Asymptotic theory of generalized information criterion for geostatistical regression model selection

Chang, Chih‐Hao; Huang, Hsin‐Cheng; Ing, Ching‐Kang

doi:10.1214/14-aos1258

Cited by 10 publications

(8 citation statements)

References 28 publications

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“…Denote the projection matrix

P_{m}^{*} = P_{m}^{F} {false|}_{\overset{θ}{false} = θ^{*}}

, the mean process

{\overset{μ}{false}}^{*} false(w false) = {\overset{μ}{false}}^{F} false(w false) {false|}_{\overset{θ}{false} = θ^{*}}

, the loss function

L_{n}^{*} false(w false) = L_{n}^{F} false(w false) {false|}_{\overset{γ}{false} = γ^{*}}

, the risk function

R_{n}^{*} false(w false) = E false(L_{n}^{*} false(w false) false)

, the SPMMA criterion

C_{n}^{*} false(w false) = C_{n}^{F} false(w false) {false|}_{\overset{γ}{false} = γ^{*}}

, the spatial autocorrelation matrix

V^{*} = \overset{V}{false} {false|}_{\overset{θ}{false} = θ^{*}}

, and the spatial covariance matrix

{normalΩ}^{*} = \overset{normalΩ}{false} {false|}_{\overset{γ}{false} = γ^{*}}

, where

\overset{γ}{false} = {false({\overset{σ}{false}}^{2}, \overset{θ}{false} false)}^{'}

is assumed to converge to γ ∗ = ( σ ∗ 2 , θ ∗ ) ′ in probability, not necessarily to the true value γ . A similar assumption can be found in Chang, Huang & Ing () and Zhang, Zou & Liang (). Assume θ ∗ >0.…”

Section: Asymptotic Optimalitysupporting

confidence: 76%

“…Equation is similar to Condition (12) of Wan, Zhang & Zou (). Equation is commonly used in the literature on model selection and averaging (Chang, Huang & Ing, ). Equation is similar to Condition (A.5) of Zhang, Zou & Carroll ().…”

Section: Asymptotic Optimalitymentioning

confidence: 99%

“…Zhu, Huang & Reyes () extended this variable selection idea and proposed a penalized least squares estimator with some asymptotic properties. Chang, Huang & Ing () studied the generalized information criterion (GIC) for geostatistical model selection and found that asymptotic properties of spatial model selection estimators can be affected by both the growth rate of the domain and the degree of smoothness of candidate regressors.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spatial Mallows model averaging for geostatistical models

2019

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Important progress has been made with model averaging methods over the past decades. For spatial data, however, the idea of model averaging has not been applied well. This article studies model averaging methods for the spatial geostatistical linear model. A spatial Mallows criterion is developed to choose weights for the model averaging estimator. The resulting estimator can achieve asymptotic optimality in terms of L2 loss. Simulation experiments reveal that our proposed estimator is superior to the model averaging estimator by the Mallows criterion developed for ordinary linear models [Hansen, 2007] and the model selection estimator using the corrected Akaike's information criterion, developed for geostatistical linear models [Hoeting et al., 2006]. The Canadian Journal of Statistics 47: 336–351; 2019 © 2019 Statistical Society of Canada

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“…Denote the projection matrix

P_{m}^{*} = P_{m}^{F} {false|}_{\overset{θ}{false} = θ^{*}}

, the mean process

{\overset{μ}{false}}^{*} false(w false) = {\overset{μ}{false}}^{F} false(w false) {false|}_{\overset{θ}{false} = θ^{*}}

, the loss function

L_{n}^{*} false(w false) = L_{n}^{F} false(w false) {false|}_{\overset{γ}{false} = γ^{*}}

, the risk function

R_{n}^{*} false(w false) = E false(L_{n}^{*} false(w false) false)

, the SPMMA criterion

C_{n}^{*} false(w false) = C_{n}^{F} false(w false) {false|}_{\overset{γ}{false} = γ^{*}}

, the spatial autocorrelation matrix

V^{*} = \overset{V}{false} {false|}_{\overset{θ}{false} = θ^{*}}

, and the spatial covariance matrix

{normalΩ}^{*} = \overset{normalΩ}{false} {false|}_{\overset{γ}{false} = γ^{*}}

, where

\overset{γ}{false} = {false({\overset{σ}{false}}^{2}, \overset{θ}{false} false)}^{'}

Section: Asymptotic Optimalitysupporting

confidence: 76%

Section: Asymptotic Optimalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Spatial Mallows model averaging for geostatistical models

2019

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“…Our work has important relations to several research topics. First, our theory can be viewed as the Bayesian counterpart of the frequentist fixed-domain asymptotic theory on the maximum likelihood estimator in Ying [80], Ying [81], Zhang [82], Chen et al [14], Loh [46], Du et al [21], Wang and Loh [75], Kaufman and Shaby [38], Chang et al [12], Velandia et al [73], Bachoc et al [3], Bachoc and Lagnoux [4], etc. Second, our posterior asymptotic efficiency result is a counterpart of Stein's work in the Bayesian setup and guarantees the optimal estimation of prediction MSE.…”

Section: Our Contributionsmentioning

confidence: 99%

Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

Li¹

2020

Preprint

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Gaussian process models typically contain finite dimensional parameters in the covariance functions that need to be estimated from the data. We study the Bayesian fixed-domain asymptotic properties of the covariance parameters in a Gaussian process with an isotropic Matérn covariance function, which has many applications in spatial statistics. Under fixeddomain asymptotics, it is well known that when the dimension of data is less than or equal to three, the microergodic parameter can be consistently estimated with asymptotic normality while the variance parameter and the range (or length-scale) parameter cannot. Motivated by the frequentist theory, we prove a Bernstein-von Mises theorem for the covariance parameters in isotropic Matérn covariance functions. We show that under fixed-domain asymptotics, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors as the sample size goes to infinity. The posterior of the microergodic parameter converges in total variation norm to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not necessarily converge to any degenerate distribution in general. Our theory allows an unbounded prior support for the range parameter on the whole positive real line. Furthermore, we propose a new property called the posterior asymptotic efficiency in linear prediction, and show that the Bayesian kriging predictor at a new spatial location with covariance parameters randomly drawn from their posterior distribution has the same prediction mean squared error as if the true parameters were known. In the special case of one-dimensional Ornstein-Uhlenbeck process, we derive an explicit form for the limiting posterior distribution of the range parameter and an explicit posterior convergence rate for the posterior asymptotic efficiency in linear prediction. We verify these asymptotic results in numerical examples.

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“…For example, when (1.4) holds and p n ≥ 1 is a fixed integer, the convergence rate of θ obtained in Theorem 2.1 plays an indispensable role in analyzing the convergence rate of the ML estimator, β( θ), of β. Recently, by making use of Theorems 2.1 and 2.2, Chang, Huang and Ing [4] established the first model selection consistency result under the mixed domain asymptotic framework. Moreover, some technical results established in the proofs of Theorems 2.1 and 2.2 have been used by Chang, Huang and Ing [4] to develop a model selection consistency result under a misspecified covariance model.…”

Section: Central Limit Theorems and Rates Of Convergencementioning

confidence: 99%

Mixed domain asymptotics for a stochastic process model with time trend and measurement error

Chang¹,

Huang²,

Ing³

2017

Bernoulli

Self Cite

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We consider a stochastic process model with time trend and measurement error. We establish consistency and derive the limiting distributions of the maximum likelihood (ML) estimators of the covariance function parameters under a general asymptotic framework, including both the fixed domain and the increasing domain frameworks, even when the time trend model is misspecified or its complexity increases with the sample size. In particular, the convergence rates of the ML estimators are thoroughly characterized in terms of the growing rate of the domain and the degree of model misspecification/complexity.

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Asymptotic theory of generalized information criterion for geostatistical regression model selection

Cited by 10 publications

References 28 publications

Spatial Mallows model averaging for geostatistical models

Spatial Mallows model averaging for geostatistical models

Bayesian Fixed-domain Asymptotics: Bernstein-von Mises Theorem for Covariance Parameters in a Gaussian Process Model

Mixed domain asymptotics for a stochastic process model with time trend and measurement error

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