Schwarz type model comparison for LAQ models

Abstract

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...

Data driven time scale in Gaussian quasi-likelihood inference

“…The proof given in Section 5.5 goes through as in [4] under essentially weaker conditions due to Theorem A.1.…”

“…We here consider consistent model selection by the (quasi-)Bayesian information criterion ((Q)BIC for short) studied in [4]; previously, the correct form of the classical Schwarz's BIC for ergodic diffusion observed at high frequency was given in [4, Theorems 3.7, 4.5, and 4.6] when h is given a priori.…”

“…We will introduce the BIC type statistics through the stochastic expansion of the proposed mGQLF. In cases where the candidate models are given by the ergodic diffusion models with known h, [4] has introduced BIC type statistics, and studied their model-selection consistency. We note that many authors have investigated the information criteria concerning sampled data from stochastic process models; see, for example [18], [20], [21], [22], and [25].…”

Adaptive test for ergodic diffusions plus noise

“…[14] proposes likelihood-ratio-type test statistic for discretely observed ergodic diffusions to examine parametric hypotheses such as [6] and shows the convergence in law of test statistics and the consistency of the test. As another approach, [18] researches contrast-based information criterion for ergodic diffusion processes with discretised observation scheme (see also [8] and [3]). These instruments to examine goodness-of-fit are important to see whether we are motivated to use the flexible modelling with stochastic differential equations.…”

Hybrid estimation for ergodic diffusion processes based on noisy discrete observations

“…Hence, Kaino et al (2017) and Kaino and Uchida (2018a,b) propose hybrid multi-step estimation procedure for diffusion processes where initial values in optimisation are derived from Bayes type estimation with reduced sample sizes and the sequential optimisation with these initial values is implemented, which inherits the idea of hybrid multi-step estimation for diffusion processes with full sample sizes in Kamatani and Uchida (2015) (see also Kutoyants, 2017). In this research, we also consider hybrid multi-step estimation and apply the idea into inference problem, in particular, PLDI for the quasi likelihood functions and the convergence of moments of estimators for discretely and noisily observed ergodic diffusion processes since PLDI and convergence of moment of estimators are key tools to show the mathematical validity of information criteria for model selection problems (see Uchida, 2010;Fujii and Uchida, 2014;Eguchi and Masuda, 2018).…”