Information Criteria for Multivariate CARMA Processes

Fasen, Vicky; Kimmig, Sebastian

doi:10.48550/arxiv.1505.00901

Cited by 2 publications

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“…We only prove (5.2) because (5.3) can be handled analogously. We basically follow Fasen and Kimmig [16].…”

Section: It Follows Thatmentioning

confidence: 99%

Schwarz type model comparison for LAQ models

Eguchi¹,

Masuda²

2018

Bernoulli

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For model-specification purpose, we study asymptotic behavior of the marginal quasi-log likelihood associated with a family of locally asymptotically quadratic (LAQ) statistical experiments. Our result entails a far-reaching extension of applicable scope of the classical approximate Bayesian model comparison due to Schwarz, with frequentist-view theoretical foundation. In particular, the proposed statistics can deal with both ergodic and non-ergodic stochastic-process models, where the corresponding M -estimator is of multi-scaling type and the asymptotic quasi-information matrix is random. Focusing on the ergodic diffusion model, we also deduce the consistency of the multistage optimal-model selection where we may select an optimal sub-model structure step by step, so that computational cost can be much reduced. We illustrate the proposed method by the Gaussian quasi-likelihood for diffusion-type models in details, together with several numerical experiments.Date: November 5, 2018. Key words and phrases. Approximate Bayesian model comparison, Gaussian quasi-likelihood, locally asymptotically quadratic family, quasi-likelihood, Schwarz's criterion. 1• The measurable function x → H m,n (x|θ m ) for each θ m ∈ Θ m defines a logarithmic regular conditional probability density of L(X n |θ m ) with respect to µ n (dx).Each M m may be misspecified in the sense that the true data generating model g n (x) does not belong to the family {exp{H m,n (·|θ m )}| θ m ∈ Θ m }; we will, however, assume suitable regularity conditions for the associated statistical random fields to have a suitable asymptotic behavior.Concerning the model M m , the random function θ m → exp{H m,n (X n |θ m )}, assumed to be a.s. welldefined, is referred to as the quasi-likelihood of L(X n |θ m ). The quasi-maximum likelihood estimator (QMLE)θ m,n associated with H m,n is defined to be any maximizer of H m,n :For simplicity we will assume the a.s. continuity of H m,n over the compact setΘ m , so thatθ m,n always exists.Our objective includes estimators of multi-scaling type, meaning that the components ofθ m,n converges at different rates, which can often occur when considering high-frequency asymptotics. A typical example is the Gaussian quasi-likelihood estimation of ergodic diffusion process: see [24], also Section 4.2. Let K m ∈ N be a given number, which represents the number of the components having different convergence rates in M m , and assume that the mth-model parameter vector is divided into K m parts:Then the QMLE in the mth model takes the formθ m,n = (θ m,1,n , . . . ,θ m,Km,n ). The optimal value of θ m associated with H m,n , to be precisely defined later on, is denoted by θ m,0 = (θ m,1,0 , . . . , θ m,Km,0 ), θ m,k,0 ∈ Θ m,k . The rate matrix in the model M m is then given in the form A m,n (θ m,0 ) = diag a m,1,n (θ m,0 )I pm,1 , . . . , a m,Km,n (θ m,0 )I pm,K m , (2.1)where I p denotes the p-dimensional identity matrix and a m,k,n (θ m,0 ) is a deterministic sequence satisfying that a m,k,n (θ m,0 ) → 0, a m,i,n (θ m,0 )/a m,j,n...

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“…We only prove (5.2) because (5.3) can be handled analogously. We basically follow Fasen and Kimmig [16].…”

Section: It Follows Thatmentioning

confidence: 99%

Schwarz type model comparison for LAQ models

Eguchi¹,

Masuda²

2018

Bernoulli

View full text Add to dashboard Cite

show abstract

“…Recall that θ m,0 and m 0 are given by {θ m,0 } = argmax θ∈Θ H m,0 (θ) and {m 0 } = argmin m∈M dim(Θ m ), respectively, where M = argmax m∈{1,...,M } H m,0 (θ m,0 ). Fasen and Kimmig [16] proved the model selection consistency of likelihood-based information criteria, which include AIC and BIC, for multivariate continuous-time ARMA processes. We basically follow their scenario for the proof of Theorem 3.4.…”

Section: Proofsmentioning

confidence: 99%

“…Bozdogan [6] showed that the Akaike information criterion (AIC, Akaike [1], [2]) has a positive probability of overestimating the true dimension. Casella et al [8] and Fasen and Kimmig [16] as well as the references therein studied the model selection consistency of the BIC. Moreover, various extensions of the AIC and BIC have been introduced; for example, the extended BIC for large model spaces (Chen and Chen [10]), generalized information criterion (Konishi and Kitagawa [21]), generalized BIC in misspecified GLMs for independent data (Lv and Liu [23]), and information criteria for stochastic processes (e.g., Sei and Komaki [28] and Uchida [29]).…”

Section: Introductionmentioning

confidence: 99%

Model comparison for generalized linear models with dependent observations

Eguchi

2018

Econometrics and Statistics

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Abstract. The stochastic expansion of the marginal quasi-likelihood function associated with a class of generalized linear models is shown. Based on the expansion, a quasi-Bayesian information criterion is proposed that is able to deal with misspecified models and dependent data, resulting in a theoretical extension of the classical Schwarz's Bayesian information criterion. It is also proved that the proposed criterion has model selection consistency with respect to the optimal model. Some illustrative numerical examples and a real data example are presented.

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Information Criteria for Multivariate CARMA Processes

Cited by 2 publications

References 0 publications

Schwarz type model comparison for LAQ models

Schwarz type model comparison for LAQ models

Model comparison for generalized linear models with dependent observations

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