Statistical inference for critical continuous state and continuous time branching processes with immigration

Abstract

We study asymptotic behavior of conditional least squares estimators for critical continuous state and continuous time branching processes with immigration based on discrete time (low frequency) observations.Keywords Continuous state and continuous time branching processes with immigration · Branching and immigration mechanisms · Conditional least squares estimator · Non-normal asymptotic limit behaviour Mathematics Subject Classification 62F12 · 60J80

Statistical inference for 2-type doubly symmetric critical irreducible continuous state and continuous time branching processes with immigration

“…see the proof of Theorem 3.4 in Barczy et al[4]), Cv, v > 0 and P Y t dt > 0 = 1, we obtain(3.21).In order to prove(3.22), first note that, under the additional condition Cv, v = 0, we have(4.14) V 0 v, v = 0 if and only if U 2 ν(dz)…”

“…. }, supposing that c, µ and ν are known, Barczy et al [4] derived (non-weighted) CLS estimator ( b n , β n ), of ( b, β), where β := β + ∞ 0 z ν(dz). In the critical case, under some moment assumptions, it has been shown that n( b n − b), β n − β has a non-normal limit.…”

Nearly unstable integer‐valued ARCH process and unit root testing

“…They showed that, under nearly nonstationarity and assuming finite second moment for $$$ {\epsilon}_t $$$, the CLS estimator weakly converges to a normal distribution at the rate $$$ {n}^{3/2} $$$. Other related works dealing with nearly unstable INAR (Galton–Watson/branching) processes are due to Wei and Winnicki (1990), Winnicki (1991), Ispány et al (2005, 2014), Rahimov (2007, 2008, 2009), Drost et al (2009), Barczy et al (2011), Barczy et al (2014, 2016), and Guo and Zhang (2014). Practical situations demonstrating evidence of a nearly unstable INAR model are discussed for instance by Hellström (2001).…”

Nearly Unstable Integer-Valued ARCH Process and Unit Root Testing

“…They showed that, under nearly non-stationarity and assuming finite second moment for t , the CLS estimator weakly converges to a normal distribution at the rate n 3/2 . Other related works dealing with nearly unstable INAR (Galton-Watson/branching) processes are due to Wei and Winnicki (1990), Winnicki (1991), Ispány, Pap and Van Zuijlen (2005), Rahimov (2007), Rahimov (2008), Drost, Van Den Akker and Werker (2009), Rahimov (2009), Barczy, Ispány and Pap (2011), Ispány, Körmendi and Pap (2014), Barczy, Ispány and Pap (2014), Guo and Zhang (2014), and Barczy, Körmendi and Pap (2016). Practical situations demonstrating evidence of a nearly unstable INAR model are discussed for instance by Hellström (2001).…”