×

Detecting and estimating intensity of jumps for discretely observed \(\mathrm{ARMA}D(1,1)\) processes. (English) Zbl 1381.62246

Summary: We consider \(n\) equidistributed random functions, defined on [0,1], and admitting fixed or random jumps, the context being \(D [0,1]\)-valued ARMA(1,1) processes. We begin with properties of ARMA\(D(1,1)\) processes. Next, different scenarios are considered: fixed instants with a given but unknown probability of jumps (the deterministic case), random instants with ordered intensities (the random case), and random instants with non ordered intensities (the completely random case). By using discrete data and for each scenario, we identify the instants of jumps, whose number is either random or fixed, and then estimate their intensity.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F17 Functional limit theorems; invariance principles

Software:

fda (R)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aït-Sahalia, Y.; Jacod, J., Fisher’s information for discretely sampled Lévy processes, Econometrica, 76, 4, 727-761 (2008) · Zbl 1144.62070
[2] Aït-Sahalia, Y.; Jacod, J., Estimating the degree of activity of jumps in high frequency data, Ann. Statist., 37, 5A, 2202-2244 (2009) · Zbl 1173.62060
[3] Antoniadis, A.; Brossat, X.; Cugliari, J.; Poggi, J.-M., Prévision d’un processus à valeurs fonctionnelles en présence de non stationnarités. Application à la consommation d’électricité, J. SFdS, 153, 2, 52-78 (2012) · Zbl 1316.62171
[4] Barndorff-Nielsen, O. E.; Benth, F. E.; Veraart, A. E., Modelling electricity futures by ambit fields, Adv. Appl. Probab., 46, 3, 719-745 (2014) · Zbl 1304.91213
[5] Besse, P.; Cardot, H.; Stephenson, D., Autoregressive forecasting of some climatic variation, Scand. J. Stat., 27, 4, 673-687 (2000) · Zbl 0962.62089
[6] Billingsley, P., (Convergence of Probability Measures. Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics (1999), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York) · Zbl 0944.60003
[7] Blanke, D.; Bosq, D., Exponential bounds for intensity of jumps, Math. Methods Statist., 23, 3, 1-17 (2014) · Zbl 1321.60063
[8] Bollerslev, T.; Todorov, V., Estimation of jump tails, Econometrica, 79, 6, 1727-1783 (2011) · Zbl 1241.91136
[9] Bongiorno, E. G.; Salinelli, E.; Goia, A.; Vieu, P., Contributions in Infinite-Dimensional Statistics and Related Topics (2014), Società Editrice Esculapio
[10] Borisov, A. V., Analysis and estimation of the states of special Markov jump processes. II. Optimal filtering in the presence of Wiener noise, Avtomat. i Telemekh., 5, 61-76 (2004)
[11] Bosq, D., (Linear Processes in Function Spaces. Linear Processes in Function Spaces, Lecture Notes in Statistics, vol. 149 (2000), Springer-Verlag: Springer-Verlag New-York) · Zbl 0962.60004
[12] Bosq, D., Estimating and detecting jumps. Applications to \(D [0, 1]\)-valued linear processes, (Hallin, M.; Mason, D. M.; Pfeifer, D.; Steinebach, J. G., Festschrift in Honour of Paul Deheuvels (2015), Springer), 41-66, (Chapter 4) · Zbl 1328.60092
[13] Bosq, D.; Blanke, D., (Prediction and Inference in Large Dimensions. Prediction and Inference in Large Dimensions, Wiley Series in Probability and Statistics (2007), John Wiley & Sons, Ltd.: John Wiley & Sons, Ltd. Chichester), Dunod, Paris · Zbl 1183.62157
[14] Brockwell, P. J.; Davis, R. A.; Yang, Y., Estimation for nonnegative Lévy-driven Ornstein-Uhlenbeck processes, J. Appl. Probab., 44, 4, 977-989 (2007) · Zbl 1513.62162
[15] Cardot, H.; Crambes, C.; Sarda, P., Ozone pollution forecasting using conditional mean and conditional quantiles with functional covariates, (Statistical Methods for Biostatistics and Related Fields (2007), Springer: Springer Berlin), 221-243
[16] Chiquet, J.; Limnios, N., Dynamical systems with semi-Markovian perturbations and their use in structural reliability, (Stochastic Reliability and Maintenance Modeling. Stochastic Reliability and Maintenance Modeling, Springer Ser. Reliab. Eng., vol. 9 (2013), Springer: Springer London), 191-218 · Zbl 1365.60075
[17] Clément, E.; Delattre, S.; Gloter, A., Asymptotic lower bounds in estimating jumps, Bernoulli, 20, 3, 1059-1096 (2014) · Zbl 1401.62136
[18] Comte, F.; Duval, C.; Genon-Catalot, V., Nonparametric density estimation in compound Poisson processes using convolution power estimators, Metrika, 77, 1, 163-183 (2014) · Zbl 1282.62088
[19] Comte, F.; Genon-Catalot, V., Nonparametric adaptive estimation for pure jump Lévy processes, Ann. Inst. Henri Poincaré Probab. Stat., 46, 3, 595-617 (2010) · Zbl 1201.62042
[20] Cont, R.; Tankov, P., (Financial Modelling with Jump Processes. Financial Modelling with Jump Processes, Chapman & Hall/CRC Financial Mathematics Series (2004), Chapman & Hall/CRC: Chapman & Hall/CRC Boca Raton, FL) · Zbl 1052.91043
[21] Crambes, C.; Mas, A., Asymptotics of prediction in functional linear regression with functional outputs, Bernoulli, 19, 5B, 2627-2651 (2013) · Zbl 1280.62084
[22] Dabo-Niang, S.; Laksaci, A., Nonparametric quantile regression estimation for functional dependent data, Comm. Statist. Theory Methods, 41, 7, 1254-1268 (2012) · Zbl 1319.62086
[23] Damon, J.; Guillas, S., Estimation and simulation of autoregressive hilbertian processes with exogenous variables, Stat. Inference Stoch. Process., 8, 2, 185-204 (2005) · Zbl 1076.62087
[24] de Saporta, B.; Yao, J.-F., Tail of a linear diffusion with Markov switching, Ann. Appl. Probab., 15, 1B, 992-1018 (2005) · Zbl 1064.60174
[25] Djebali, S.; Górniewicz, L.; Ouahab, A., Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces, Nonlinear Anal., 74, 6, 2141-2169 (2011) · Zbl 1215.34070
[26] Duval, C., Density estimation for compound Poisson processes from discrete data, Stochastic Process. Appl., 123, 11, 3963-3986 (2013) · Zbl 1320.62079
[27] El Hajj, L., Théorèmes limites pour les processus autorégressifs à valeurs dans \(D [0, 1] (2013)\), Université Paris: Université Paris 6, Paris, décembre
[28] El Hajj, L., Estimation et prévision des processus à valeurs dans \(D [0, 1]\), Ann. I.S.U.P., 58, 3, 61-76 (2014)
[29] El Karoui, N.; Gobet, E., Les outils Stochastiques des Marchés Financiers (2012), Éditions de l’École Polytechnique: Éditions de l’École Polytechnique Palaiseau
[30] (Ferraty, F.; Romain, Y., The Oxford Handbook of Functional Data Analysis (2011), Oxford University Press: Oxford University Press Oxford) · Zbl 1284.62001
[31] Ferraty, F.; Vieu, P., (Nonparametric Functional Data Analysis. Nonparametric Functional Data Analysis, Springer Series in Statistics (2006), Springer: Springer New York), Theory and practice · Zbl 1119.62046
[32] Goia, A., A functional linear model for time series prediction with exogenous variables, Statist. Probab. Lett., 82, 5, 1005-1011 (2012) · Zbl 1241.62124
[33] Guy, R.; Larédo, C.; Vergu, E., Approximation of epidemic models by diffusion processes and their statistical inference, J. Math. Biol., 70, 3, 621-646 (2015) · Zbl 1308.62040
[34] Guyon, X.; Iovleff, S.; Yao, J.-F., Linear diffusion with stationary switching regime, ESAIM Probab. Stat., 8, 25-35 (2004) · Zbl 1033.60084
[35] Horváth, L.; Kokoszka, P., (Inference for Functional Data With Applications. Inference for Functional Data With Applications, Springer Series in Statistics (2012), Springer: Springer New York) · Zbl 1279.62017
[36] Ignaccolo, R.; Ghigo, S.; Bande, S., Functional zoning for air quality, Environ. Ecol. Stat., 20, 1, 109-127 (2013)
[37] Jacq, V.; Albert, P.; Delorme, R., Le mistral. Quelques aspects des connaissances actuelles, La météorologie, 50, 30-38 (2005)
[39] Jeanblanc, M.; Yor, M.; Chesney, M., (Mathematical Methods for Financial Markets. Mathematical Methods for Financial Markets, Springer Finance (2009), Springer-Verlag London, Ltd.: Springer-Verlag London, Ltd. London) · Zbl 1205.91003
[40] (Kannan, D.; Lakshmikantham, V., Handbook of Stochastic Analysis and Applications. Handbook of Stochastic Analysis and Applications, Statistics: Textbooks and Monographs, vol. 163 (2002), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York) · Zbl 0979.00017
[41] Kargin, V.; Onatski, A., Curve forecasting by functional autoregression, J. Multivariate Anal., 99, 10, 2508-2526 (2008) · Zbl 1151.62073
[42] Koroliuk, V. S.; Limnios, N., Poisson approximation of processes with locally independent increments with Markov switching, Theory Stoch. Process., 15, 1, 40-48 (2009) · Zbl 1224.60195
[43] Kurchenko, O. O., Convergence of a sequence of random fields in the space \(D([0, 1]^d)\), Teor. I˘movı¯r. Mat. Stat., 64, 82-91 (2001) · Zbl 0998.60029
[44] Lévy, P., Fonctions aléatoires à corrélation linéaire, C. R. Acad. Sci. Paris, 242, 2095-2097 (1956) · Zbl 0070.36503
[45] Marion, J. M.; Pumo, B., Comparaison des modèles ARH(1) et ARHD(1) sur des données physiologiques, Ann. I.S.U.P., 48, 3, 29-38 (2004) · Zbl 1065.62186
[46] Nason, G., A test for second-order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series, J. R. Stat. Soc. Ser. B. Stat. Methodol., 75, 5, 879-904 (2013) · Zbl 1411.62259
[47] Pestman, W. R., Measurability of linear operators in the Skorokhod topology, Bull. Belg. Math. Soc. Simon Stevin, 2, 4, 381-388 (1995) · Zbl 0844.47016
[48] Preston, D.; Protopapas, P.; Brodley, C., Discovering arbitrary event types in time series, Stat. Anal. Data Min., 2, 5-6, 396-411 (2009)
[49] Privault, N., (Stochastic Finance. Stochastic Finance, Chapman & Hall/CRC Financial Mathematics Series (2014), CRC Press: CRC Press Boca Raton, FL), An introduction with market examples · Zbl 1294.91003
[50] Ramsay, J. O.; Silverman, B., Functional Data Analysis (2005), Springer Verlag · Zbl 1079.62006
[51] Shimizu, Y., Threshold selection in jump-discriminant filter for discretely observed jump processes, Stat. Methods Appl., 19, 3, 355-378 (2010) · Zbl 1332.62305
[52] Shimizu, Y.; Yoshida, N., Estimation of parameters for diffusion processes with jumps from discrete observations, Stat. Inference Stoch. Process., 9, 3, 227-277 (2006) · Zbl 1125.62089
[53] Shiryaev, A. N., (Probability. Probability, Graduate Texts in Mathematics, vol. 95 (1996), Springer-Verlag: Springer-Verlag New York), Translated from the first (1980) Russian edition by R. P. Boas
[54] Tankov, P.; Voltchkova, E., Jump-diffusion models: a practitioner’s guide, Banque et Marchés, 99, 1-24 (2009)
[55] Tanushev, N. M., Superpositions and higher order Gaussian beams, Commun. Math. Sci., 6, 2, 449-475 (2008) · Zbl 1166.35345
[56] Torgovitski, L., A Darling-Erdős-type CUSUM-procedure for functional data, Metrika, 78, 1, 1-27 (2015) · Zbl 1333.62123
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.