Multivar. Anal. 76, 226–248 (2001)
D’Auria, B., Resnick, S.I.: Data network models of burstiness. Adv. Appl. Probab. 38, 373–404 (2006)
D’Auria, B., Resnick, S.I.: The influence of dependence on data network models. Adv. Appl. Probab. 40, 60–94 (2008)
Davis, R., Klüppelberg, C., Steinkohl, C.: Statistical inference for max-stable processes in space and time. J. R. Stat. Soc. Series B 75, 791–819 (2013a)
Davis, R., Klüppelberg, C., Steinkohl, C.: Max-stable processes for modeling extremes observed in space and time. J. Kor. Stat. Soc. 42, 399–414 (2013b)
Davis, R., Mikosch, T.: Extreme value theory for GARCH processes. In: Mikosch, T., Kreiss, J-P., Davis, R., Andersen, T. G. (eds.) Handbook of Financial Time Series. Springer-Verlag, Berlin (2009a)
Davis, R., Mikosch, T.: The extremogram: A correlogram for extreme events. Bernoulli 15, 977–1009 (2009b)
Davis, R., Mikosch, T., Cribben, I.: Towards estimating extremal serial dependence via the bootstrapped extremogram. J. Econ. 170, 142–152 (2012)
Davis, R., Mikosch, T., Zhao, Y.: Measures of serial extremal dependence and their estimation. Stoch. Process. Appl. 123, 2575–2602 (2013)
Davis, R.A., Resnick, S.I.: Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13, 179–195 (1985)
Davis, R.A., Resnick, S.I.: Limit theory for the sample covariance and correlation functions of moving averages. Ann. Stat. 14, 533–558 (1986)
Davison, A.C., Huser, R., Thibaud, E.: Geostatistics of dependent and asymptotically independent extremes. Math. Geosci. 45, 511–529 (2013)
Davision, A.C., Smith, R.L.: Models for exceedances over high thresholds. J. R. Stat. Soc. Ser. B 52, 393–442 (1990)
de Haan, L.: A spectral representation for max-stable processes. Ann. Probab. 12, 1194–1204 (1984)
de Haan, L., Pereira, T.T.: Spatial extremes: models for the stationary case. Ann. Stat. 34, 146–168 (2006)
Doukhan, P., Prohl, S., Robert, C.: Subsampling weakly dependent time series and applications to extremes. Test 20, 447–479 (2011)
Drees, H.: Weighted approximations of tail processes for β-mixing random variables. Ann. Probab. 10(4), 1274–1301 (2000)
Drees, H.: Tail empirical processes under mixing conditions. In: Empirical Process Techniques for Dependent Data, pp 325–342. Birkhäuser, Boston (2002)
Drees, H.: Extreme quantile estimation for dependent data, with applications to finance. Bernoulli 9(4), 617–657 (2003)
Drees, H.: Some aspects of extreme value statistics under serial dependence. Extremes 11, 35–53 (2008)
Drees, H., Kaufmann, E.: Selecting the optimal sample fraction in univariate extreme value estimation. Stoch. Process. Appl. 75, 149–172 (1998)
Drees, H., Rootzén, H: Limit theorems for empirical processes of cluster functionals. Ann. Stat. 38, 2145–2186 (2010)
Duffy, D., McIntosh, A., Rosenstein, M., Willinger, W.: Analyzing telecommunications traffic data from working common channel signaling subnetworks. In: Proceedings of the 25th Interface, San Diego. Interface Foundation of North America (1993)
Duffy, D., McIntosh, A., Rosenstein, M., Willinger, W.: Statistical analysis of CCSN/SS7 traffic data from working CCS subnetworks. IEEE J. Select. Areas Commun. 12, 544–551 (1994)
Einmahl, J.H., Kiriliouk, A., Krajina, A., Segers, J.: An M-estimator of spatial tail dependence. J. R. Stat. Soc. Ser. B. published online (2015)
Embrechts, Mikosh, Kluppelberg: Modeling Extrema Events for Insurance and Finance. Springer-Verlag, Berlin (1997)
Epps, T.W., Epps, M.L.: The stochastic dependence of security price changes and transaction volumes. Econometrica 44, 305–321 (1976)
Fama, E.F.: The behavior of stock-market prices. J. Business 38, 34–105 (1965)
Farmer, J.D., Gillemot, L., Lillo, F., Mike, S., Sen, A.: What really causes large price changes? Quant. Finan. 4, 383–397 (2004)
Fasen, V., Klüppelberg, C., Schlather, M.: High-level dependence in time series models. Extremes 13, 1–33 (2010)
Fasen, V., Klüppelberg, C., Menzel, A.: Quantifying extreme risks. In: Klüppelberg, C., Straub, D., Welpe, I.M. (eds.) Risk – A Multidisciplinary Introduction, pp 151–181. Springer (2014)
Ferro, C. A., Segers, J.: Inference for clusters of extreme values. J. R. Stat. Soc. Ser. B 65, 545–556 (2003)
Findley, D.F., Monsell, B.C., Bell, W.R., Otto, M.C., Chen, B.: New capabilities and methods for the X-12-ARIMA seasonal adjustment program. J. Bus. Econ. Stat. 16, 127–152 (1998)
Gencay, R., Selcuk, F.: Extreme value theory and value-at-risk: Relative performance in emerging markets. Int. J. Forecast. 20, 287–303 (2004)
Gilli, M., Këllezi, E.: An application of extreme value theory for measuring financial risk. Comput. Econ. 27, 207–228 (2006)
Gomes, M.I., Figueiredo, F., Neves, M.M.: Adaptive estimation of heavy right tails: Resampling-based methods in action. Extremes 15, 463–489 (2012)
Gomes, M.I.., Guillou, A.: Extreme value theory and statistics of univariate extremes: A review. Int. Stat. Rev. 83, 263–292 (2015)
Gomes, M.I., Hall, A., Miranda, C: Subsampling techniques and the Jackknife methodology in the estimation of the extremal index. Comput. Stat. Data Anal. 52 (4), 2022–2041 (2008)
Grigoriu, M.: Applied Non-Gaussian Processes. Prentice Hall, Englewood Cliffs (1995)
Hamidieh, K., Stoev, S., Michailidis, G.: On the estimation of the extremal index based on scaling and resampling. J. Comput. Graph. Stat. 18, 731–755 (2009)
Heath, D., Resnick, S., Samorodnitsky, G: Heavy tails and long range dependence in on/off processes and associated fluid models. http://www.orie.cornell.edu/trlist/ (1997)
Hill, B.M.: A simple general approach to inference about the tail of a distribution. Ann. Stat. 3, 1163–1174 (1975)
Hill, J.B.: On functional central limit theorems for dependent, heterogeneous arrays with applications to tail index and tail dependence estimation. J. Stat. Plan. Infer. 139, 2091–2110 (2009)
Hill, J.B.: On tail index estimation for dependent, heterogeneous data. Econometric Theory 26, 1398–1436 (2010)
Hill, J.B.: Tail and nontail memory with applications to extreme value and robust statistics. Econ. Theory 27, 844–884 (2011a)
Hill, J.B.: Extremal memory of stochastic volatility with an application to tail shape inference. J. Stat. Plan. Infer. 141, 663–676 (2011b)
Hill, J.B.: Least tail-trimmed squares for infinite variance autoregressions. J. Time Ser. Anal. 34, 168–186 (2013a)
Hill, J.B.: Expected shortfall estimation and Gaussian inference for infinite variance time series. J. Finan. Econ. Published online (2013b)
Hill, J.B., Aguilar, M.: Moment condition tests for heavy tail time series. J. Econ. 172, 255–274 (2013)
Hill, J.B.: Robust estimation and inference for heavy tailed GARCH. Bernoulli 21, 1629–1669 (2015)
Hill, J.B., Shneyerov, A.: Are there common values in first-price auctions? A tail-index nonparametric test. J. Econ. 174, 144–164 (2013)
Holan, S. H., McElroy, T.S.: Tail exponent estimation via broadband log density-quantile regression. J. Stat. Plan. Infer. 140, 3693–3708 (2010)
Horváth, L., Kokoszka, P.S.: Sample autocovariances of long-memory time series. Bernoulli 14, 405–418 (2008)
Hsing, T.: Estimating the parameters of rare events. Stoch. Proc. Appl. 37, 117–139 (1991a)
Hsing, T.: On tail index estimation using dependent data. Ann. Stat. 19, 1547–1569 (1991b)
Hsing, T.: Extremal index estimation for a weakly dependent stationary sequence. Ann. Stat. 21, 2043–2071 (1993)
Hsing, T., Hüsler, J., Leadbetter, M.R.: On the exceedance point process for a stationary process. Probab. Th. Rel. Fields 78, 97–112 (1988)
Huser, R., Davison, A.C.: Space-time modelling of extreme events. J. R. Stat. Soc. Ser. B 76, 439–461 (2014)
Iglesias, E.M.: An analysis of extreme movements of exchange rates of the main currencies traded in the Foreign Exchange market. Appl. Econ. 44, 4631–4637 (2012)
Jach, A., McElroy, T.S., Politis, D.N.: Subsampling inference for the mean of heavy-tailed long memory time series. J. Time Ser. Anal. 33, 96–111 (2012)
Kabluchko, Z.: Extremes of space-time Gaussian processes. Stoch. Process. Appl. 119, 3962–3980 (2009)
Kabluchko, Z., Schlather, M., de Haan, L: Stationary max-stable fields associated to negative definite functions. Ann. Probab. 37(5), 2042–2065 (2009)
Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math 131, 207–248 (1973)
Kim, S.J., Koh, K., Boyd, S., Gorinevsky, D.: ℓ 1 Trend filtering. SIAM Rev. 51, 339–360 (2009)
Koedijk, K., Schafgans, M., De Vries, C.: The tail index of exchange rate returns. J. Int. Econ. 29, 93–108 (1990)
Kokoszka, P.S., Taqqu, M.S.: Parameter estimation for infinite variance fractional ARIMA. Ann. Stat. 24, 1880–1913 (1996)
Kokoszka, P.S., Taqqu, M.S.: Discrete time parametric models with long memory and infinite variance. Math. Comput. Model. 293, 203–215 (1999)
Kulik, R., Soulier, P.: The tail empirical process of some long memory sequences. Stoch. Process. Appl. 121, 109–134 (2011)
Kulik, R., Soulier, P.: Limit theorems for long memory stochastic volatility models with infinite variance: Partial sums and sample covariances. Adv. Appl. Probab. 44(4), 1113–1141 (2012)
Kulik, R., Soulier, P.: Estimation of limiting conditional distibutions for the heavy tailed long memory stochastic volatility process. Extremes 16, 203–239 (2013)
Kulik, R., Soulier, P.: Heavy tailed time series with extremal independence. Extremes 18, 273–299 (2015)
Kyselý, J.: A cautionary note on the use of nonparametric bootstrap for estimating uncertainties in extreme-value models. Amer. Metereol. Soc. 47, 3236–3251 (2008)
Lahiri, S.: Resampling Methods for Dependent Data. Springer, New York (2003)
Larsson, M., Resnick, S.I.: Extremal dependence measure and extremogram: The regularly varying case. Extremes 15, 231–256 (2012)
Leadbetter, M.R.: On extreme values in stationary sequences. Prob. Th. Rel. Fields 28, 289–303 (1974)
Leadbetter, M.R.: Extremes and local dependence in stationary sequences. Probab. Theory Related Fields 65, 291–306 (1983)
Leadbetter, M.R., Lindgren, G., Rootzén, H.: Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York - Berlin (1983)
Leadbetter, M.R., Nandagopalan, S.: On exceedance point processes for stationary sequences under mild oscillation restrictions. In: Hüsler, J., Reiss, R.-D. (eds.) Extreme Value Theory, pp 69–80. Springer-Verlag, New York (1989)
Leadbetter, M.R., Rootzén, H.: Extremal theory for stochastic processes. Ann. Probab. 16, 431–478 (1988)
Ledford, A., Tawn, J.: Diagnostics for dependence within time series extremes. J. R. Stat. Soc. Ser. B 65, 521–543 (2003)
Li, Y., Simmonds, D., Reeve, D.: Quantifying uncertainty in extreme values of design parameters with resampling techniques. Ocean Eng. 35, 1029–1038 (2008)
Linton, O., Xiao, Z.: Estimation of and inference about the expected shortfall for time series with infinite variance. Econ. Theory 29, 771–807 (2013)
Loynes, R.M.: Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Stat. 36, 993–999 (1965)
Mandelbrot, B.B.: The variation of certain speculative prices. J. Bus. 35, 394–419 (1963)
Mandelbrot, B.B.: Self-similar error clusters in communications systems and the concept of conditional systems and the concept of conditional stationarity. IEEE Trans. Commun. Technol. COM-13, 71–90 (1965)
Mandelbrot, B.B.: Long-run linearity, locally Gaussian processes, H-spectra and infinite variances. Int. Econ. Rev. 10, 82–113 (1969)
Mandelbrot, B.B.: Intermittent turbulence in self similar cascades; divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974)
Mandelbrot, B.B.: The Fractal Geometry of Nature. W.H.Freeman and Company (1983)
Mandelbrot, B.B., Wallis, J.R.: Noah, Joseph, and operational hydrology. Water Resour. Res. 4, 909–917 (1968)
Mandelbrot, B.B., Wallis, J.R.: Some long-run properties of geophysical records. Water Resour. Res. 5, 321–340 (1969)
Markovich, N.M.: Modeling clusters of extreme values. Extremes 17, 97–125 (2014)
Martin, D.R., Yohai, V.J.: Influence functionals for time series. Ann. Stat. 14, 781–855 (1986)
McElroy, T.S.: Exact formulas for the Hodrick-Prescott filter. Econ. J. 11, 209–217 (2008)
McElroy, T.S., Jach, A.: Tail index estimation in the presence of long memory dynamics. Comput. Stat. Data Anal. 56, 266–282 (2012a)
McElroy, T.S., Jach, A.: Subsampling inference for the autocovariances of heavy-tailed long-memory time series. J. Time Ser. Anal. 33, 935–953 (2012b)
McElroy, T.S., Nagaraja, C.H.: Tail index estimation with a fixed tuning parameter fraction. J. Stat. Plan. Infer. 170, 27–45 (2016)
McElroy, T.S., Politis, D.N.: Robust inference for the mean in the presence of serial correlation and heavy-tailed distributions. Econ. Theory 5, 1019–1039 (2002)
McElroy, T.S., Politis, D.N.: Stable marked point processes. Ann. Stat. 35, 393–419 (2007a)
McElroy, T.S., Politis, D.N: Computer-intensive rate estimation, diverging statistics, and scanning. Ann. Stat. 35, 1827–1848 (2007b)
McElroy, T.S., Politis, D.N.: Moment-based tail index estimation. J. Stat. Plan. Infer. 137, 1389–1406 (2007c)
McElroy, T.S., Politis, D.N.: Self-normalization for heavy-tailed time series with long memory. Statistica Sinica 17, 199–220 (2007d)
McNeil, A.J., Frey, R.: Estimation of tail-related risk measures for heteroscedastic financial time series: An extreme value approach. J. Empir. Finan. 7, 271–300 (2000)
Meerschaert, M.M., Scheffler, H.-P.: A simple robust estimator for the thickness of heavy tails. J. Stat. Plan. Infer. 71, 19–34 (1998)
Meerschaert, M.M., Scheffler, H.-P.: Sample cross-correlations for moving averages with regularly varying tails. J. Time Ser. Anal. 22, 481–492 (2001)
Meier-Hellstern, K., Wirth, P., Yan, Y., Hoeflin, D.: Traffic models for ISDN data users: Office automation application. In: Jensen, A., Iversen, V.B. (eds.) Teletraffic and Datatraffic in a Period of Change. Proceedings of the 13th ITC, pp 167–192. North Holland, Amsterdam (1991)
Mikosch, T.: Modeling dependence and tails of financial time series. In: Finkenstädt, B., Rootzén, H. (eds.) Extreme Values in Finance, Telecommunications, and the Environment. Chapman and Hall, Boca Raton (2003)
Mikosch, T., Stărică, C.: Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Ann. Stat. 28, 1427–1451 (2000)
Mikosch, T., Zhao, Y.: A Fourier analysis of extreme events. Bernoulli 20, 803–845 (2014)
Mikosch, T., Zhao, Y.: The integrated periodogram of a dependent extremal event sequence. Stoch. Process. Appl. 125, 3126–3169 (2015)
Nandagopalan, S.: Multivariate extremes and estimation of the extremal index. Ph.D, Thesis, Univ. North Carolina at Chapel Hill (1990)
Newcomb, S.: A generalized theory of the combination of observations so as to obtain the best result. Amer. J. Math. 8, 343–366 (1886)
Pandey, M.D., Van Gelder, P.H.A.J.M., Vrijling, J.K.: Bootstrap simulations for evaluating the uncertainty associated with peaks-over-trheshold estimates of extreme wind velocity. Environmetrics 14, 27–43 (2003)
Phillips, P.C.B.: Time series regression with a unit root and infinite variance errors. Econ. Theory 6, 44–62 (1990)
Politis, D.N.: A new approach on estimation of the tail index. C.R. Acad. Sci. Paris, Ser. I 335, 279–282 (2002)
Politis, D.N., Romano, J.P., Wolf, M.: Subsampling. Springer, New York (1999)
Reiss, R.-D., Thomas, M.: Statistical Analysis of Extreme Values. Basel, Birkhäuser (1997)
Resnick, S.: Special invited paper: Heavy tail modeling and teletraffic data. Ann. Stat. 25(5), 1805–1849 (1997)
Resnick, S.: The extremal dependence measure and asymptotic independence. Stoch. Models 20, 205–227 (2004)
Resnick, S.: Heavy-tail Phenomena: Probabilisitc and Statistical Modeling. Springer, New York (2007)
Resnick, S., Stărică, C.: Consistency of Hill’s estimator for dependent data. J. Appl. Probab. 32, 139–167 (1995)
Resnick, S., Stărică, C.: Tail index estimation for dependent data. Ann. Appl. Probab. 8(4), 1156–1183 (1998)
Robert, C.Y., Segers, J., Ferro, C.A.T.: A sliding blocks estimator for the extremal index. Electron. J. Stat. 3, 993–1020 (2009)
Robert, C.Y.: Asymptotic distributions for the intervals estimators of the extremal index and the cluster-size probabilities. J. Stat. Plann. Infer. 139, 3288–3309 (2009a)
Robert, C.Y.: Inference for the limiting cluster size distribution of extreme values. Ann. Stat. 37, 271–310 (2009b)
Rootzén, H.: Extremes of moving averages of stable processes. Ann. Probab. 6, 847–869 (1978)
Rootzén, H.: Weak convergence of the tail empirical process for dependent sequences. Stoch. Proces. Appl. 119(2), 468–490 (2009)
Rossi, E., Santucci de Magistris, P: Long memory and tail dependence in trading volume and volatility. J. Empir. Finan. 22, 94–112 (2013)
Schlather, M.: Models for stationary max-stable random fields. Extremes 5(1), 33–44 (2002)
Sibuya, M.: Bivariate extreme statistics, I. Ann. Inst. Stat. Math. 11, 195–210 (1959)
Smith, R.L.: Extreme value analysis of environmental time series: An example based on ozone data. Stat. Sci. 4, 367–393 (1989)
Smith, R.L., Tawn, J.A., Coles, S.G.: Markov chain models for threshold exceedances. Biometrika 84, 249–268 (1997)
Smith, R.L., Weissman, I.: Estimating the extremal index. J. Roy. Stat. Soc. Ser. B 56, 515–528 (1994)
Stigler, S.M.: Simon Newcomb, Percy Daniell, and the history of robust estimation 1885–1920. J. Amer. Stat. Assoc. 68, 872–879 (1973)
Stoev, S., Michailidis, G., Taqqu, M.: Estimating heavy-tail exponents through max self-similarity. IEEE Trans. Inf. Theory 57, 1615–1635 (2011)
Tesfaye, Y.G., Meerschaert, M.M., Anderson, P.L.: Identification of periodic autoregressive moving average models and their application to the modeling of river flows. Water Resour. Res. 42, 1–11 (2006)
Thibaud, E., Mutzner, R., Davison, A.C.: Threshold modeling of extreme spatial rainfall. Water Resour. Res. 49, 4633–4644 (2013)
Trimbur, T.M.: Stochastic level shifts and outliers and the dynamics of oil price movements. Int. J. Forecast. 26, 162–179 (2010)
Weller, G.B., Cooley, D.S., Sain, S.R.: An investigation of the pineapple express phenomenon via bivariate extreme value theory. Environmetrics 23, 420–439 (2012)
Willinger, W., Taqqu, M., Leland, W., Wilson, D.: Self-similarity in high-speed packet traffic: Analysis and modeling of ethernet traffic measurements. Stat. Sci. 10, 67–85 (1995)
Willinger, W., Taqqu, M., Sherman, R., Wilson, D.: Self-similarity through high-variability: Statistical analysis of Ethernet LAN traffic at the source level. IEEE/ACM Trans. Network. 5, 71–96 (1997)
Xiao, Z., Koenker, R.: Conditional quantile estimation for generalized autoregressive conditional heteroscedasticity models. J. Amer. Stat. Assoc. 104, 1696–1712 (2009)
Yamada, H., Jin, L.: Japans output gap estimation and ℓ 1 trend filtering. Empir. Econ. 45, 81–88 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
McElroy, T. On the measurement and treatment of extremes in time series. Extremes 19, 467–490 (2016). https://doi.org/10.1007/s10687-016-0254-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10687-016-0254-4