## Bernoulli

• Bernoulli
• Volume 20, Number 4 (2014), 2247-2277.

### Asymptotic behavior of CLS estimators for 2-type doubly symmetric critical Galton–Watson processes with immigration

#### Abstract

In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means $(\alpha,\beta)$ and of the criticality parameter $\varrho:=\alpha+\beta$ for a 2-type critical doubly symmetric positively regular Galton–Watson branching process with immigration is described.

#### Article information

Source
Bernoulli, Volume 20, Number 4 (2014), 2247-2277.

Dates
First available in Project Euclid: 19 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1411134459

Digital Object Identifier
doi:10.3150/13-BEJ556

Mathematical Reviews number (MathSciNet)
MR3263104

Zentralblatt MATH identifier
1321.60178

#### Citation

Ispány, Márton; Körmendi, Kristóf; Pap, Gyula. Asymptotic behavior of CLS estimators for 2-type doubly symmetric critical Galton–Watson processes with immigration. Bernoulli 20 (2014), no. 4, 2247--2277. doi:10.3150/13-BEJ556. https://projecteuclid.org/euclid.bj/1411134459

#### References

• [1] Athreya, K.B. and Ney, P.E. (1972). Branching Processes. New York: Springer.
• [2] Barczy, M., Ispány, M. and Pap, G. (2012). Asymptotic behavior of CLS estimators for unstable $\mathrm{INAR}(2)$ models. Available at http://arxiv.org/abs/1202.1617.
• [3] Barczy, M., Ispány, M. and Pap, G. (2011). Asymptotic behavior of unstable $\mathrm{INAR}(p)$ processes. Stochastic Process. Appl. 121 583–608.
• [4] Guttorp, P. (1991). Statistical Inference for Branching Processes. Wiley Series in Probability and Mathematical Statistics. New York: Wiley.
• [5] Hall, P. and Yao, Q. (2003). Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71 285–317.
• [6] Hamilton, J.D. (1994). Time Series Analysis. Princeton, NJ: Princeton Univ. Press.
• [7] Horn, R.A. and Johnson, C.R. (1985). Matrix Analysis. Cambridge: Cambridge Univ. Press.
• [8] Ispány, M., Körmendi, K. and Pap, G. (2012). Asymptotic behavior of CLS estimators for 2-type critical Galton–Watson processes with immigration. Available at http://arxiv.org/abs/1210.8315.
• [9] Ispány, M. and Pap, G. (2012). Asymptotic behavior of critical primitive multi-type branching processes with immigration. Available at http://arxiv.org/abs/1205.0388.
• [10] Ispány, M. and Pap, G. (2010). A note on weak convergence of random step processes. Acta Math. Hungar. 126 381–395.
• [11] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Berlin: Springer.
• [12] Kallenberg, O. (1997). Foundations of Modern Probability. Probability and Its Applications (New York). New York: Springer.
• [13] Kesten, H. and Stigum, B.P. (1966). A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37 1211–1223.
• [14] Mikosch, T. and Straumann, D. (2002). Whittle estimation in a heavy-tailed $\mathrm{GARCH}(1,1)$ model. Stochastic Process. Appl. 100 187–222.
• [15] Musiela, M. and Rutkowski, M. (1997). Martingale Methods in Financial Modelling. Applications of Mathematics (New York) 36. Berlin: Springer.
• [16] Quine, M.P. (1970). The multi-type Galton–Watson process with immigration. J. Appl. Probability 7 411–422.
• [17] Revuz, D. and Yor, M. (2001). Continuous Martingales and Brownian Motion, 3rd ed., corrected 2nd printing. Berlin: Springer.
• [18] Shete, S. and Sriram, T.N. (2003). A note on estimation in multitype supercritical branching processes with immigration. Sankhyā 65 107–121.
• [19] Tanaka, K. (1996). Time Series Analysis: Nonstationary and Noninvertible Distribution Theory. Wiley Series in Probability and Statistics. New York: Wiley.
• [20] Wei, C.Z. and Winnicki, J. (1989). Some asymptotic results for the branching process with immigration. Stochastic Process. Appl. 31 261–282.
• [21] Wei, C.Z. and Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Ann. Statist. 18 1757–1773.
• [22] Winnicki, J. (1991). Estimation of the variances in the branching process with immigration. Probab. Theory Related Fields 88 77–106.