Estimation theory for multitype branching processes

Søren Asmussen
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Søren Asmussen: University of Copenhagen, Institute of Mathematical Statistics

A chapter in Semi-Markov Models, 1986, pp 385-395 from Springer

Abstract: Abstract We consider a p-type Galton-Watson process % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qadaGadaWdaeaapeGaamOwa8aadaWgaaWcbaWdbiaad6gaa8aabeaa % aOWdbiaawUhacaGL9baapaWaaSbaaSqaa8qacaWGUbGaeyicI4SaeS % yfHukapaqabaaaaa!3EE7! $$ {\left\{ {{Z_n}} \right\}_{n \in {\Bbb N}}}$$ i.e. Zn= (Zn(1)...Zn((p)), % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaBa % aaleaacaWGUbaabeaakiabg2da9maabmaabaGaamOwamaaBaaaleaa % caWGUbaabeaakmaabmaabaGaaGymaaGaayjkaiaawMcaaiablAcilj % aadQfadaWgaaWcbaGaamOBaaqabaGcdaqadaqaaiaadchaaiaawIca % caGLPaaaaiaawIcacaGLPaaacaGGSaGaaGjbVlaadQfadaWgaaWcba % GaamOBaiabgUcaRiaaigdaaeqaaOGaeyypa0ZaaCbmaeaacqqHJoWu % aSqaaiaadMgacqGH9aqpcaWGSbaabaGaamiCaaaakmaaxadabaGaeu % 4OdmfaleaacaWGRbGaeyypa0JaamiBaaqaaiaadQfadaWgaaadbaGa % amOBaaqabaWcdaqadaqaaiaadMgaaiaawIcacaGLPaaaaaGccaWGAb % Waa0baaSqaaiaad6gacaGGSaGaam4AaaqaamaabmaabaGaamyAaaGa % ayjkaiaawMcaaaaaaaa!608B! $$ {Z_n} = \left( {{Z_n}\left( 1 \right) \ldots {Z_n}\left( p \right)} \right),\;{Z_{n + 1}} = \mathop \Sigma \limits_{i = l}^p \mathop \Sigma \limits_{k = l}^{{Z_n}\left( i \right)} Z_{n,k}^{\left( i \right)}$$ where the % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGAbWdamaaBaaaleaapeGaamOBaiabgUcaRiaaigdaa8aabeaa % k8qacqGH9aqppaWaaCbmaeaapeGaeu4Odmfal8aabaWdbiaadMgacq % GH9aqpcaaIXaaapaqaa8qacaWGWbaaaOWdamaaxadabaWdbiabfo6a % tbWcpaqaa8qacaWGRbGaeyypa0JaaGymaaWdaeaapeGaamOwa8aada % WgaaadbaWdbiaad6gaa8aabeaal8qadaqadaWdaeaapeGaamyAaaGa % ayjkaiaawMcaaaaakiaadQfapaWaa0baaSqaa8qacaWGUbGaaiilai % aadUgaa8aabaWdbmaabmaapaqaa8qacaWGPbaacaGLOaGaayzkaaaa % aaaa!5129! $$ {Z_{n + 1}} = \mathop \Sigma \limits_{i = 1}^p \mathop \Sigma \limits_{k = 1}^{{Z_n}\left( i \right)} Z_{n,k}^{\left( i \right)}$$ are independent for all n, i, k and with the same distribution for any fixed i.

Keywords: Maximum Likelihood Estimator; Asymptotic Property; Estimation Theory; Type Vector; Jordan Canonical Form (search for similar items in EconPapers)
Date: 1986
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DOI: 10.1007/978-1-4899-0574-1_21

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