zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Asymptotic behavior of unstable INAR($p$) processes. (English) Zbl 1241.62122
The authors investigate the asymptotic properties of integer-valued autoregressive model $INAR(p)$ determined by the equation $$X_k=\alpha_1\circ X_{k-1}+\cdots+\alpha_p\circ X_{k-p}+\varepsilon_k,k\in\mathbb N,$$ where $\alpha\circ X=\sum_{j=1}^X\xi_j$ if $X>0$, $\alpha\circ X=0$ if $X>0$, $\xi_j,\ j\in\mathbb N$, are i.i.d. Bernoulli random variables with mean $\alpha\in[0,1]$, $\varepsilon_k$ are i.i.d. non-negative integer-valued random variables with the mean $\mu_{\varepsilon}$, and $\alpha_1,\dots,\alpha_p\in[0,1]$. Under the assumption that the second moment of the innovation $\varepsilon_k$ is finite the authors proved that the sequence of appropriately scaled random step functions $X^n_t=X_{[nt]}/n$, $t\in\mathbb R_+$, $n\in\mathbb N$, formed from an unstable $INAR(p)$ process ($\alpha_1+\cdots+\alpha_p=1$) converges weakly towards a squared Bessel process determined by the stochastic differential equation $dX_t=\left( \mu_{\varepsilon}dt+\sqrt{\sigma^2_{\alpha}X_t^+}dW_t \right)/\varphi'(1)$, $X_0=0$, $t\in\mathbb R_+$, where $\varphi'(1)=\alpha_1+2\alpha_2+\cdots+p\alpha_p>0$, $\sigma^2_{\alpha}=\alpha_1(1-\alpha_1)+2\alpha_2+\cdots+p\alpha_p(1-\alpha_p)$, $W_t,\ t\in\mathbb R_+$, is a standard Wiener process. This limit process is a continuous branching process also known as square-root process or Cox-Ingersoll-Ross process. This asymptotic behavior of unstable $INAR(p)$ models is quite different from that of familiar (real-valued) unstable autoregressive processes of order $p$ ($AR(p)$ models). An application to Boston armed robberies data is presented.

MSC:
62M10Time series, auto-correlation, regression, etc. (statistics)
60J80Branching processes
60F99Limit theorems (probability)
62P25Applications of statistics to social sciences
WorldCat.org
Full Text: DOI arXiv
References:
[1] Al-Osh, M. A.; Alzaid, A. A.: First order integer-valued autoregressive $INAR(1)$ process, J. time ser. Anal. 8, No. 3, 261-275 (1987) · Zbl 0617.62096 · doi:10.1111/j.1467-9892.1987.tb00438.x
[2] Al-Osh, M. A.; Alzaid, A. A.: An integer-valued pth-order autoregressive structure $(INAR(p))$ process, J. appl. Probab. 27, No. 2, 314-324 (1990) · Zbl 0704.62081 · doi:10.2307/3214650
[3] M. Barczy, M. Ispány, G. Pap, Asymptotic behavior of unstable INAR(p) processes, 2009. Available on the arXiv: http://arxiv.org/abs/0908.4560. · Zbl 1241.62122
[4] Billingsley, P.: Convergence of probability measures, (1999) · Zbl 0944.60003
[5] Böckenholt, U.: Mixed $INAR(1)$ Poisson regression models: analyzing heterogeneity and serial dependencies in longitudinal count data, J. econometrics 89, 317-338 (1999) · Zbl 0958.62110 · doi:10.1016/S0304-4076(98)00069-4
[6] Box, G. E. P.; Tiao, G. C.: Intervention analysis with applications to economic and environmental problems, J. amer. Statist. assoc. 70, No. 349, 70-79 (1975) · Zbl 0316.62045 · doi:10.2307/2285379
[7] Brännäs, K.; Hellström, J.: Generalized integer-valued autoregression, Econometric rev. 20, 425-443 (2001) · Zbl 1077.62530 · doi:10.1081/ETC-100106998
[8] K. Brännäs, Q. Shahiduzzaman, Integer-valued moving average modelling of the number of transactions in stocks, Umea Economic Studies 637, University of Umeå, 2004.
[9] Brualdi, R. A.; Cvetković, D.: A combinatorial approach to matrix theory and its applications, (2009) · Zbl 1155.15003
[10] Chan, N. H.; Wei, C. Z.: Limiting distributions of least squares estimates of unstable autoregressive processes, Ann. statist. 16, 367-401 (1988) · Zbl 0666.62019 · doi:10.1214/aos/1176350711
[11] Deutsch, S. J.; Alt, F. B.: The effect of massachusetts’ gun control law on gun-related crimes in the city of Boston, Eval. Q. 1, No. 4, 543-568 (1977)
[12] Drost, F. C.; Den Akker, R. V.; Werker, B. J. M.: The asymptotic structure of nearly unstable non-negative integer-valued $AR(1)$ models, Bernoulli 15, No. 2, 297-324 (2009) · Zbl 1200.62105 · doi:10.3150/08-BEJ153
[13] Du, J. G.; Li, Y.: The integer valued autoregressive $(INAR(p))$ model, J. time ser. Anal. 12, No. 2, 129-142 (1991) · Zbl 0727.62084 · doi:10.1111/j.1467-9892.1991.tb00073.x
[14] Enciso-Mora, V.; Neal, P.; Rao, T. Subba: Efficient order selection algorithms for integer-valued ARMA processes, J. time ser. Anal. 30, No. 1, 1-18 (2009) · Zbl 1224.62053 · doi:10.1111/j.1467-9892.2008.00592.x
[15] Ethier, S. N.; Kurtz, T. G.: Markov processes, (1986)
[16] Franke, J.; Seligmann, T.: Conditional maximum-likelihood estimates for $INAR(1)$ processes and their applications to modelling epileptic seizure counts, Developments in time series, 310-330 (1993) · Zbl 0878.62080
[17] J. Franke, T. Subba Rao, Multivariate first order integer valued autoregressions, Technical Report, Math. Dep. UMIST, England, 1995.
[18] Gauthier, G.; Latour, A.: Convergence FORTE des estimateurs des paramétres d’un processus $GENAR(p)$, Ann. sci. Math. Québec 18, No. 1, 49-71 (1994) · Zbl 0852.62082 · http://www.lacim.uqam.ca/~annales/volumes/18-1/49.html
[19] Gourieroux, C.; Jasiak, J.: Heterogeneous $INAR(1)$ model with application to car insurance, Insurance math. Econom. 34, 177-192 (2004) · Zbl 1107.62110 · doi:10.1016/j.insmatheco.2003.11.005
[20] Hay, R.; Mccleary, R.: Box--tiao times series models for impact assessment, Eval. Q. 3, No. 2, 277-314 (1979)
[21] Hellström, J.: Unit root testing in integer-valued $AR(1)$ models, Econom. lett. 70, 9-14 (2001) · Zbl 0968.91029 · doi:10.1016/S0165-1765(00)00344-X
[22] Horn, R. A.; Johnson, Ch.R.: Matrix analysis, (1985) · Zbl 0576.15001
[23] Ispány, M.; Pap, G.; Van Zuijlen, M. C. A.: Asymptotic inference for nearly unstable $INAR(1)$ models, J. appl. Probab. 40, No. 3, 750-765 (2003) · Zbl 1042.62080 · doi:10.1239/jap/1059060900
[24] Ispány, M.; Pap, G.: A note on weak convergence of step processes, Acta math. Hungar. 126, No. 4, 381-395 (2010) · Zbl 1274.60109
[25] Jacod, J.; Shiryaev, A. N.: Limit theorems for stochastic processes, (2003) · Zbl 1018.60002
[26] Jeganathan, P.: On the asymptotic behavior of least squares estimators in AR time series with roots near the unit circle, Econom. theory 7, 269-306 (1991)
[27] Kallenberg, O.: Foundations of modern probability, (1997) · Zbl 0892.60001
[28] Karatzas, I.; Shreve, S. E.: Brownian motion and stochastic calculus, (1991) · Zbl 0734.60060
[29] Klimko, L. A.; Nelson, P. I.: On conditional least squares estimation for stochastic processes, Ann. statist. 6, No. 3, 629-642 (1978) · Zbl 0383.62055 · doi:10.1214/aos/1176344207
[30] Latour, A.: The multivariate $GINAR(p)$ process, Adv. in appl. Probab. 29, 228-248 (1997) · Zbl 0871.62073 · doi:10.2307/1427868
[31] Latour, A.: Existence and stochastic structure of a non-negative integer-valued autoregressive processes, J. time ser. Anal. 19, No. 4, 439-455 (1998) · Zbl 1127.62402 · doi:10.1111/1467-9892.00102
[32] B.P.M. McCabe, G.M. Martin, D. Harris, Optimal probabilistic forecasts for counts, Monash University, Working Paper 7/09, 2009.
[33] Mckenzie, E.: Some simple models for discrete variate time series, Water resour. Bull. 21, 645-650 (1985)
[34] O’donovan, T. M.: Short term forecasting: an introduction to the box--Jenkins approach, (1983) · Zbl 0565.62078
[35] Pavlopoulos, H.; Karlis, D.: $INAR(1)$ modeling of overdispersed count series with an environmental application, Environmetrics 19, 369-393 (2008)
[36] Phillips, P. C. B.; Xiao, Z.: A primer on unit root testing, J. econom. Surv. 12, 423-470 (1998)
[37] Revuz, D.; Yor, M.: Continuous martingales and Brownian motion, (2001) · Zbl 1087.60040
[38] Rudholm, N.: Entry and the number of firms in the swedish pharmaceuticals market, Rev. ind. Organ. 19, 351-364 (2001)
[39] Steutel, F.; Van Harn, K.: Discrete analogues of self-decomposability and stability, Ann. probab. 7, 893-899 (1979) · Zbl 0418.60020 · doi:10.1214/aop/1176994950
[40] Steutel, F. W.; Van Harn, K.: Infinite divisibility of probability distributions on the real line, (2004) · Zbl 1063.60001
[41] Thyregod, P.; Carstensen, J.; Madsen, H.; Arnbjerg-Nielsen, K.: Integer valued autoregressive models for tipping bucket rainfall measurements, Environmetrics 10, 395-411 (1999)
[42] Van Der Meer, T.; Pap, G.; Van Zuijlen, M. C. A.: Asymptotic inference for nearly unstable $AR(p)$ processes, Econom. theory 15, 184-217 (1999) · Zbl 0967.62072 · doi:10.1017/S0266466699152034
[43] Wei, C. Z.; Winnicki, J.: Some asymptotic results for the branching process with immigration, Stochastic process. Appl. 31, No. 2, 261-282 (1989) · Zbl 0673.60092 · doi:10.1016/0304-4149(89)90092-6
[44] Wei, C. Z.; Winnicki, J.: Estimation of the means in the branching process with immigration, Ann. statist. 18, No. 4, 1757-1773 (1990) · Zbl 0736.62071 · doi:10.1214/aos/1176347876
[45] Weiß, C. H.: Thinning operations for modelling time series of counts--a survey, Asta adv. Stat. anal. 92, No. 3, 319-341 (2008)