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| Év szerint | Hónap szerint | Ugrás a hónaphoz | |||||||
Barczy Mátyás: Stable convergence of CLS estimators for supercritical continuous state and continuous time branching processes with immigration |
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| Abstract: We study the asymptotic behavior of conditional least squares (CLS) estimators of drift parameters for supercritical continuous state and continuous time branching processes with immigration (CBI processes) based on discrete time (low frequency) observations. According to our knowledge, results on stable (mixing) convergence of CLS estimators for parameters of CBI processes are not available in the literature, all the existing results state convergence in distribution of the appropriately normalized CLS estimators in question. For supercritical discrete time Galton-Watson branching processes, Häusler and Luschgy [3] proved stable convergence of the CLS estimator of the offspring mean under non-extinction, which served us as a motivation for investigating the problem for CBI processes. In case of a nontrivial immigration mechanism, under second order moment conditions on the initial law and on the branching and immigration mechanisms, we describe the asymptotic behavior of the CLS estimator of (transformed) drift parameters for a supercritical CBI process, by proving stable convergence. The limit distribution is mixed normal, except a particular case. Our results immediately yield convergence in distribution of the appropriately normalized CLS estimators in question, since stable convergence implies convergence in distribution. The main step of our proof is to establish a stable limit theorem for a martingale associated to the supercritical CBI process in question. At this point, we use a multidimensional analogue of a one-dimensional stable limit theorem due to Häusler and Luschgy [3, Chapter 7] for so called explosive stochastic processes. In fact, our multidimensional analogue may be interesting in its own right as well. If time permits, then, as special cases, we present multidimensional stable limit theorems involving multidimensional normal-, Cauchy- and stable distributions as well. The talk is based on the papers Barczy and Pap [1] and Barczy [2]. [1] Barczy, M., Pap, G. (2023) A multidimensional stable limit theorem, Filomat 37(11), 3493-3512. [2] Barczy, M. (2023+) Stable convergence of conditional least squares estimators for supercritical continuous state and continuous time branching processes with immigration, Arxiv: 2207.14056. [3] Häusler, E., Luschgy, H. (2015) Stable Convergence and Stable Limit Theorems, Springer, Cham. |
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| Hely : Szeged, Aradi vértanúk tere 1., Riesz terem | ||||
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