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A prediction-residual approach for identifying rare events in periodic time series. (English) Zbl 1294.62201

Summary: Many environmental time series have seasonal structures. This article presents an approach to identifying the rare events of such series based on time-series residuals. The methods justify the application of classical peaks over threshold methods to estimated versions of the one-step-ahead prediction errors of the series. Such methods enable the seasonal means, variances and autocorrelations of the series to be taken into account. Even in stationary settings, the proposed strategy is useful as it bypasses the need for blocking runs of extremes. The mathematics are justified via a limit theorem for a periodic autoregressive moving-average time series. A detailed application to a daily temperature series from Griffin, Georgia, is pursued.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G70 Extreme value theory; extremal stochastic processes
62P12 Applications of statistics to environmental and related topics

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[1] Adams, Parameter Estimation for Periodic ARMA Models, Journal of Time Series Analysis 16 pp 127– (1995) · Zbl 0814.62050
[2] Anderson, Asymptotic results for periodic autoregressive moving-average processes, Journal of Time Series Analysis 14 pp 1– (1993) · Zbl 0767.62071
[3] Anderson, Innovations algorithm for periodically stationary time series, Stochastic Processes and their Applications 83 pp 149– (1999) · Zbl 0995.62082
[4] Anderson, Fourier-PARMA models and their application to river flows, Journal of Hydrologic Engineering 12 pp 462– (2007)
[5] Anderson, Innovations algorithm asymptotics for periodically stationary time series with heavy tails, Journal of Multivariate Analysis 99 pp 94– (2008) · Zbl 1133.62070
[6] Anton, Total ozone and solar erythemal irradiance in Southwestern Spain, Geophysical Research Letters 35 (2008)
[7] Ballerini, Extreme value theory for processes with periodic variances, Stochastic Models 5 pp 45– (1989) · Zbl 0669.60030
[8] Basawa, Large sample properties of parameter estimates from periodic ARMA models, Journal of Time Series Analysis 22 pp 651– (2001) · Zbl 0984.62062
[9] Bloomfield, Periodic correlation in stratospheric ozone data, Journal of Time Series Analysis 15 pp 127– (1994) · Zbl 0794.62065
[10] Brockwell, Time Series: Theory and Methods (1991) · Zbl 0709.62080
[11] Chernick, Calculating the extremal index for a class of stationary sequences, Advances in Applied Probability 23 pp 835– (1991) · Zbl 0741.60042
[12] Coles, An Introduction to to Statistical Modelling of Extreme Values (2001) · Zbl 0980.62043
[13] Davison, Models for exceedences over high thresholds (with discussion), Journal of the Royal Statistical Society, Series B 52 pp 393– (1990)
[14] Eastoe, Modelling non-stationary extremes with application to surface level ozone, Journal of the Royal Statistical Society, Series C 58 pp 22– (2008)
[15] Embrechts, Modeling Extremal Events (1997)
[16] Ferreira, The extremal index of sub-sampled periodic sequences with strong local dependence, REVSTAT - Statistical Journal 1 pp 15– (2003) · Zbl 1055.62054
[17] Gardner, Cyclostationarity: half a century of research, Signal Processing 86 pp 639– (2006) · Zbl 1163.94338
[18] Horowitz, Extreme values for a nonstationary process: an application to air quality analysis, Technometrics 22 pp 469– (1980)
[19] Hsing, On exceedance point processes from a stationary sequence, Probability and Related Fields 78 pp 97– (1988) · Zbl 0619.60054
[20] Hurd, Graphical methods for determining the presence of periodic correlation, Journal of Time Series Analysis 12 pp 337– (1991) · Zbl 04503458
[21] Jones, Time series with periodic structure, Biometrika 54 pp 403– (1967) · Zbl 0153.47706
[22] Koch, A composite study on the structure and formation of ozone miniholes and minihighs over Central Europe, Geophysical Research Letters 32 (2005)
[23] Konstant, Extreme values of the cyclostationary Gaussian random process, Journal of Applied Probability 30 pp 82– (1993) · Zbl 0768.60048
[24] Lax, Functional Analysis (2002)
[25] Leadbetter, Extremes and Related Properties of Random Sequences and Processes (1983)
[26] Lund, Recursive prediction and likelihood evaluation for periodic ARMA models, Journal of Time Series Analysis 21 pp 75– (2000) · Zbl 0974.62085
[27] Lund, Parsimonious periodic time series modeling, Australian & New Zealand Journal of Statistics 48 pp 33– (2005) · Zbl 1109.62079
[28] MacCluer, Elementary Functional Analysis (2009) · Zbl 1170.46002
[29] Obeysekera, Modeling of aggregated hydrologic time series, Journal of Hydrology 86 pp 197– (1986)
[30] Pickands, Statistical inference using extreme order statistics, Annals of Statistics 3 pp 119– (1975) · Zbl 0312.62038
[31] Resnick, Extreme Values, Regular Variation, and Point Processes (1987) · Zbl 0633.60001
[32] Scotto, Extremes for solutions to stochastic difference equations with regularly varying tails, REVSTAT - Statistical Journal 3 pp 229– (2007) · Zbl 1149.62041
[33] Shao, Computation and characterization of autocorrelations and partial autocorrelations in periodic ARMA models, Journal of Time Series Analysis 25 pp 359– (2004) · Zbl 1062.62203
[34] Vecchia, Periodic autoregressive-moving average (PARMA) modeling with applications to water resources, Water Resources Bulletin 21 pp 721– (1985)
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