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Parameter estimation and diagnostic tests for INMA(1) processes. (English) Zbl 1460.62142

Summary: The INMA(1) model, an integer-valued counterpart to the usual moving-average model of order 1, gained recently importance for insurance applications. After a comprehensive discussion of stochastic properties of the INMA(1) model, we develop diagnostic tests regarding the marginal distribution (overdispersion, zero inflation) and the autocorrelation structure. We also derive formulae for correcting the bias of point estimators and for constructing joint confidence regions. These inferential approaches rely on asymptotic properties, the finite-sample performance of which is investigated with simulations. A real-data example illustrates the application of the novel diagnostic tools.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F03 Parametric hypothesis testing
62F10 Point estimation
62J20 Diagnostics, and linear inference and regression
60G10 Stationary stochastic processes
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[1] Al-Osh, Ma; Alzaid, Aa, Integer-valued moving average (INMA) process, Stat Pap, 29, 1, 281-300 (1988) · Zbl 0654.62074
[2] Brännäs, K.; Hall, A., Estimation in integer-valued moving average models, Appl Stoch Models Bus Ind, 17, 3, 277-291 (2001) · Zbl 0979.62066
[3] Brännäs, K.; Quoreshi, Amms, Integer-valued moving average modelling of the number of transactions in stocks, Appl Financ Econ, 20, 18, 1429-1440 (2010)
[4] Cossette, H.; Marceau, E.; Maume-Deschamps, V., Discrete-time risk models based on time series for count random variables, ASTIN Bull J IAA, 40, 1, 123-150 (2010) · Zbl 1230.91071
[5] Cossette, H.; Marceau, E.; Toureille, F., Risk models based on time series for count random variables, Insur Math Econ, 48, 1, 19-28 (2011) · Zbl 1218.91074
[6] Davis, Ra; Holan, Sh; Lund, R.; Ravishanker, N., Handbook of discrete-valued time series (2016), Boca Raton: Chapman & Hall, Boca Raton
[7] Freeland RK (1998) Statistical analysis of discrete time series with applications to the analysis of workers compensation claims data. Ph.D. thesis, University of British Columbia, Canada. https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0088709
[8] Hall, A., Extremes of integer-valued moving averages models with regularly varying tails, Extremes, 4, 3, 219-239 (2001) · Zbl 1053.62064
[9] Hall, A., Extremes of integer-valued moving averages models with exponential type tails, Extremes, 6, 4, 361-379 (2003) · Zbl 1088.60050
[10] Hall, A.; Scotto, Mg; Cruz, Jp, Extremes of integer-valued moving average sequences, Test, 19, 2, 359-374 (2010) · Zbl 1203.60056
[11] Hu, X.; Zhang, L.; Sun, W., Risk model based on the first-order integer-valued moving average process with compound Poisson distributed innovations, Scand Actuar J, 5, 412-425 (2018) · Zbl 1416.91190
[12] Ibragimov, I., Some limit theorems for stationary processes, Theory Probab Appl, 7, 4, 349-382 (1962) · Zbl 0119.14204
[13] Mckenzie, E., Some simple models for discrete variate time series, Water Resour Bull, 21, 4, 645-650 (1985)
[14] Mckenzie, E., Some ARMA models for dependent sequences of Poisson counts, Adv Appl Probab, 20, 4, 822-835 (1988) · Zbl 0664.62089
[15] Romano, Jp; Thombs, La, Inference for autocorrelations under weak assumptions, J Am Stat Assoc, 91, 434, 590-600 (1996) · Zbl 0868.62071
[16] Schweer, S.; Weiß, Ch, Compound Poisson INAR(1) processes: stochastic properties and testing for overdispersion, Comput Stat Data Anal, 77, 267-284 (2014) · Zbl 1506.62162
[17] Steutel, Fw; Van Harn, K., Discrete analogues of self-decomposability and stability, Ann Probab, 7, 5, 893-899 (1979) · Zbl 0418.60020
[18] Weiß, Ch, Serial dependence and regression of Poisson INARMA models, J Stat Plan Inference, 138, 10, 2975-2990 (2008) · Zbl 1140.62069
[19] Weiß, Ch, An introduction to discrete-valued time series (2018), Chichester: Wiley, Chichester · Zbl 1407.62009
[20] Weiß, Ch; Homburg, A.; Puig, P., Testing for zero inflation and overdispersion in INAR(1) models, Stat Pap (2016) · Zbl 1420.62401
[21] Yu K, Zou H (2015) The combined Poisson INMA(q) models for time series of counts. J Appl Math. Article ID 457842 · Zbl 1435.62342
[22] Zhang, L.; Hu, X.; Duan, B., Optimal reinsurance under adjustment coefficient measure in a discrete risk model based on Poisson MA(1) process, Scand Actuar J, 5, 455-467 (2015) · Zbl 1401.91213
[23] Zou, H.; Yu, K., First order threshold integer-valued moving average processes, Dyn Contin Discrete Impuls Sys Ser B, 21, 2-3, 197-205 (2014) · Zbl 1333.65016
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