Abstract
It is frequent to encounter a time series of counts which are small in value and show a trend having relatively large fluctuation. To handle such a non-stationary integer-valued time series with a large dispersion, we introduce a new process called integer-valued autoregressive process of order p with signed binomial thinning (INARS(p)). This INARS(p) uniquely exists and is stationary under the same stationary condition as in the AR(p) process. We provide the properties of the INARS(p) as well as the asymptotic normality of the estimates of the model parameters. This new process includes previous integer-valued autoregressive processes as special cases. To preserve integer-valued nature of the INARS(p) and to avoid difficulty in deriving the distributional properties of the forecasts, we propose a bootstrap approach for deriving forecasts and confidence intervals. We apply the INARS(p) to the frequency of new patients diagnosed with acquired immunodeficiency syndrome (AIDS) in Baltimore, Maryland, U.S. during the period of 108 months from January 1993 to December 2001.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alonso AM, Peña D, Romo J (2002) Forecasting time series with sieve bootstrap. J Stat Plan Infer 100:1–11
Al-Osh MA, Alzaid AA (1987) First-order integer-valued autoregressive (INAR(1)) process. J~Time Ser Anal 8:261–275
Aly AA, Bouzar N (1994) On some integer-valued autoregressive moving average models. J Multivariate Anal 50:132–151
Alzaid AA, Al-Osh M (1990) An integer-valued pth-order autoregressive structure (INAR(p)) process. J Appl Probab 27:314–324
Anderson RM, Grenfell BT (1984) Oscillatory fluctuations in the incidence of infectious disease and impact of vaccination: time series analysis. J Hyg-Cambridge 93:587–608
Breslow N (1990) Tests of hypotheses in overdispersed Poisson regression and other quasi-likelihood models. J Am Stat Assoc 85:565–571
Cardinal M, Roy R, Lambert J (1999) On the application of integer-valued time series models for the analysis of disease incidence. Stat Med 18:2025–2039
Center for Disease Control and Prevention (CDC 2001) AIDS Public Information Data Set
Cordeiro GM, Botter DA (2001) Second-order of maximum likelihood estimates in overdispersed generalized linear models. Stat Probabil Lett 55:269–280
Davis RA, Dunsmuir WTM, Wang Y (2000) On autocorrelation in a Poisson regression model. Biometrika 87:491–505
Dey DK, Gelfand AE, Peng F (1997) Overdispersed generalized linear models. J Stat Plan Infer 64:93–107
Durrett R (1991) Probability. Wadsworth, Belmont, CA, pp 292–308
Findley DF (1986) On bootstrap estimates of forecast mean square errors for autoregressive processes. In: Allen DM (ed) Computer science and statistics: the interface NorthHolland Amsterdam, pp 11–17
Freeland RK, McCabe BPM (2004) Forecasting discrete valued low count time series. Int J Forecasting 20:427–434
Grigoletto M (1998) Bootstrap prediction intervals for autoregressions: some alternative. Int J Forecasting 14:447–456
Jin-Guan D, Yuan L (1991) The integer-valued autoregressive (INAR(p)) model. J Time Ser Anal 12:129–142
Jung RC, Tremayne AR (2006) Coherent forecasting in integer time series models. Int J Forecasting 22:223–238
Klimko LA, Nelson PI (1978) On conditional least squares estimation for stochastic processes. Ann Stat 6:629–642
Masarotto G (1990) Bootstrap prediction intervals for autoregressions. Int J Forecasting 6:229–239
McCormick WP, Park YS (1997) An analysis of Poisson moving-average processes. Probab Eng Inform Sci 11:487–507
McCullagh P, Nelder JA (1989) Generalized linear models. Chapman & Hall, London
McKenzie E (1986) Autoregressive moving-average processes with negative binomial and geometric marginal distributions. Adv Appl Probab 18:679–705
Park YS, Oh CH (1997) Some asymptotic properties in INAR(1) processes with Poisson marginals. Stat Pap 38:287–302
Pascual L, Romo J, Ruiz E (2004) Bootstrap predictive inference for ARIMA processes. J Time Ser Anal 25:449–465
Thombs LA, Schucany WR (1990) Bootstrap prediction intervals for autoregression. J Am Stat Assoc 85:486–492
Waiter L, Richardson S, Hubert B (1991) A time series construction of an alert threshold with application to S. bovismorbificans in France. Stat Med 10:1493–1509
Zaidi AA, Schnell DJ, Reynolds G (1989) Time series analysis of sypholis surveillance data. Stat Med 8:353–362
Zeger SL, Qaqish B (1988) Markov regression models for time series: a Quasi-Likelihood approach. Biometrics 44:1019–1031
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kim, HY., Park, Y. A non-stationary integer-valued autoregressive model. Statistical Papers 49, 485–502 (2008). https://doi.org/10.1007/s00362-006-0028-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-006-0028-1