A comparison of model selection criteria

There has been significant new work published recently on the subject of model selection. Notably Rissanen (1986, 1987, 1988) has introduced new criteria based on the notion of stochastic complexity and Hurvich and Tsai(1989) have introduced a bias corrected version of Akaike's information criterion. In this paper, a Monte Carlo study is conducted to evaluate the relative performance of these new model selection criteria against the commonly used alternatives. In addition, we compare the performance of all the criteria in a number of situations not considered in earlier studies: robustness to distributional assumptions, collinearity among regressors, and non-stationarity in a time series. The evaluation is based on the number of times the correct model is chosen and the out of sample prediction error. The results of this study suggest that Rissanen's criteria are sensitive to the assumptions and choices that need to made in their application, and so are sometimes unreliable. While many of the criteria often perform satisfactorily, across experiments the Schwartz Bayesian Information Criterion (and the related Bayesian Estimation Criterion of Geweke-Meese) seem to consistently outperfom the other alternatives considered.     

Adaptive Lasso Variable Selection for the Accelerated Failure Models

This article considers the adaptive lasso procedure for the accelerated failure time model with multiple covariates based on weighted least squares method, which uses Kaplan-Meier weights to account for censoring. The adaptive lasso method can complete the variable selection and model estimation simultaneously. Under some mild conditions, the estimator is shown to have sparse and oracle properties. We use Bayesian Information Criterion (BIC) for tuning parameter selection, and a bootstrap variance approach for standard error. Simulation studies and two real data examples are carried out to investigate the performance of the proposed method.     

非正态分布下具有自回归误差项的空间自回归模型变量选择研究

将变量选择引入空间计量模型,讨论具有自回归误差项的空间自回归模型的变量选择问题。在残差非正态独立同分布的条件下,通过最大化信息熵,提出空间信息准则,并证明其在该模型变量选择中具有一致性。模拟研究结果表明:无论对单个系数还是对全部系数,空间信息准则都能很好识别,且与经典的赤池准则相比具有较大的优势。因此,空间信息准则是一种更为有效的变量选择方法。     

Bayesian variable selection for the Cox regression model with missing covariates

In this paper, we develop Bayesian methodology and computational algorithms for variable subset selection in Cox proportional hazards models with missing covariate data. A new joint semi-conjugate prior for the piecewise exponential model is proposed in the presence of missing covariates and its properties are examined. The covariates are assumed to be missing at random (MAR). Under this new prior, a version of the Deviance Information Criterion (DIC) is proposed for Bayesian variable subset selection in the presence of missing covariates. Monte Carlo methods are developed for computing the DICs for all possible subset models in the model space. A Bone Marrow Transplant (BMT) dataset is used to illustrate the proposed methodology.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. The likelihood function is utilized to compute not only some Bayesian model selection measures but also to develop Bayesian case-deletion influence diagnostics based on the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

Consistent Bayesian information criterion based on a mixture prior for possibly high-dimensional multivariate linear regression models

In the problem of selecting variables in a multivariate linear regression model, we derive new Bayesian information criteria based on a prior mixing a smooth distribution and a delta distribution. Each of them can be interpreted as a fusion of the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Inheriting their asymptotic properties, our information criteria are consistent in variable selection in both the large-sample and the high-dimensional asymptotic frameworks. In numerical simulations, variable selection methods based on our information criteria choose the true set of variables with high probability in most cases.     

Autoregressive model selection based on a prediction perspective

The autoregressive (AR) model is a popular method for fitting and prediction in analyzing time-dependent data, where selecting an accurate model among considered orders is a crucial issue. Two commonly used selection criteria are the Akaike information criterion and the Bayesian information criterion. However, the two criteria are known to suffer potential problems regarding overfit and underfit, respectively. Therefore, using them would perform well in some situations, but poorly in others. In this paper, we propose a new criterion in terms of the prediction perspective based on the concept of generalized degrees of freedom for AR model selection. We derive an approximately unbiased estimator of mean-squared prediction errors based on a data perturbation technique for selecting the order parameter, where the estimation uncertainty involved in a modeling procedure is considered. Some numerical experiments are performed to illustrate the superiority of the proposed method over some commonly used order selection criteria. Finally, the methodology is applied to a real data example to predict the weekly rate of return on the stock price of Taiwan Semiconductor Manufacturing Company and the results indicate that the proposed method is satisfactory.     

The FGM Long-Term Bivariate Survival Copula Model: Modeling,Bayesian Estimation,and Case Influence Diagnostics

In this article, we propose a bivariate long-term distribution based on the Farlie-Gumbel-Morgenstern copula model. The proposed model allows for the presence of censored data and covariates. For inferential purposes, a Bayesian approach via Markov Chain Monte Carlo (MCMC) were considered. Further, some discussions on the model selection criteria are given. In order to examine outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback-Leibler divergence. The newly developed procedures are illustrated on artificial and real data.     

Information criteria for inhomogeneous spatial point processes

The theoretical foundation for a number of model selection criteria is established in the context of inhomogeneous point processes and under various asymptotic settings: infill, increasing domain and combinations of these. For inhomogeneous Poisson processes we consider Akaike's information criterion and the Bayesian information criterion, and in particular we identify the point process analogue of ‘sample size’ needed for the Bayesian information criterion. Considering general inhomogeneous point processes we derive new composite likelihood and composite Bayesian information criteria for selecting a regression model for the intensity function. The proposed model selection criteria are evaluated using simulations of Poisson processes and cluster point processes.     

Model-based classification using latent Gaussian mixture models

A novel model-based classification technique is introduced based on parsimonious Gaussian mixture models (PGMMs). PGMMs, which were introduced recently as a model-based clustering technique, arise from a generalization of the mixtures of factor analyzers model and are based on a latent Gaussian mixture model. In this paper, this mixture modelling structure is used for model-based classification and the particular area of application is food authenticity. Model-based classification is performed by jointly modelling data with known and unknown group memberships within a likelihood framework and then estimating parameters, including the unknown group memberships, within an alternating expectation-conditional maximization framework. Model selection is carried out using the Bayesian information criteria and the quality of the maximum a posteriori classifications is summarized using the misclassification rate and the adjusted Rand index. This new model-based classification technique gives excellent classification performance when applied to real food authenticity data on the chemical properties of olive oils from nine areas of Italy.     

Bayesian zero-inflated generalized Poisson regression model: estimation and case influence diagnostics

Count data with excess zeros arises in many contexts. Here our concern is to develop a Bayesian analysis for the zero-inflated generalized Poisson (ZIGP) regression model to address this problem. This model provides a useful generalization of zero-inflated Poisson model since the generalized Poisson distribution is overdispersed/underdispersed relative to Poisson. Due to the complexity of the ZIGP model, Markov chain Monte Carlo methods are used to develop a Bayesian procedure for the considered model. Additionally, some discussions on the model selection criteria are presented and a Bayesian case deletion influence diagnostics is investigated for the joint posterior distribution based on the Kullback–Leibler divergence. Finally, a simulation study and a psychological example are given to illustrate our methodology.     

Nonlinear Logistic Discrimination Via Regularized Gaussian Basis Expansions

We consider the problem of constructing multi-class classification methods for analyzing data with complex structure. A nonlinear logistic discriminant model is introduced based on Gaussian basis functions constructed by the self-organizing map. In order to select adjusted parameters, we employ model selection criteria derived from information-theoretic and Bayesian approaches. Numerical examples are conducted to investigate the performance of the proposed multi-class discriminant procedure. Our modeling procedure is also applied to protein structure recognition in life science. The results indicate the effectiveness of our strategy in terms of prediction accuracy.     

Regularized Bayesian quantile regression

A number of nonstationary models have been developed to estimate extreme events as function of covariates. A quantile regression (QR) model is a statistical approach intended to estimate and conduct inference about the conditional quantile functions. In this article, we focus on the simultaneous variable selection and parameter estimation through penalized quantile regression. We conducted a comparison of regularized Quantile Regression model with B-Splines in Bayesian framework. Regularization is based on penalty and aims to favor parsimonious model, especially in the case of large dimension space. The prior distributions related to the penalties are detailed. Five penalties (Lasso, Ridge, SCAD0, SCAD1 and SCAD2) are considered with their equivalent expressions in Bayesian framework. The regularized quantile estimates are then compared to the maximum likelihood estimates with respect to the sample size. A Markov Chain Monte Carlo (MCMC) algorithms are developed for each hierarchical model to simulate the conditional posterior distribution of the quantiles. Results indicate that the SCAD0 and Lasso have the best performance for quantile estimation according to Relative Mean Biais (RMB) and the Relative Mean-Error (RME) criteria, especially in the case of heavy distributed errors. A case study of the annual maximum precipitation at Charlo, Eastern Canada, with the Pacific North Atlantic climate index as covariate is presented.     

Selection among open population capture-recapture models when capture probabilities are heterogeneous

Selection of a parsimonious model as a basis for statistical inference from capture-recapture data is critical, especially when using open models in the analysis of multiple, interrelated data sets (e.g. males and females, with two to three age classes, over three to five areas and 10-15 years). The global (i.e. most general) model for such data sets might contain hundreds of survival and recapture parameters. Here, we focus on a series of nested models of the Cormack-Jolly-Seber type wherein the likelihood arises from products of multinomial distributions whose cell probabilities are reparameterized in terms of survival ( phi ) and mean capture ( p ) probabilities. This paper presents numerical results on two information-theoretic methods for model selection when the capture probabilities are heterogeneous over individual animals: Akaike's Information Criterion (AIC) and a dimension-consistent criterion (CAIC), derived from a Bayesian viewpoint. Quality of model selection was evaluated based on the relative Euclidian distance between standardized theta and theta (parameter theta is vector-valued and contains the survival ( phi ) and mean capture ( p ) probabilities); this quantity (RSS = sigma{(theta i - theta i )/ theta i } 2 ) is a sum of squared bias and variance. Thus, the quality of inference (RSS) was judged by comparing the performance of the two information criteria and the use of the true model (used to generate the data), in relation to the model that provided the smallest RSS. We found that heterogeneity in the capture probabilities had a negligible effect on model selection using AIC or CAIC. Model size increased as sample size increased with both AIC- and CAIC-selected models.     

Empirical information criteria for time series forecasting model selection

In this article, we propose a new empirical information criterion (EIC) for model selection which penalizes the likelihood of the data by a non-linear function of the number of parameters in the model. It is designed to be used where there are a large number of time series to be forecast. However, a bootstrap version of the EIC can be used where there is a single time series to be forecast. The EIC provides a data-driven model selection tool that can be tuned to the particular forecasting task.

We compare the EIC with other model selection criteria including Akaike’s information criterion (AIC) and Schwarz’s Bayesian information criterion (BIC). The comparisons show that for the M3 forecasting competition data, the EIC outperforms both the AIC and BIC, particularly for longer forecast horizons. We also compare the criteria on simulated data and find that the EIC does better than existing criteria in that case also.     

A Bayesian criterion for selecting super saturated screening designs

Super-saturated designs in which the number of factors under investigation exceeds the number of experimental runs have been suggested for screening experiments initiated to identify important factors for future study. Most of the designs suggested in the literature are based on natural but ad hoc criteria. The “average s²” criteria introduced by Booth and Cox (Technometrics 4 (1962) 489) is a popular choice. Here, a decision theoretic approach is pursued leading to an optimality criterion based on misclassification probabilities in a Bayesian model. In certain cases, designs optimal under the average s² criterion are also optimal for the new criterion. Necessary conditions for this to occur are presented. In addition, the new criterion often provides a strict preference between designs tied under the average s² criterion, which is advantageous in numerical search as it reduces the number of local minima.     

Exact posterior distributions and model selection criteria for multiple change-point detection problems

In segmentation problems, inference on change-point position and model selection are two difficult issues due to the discrete nature of change-points. In a Bayesian context, we derive exact, explicit and tractable formulae for the posterior distribution of variables such as the number of change-points or their positions. We also demonstrate that several classical Bayesian model selection criteria can be computed exactly. All these results are based on an efficient strategy to explore the whole segmentation space, which is very large. We illustrate our methodology on both simulated data and a comparative genomic hybridization profile.     

Stochastic volatility in mean models with scale mixtures of normal distributions and correlated errors: A Bayesian approach

A stochastic volatility in mean model with correlated errors using the symmetrical class of scale mixtures of normal distributions is introduced in this article. The scale mixture of normal distributions is an attractive class of symmetric distributions that includes the normal, Student-t, slash and contaminated normal distributions as special cases, providing a robust alternative to estimation in stochastic volatility in mean models in the absence of normality. Using a Bayesian paradigm, an efficient method based on Markov chain Monte Carlo (MCMC) is developed for parameter estimation. The methods developed are applied to analyze daily stock return data from the São Paulo Stock, Mercantile & Futures Exchange index (IBOVESPA). The Bayesian predictive information criteria (BPIC) and the logarithm of the marginal likelihood are used as model selection criteria. The results reveal that the stochastic volatility in mean model with correlated errors and slash distribution provides a significant improvement in model fit for the IBOVESPA data over the usual normal model.     

Model selection: A Lagrange optimization approach

This paper proposes an adaptive model selection criterion with a data-driven penalty term. We treat model selection as an equality constrained minimization problem and develop an adaptive model selection procedure based on the Lagrange optimization method. In contrast to Akaike's information criterion (AIC), Bayesian information criterion (BIC) and most other existing criteria, this new criterion is to minimize the model size and take a measure of lack-of-fit as an adaptive penalty. Both theoretical results and simulations illustrate the power of this criterion with respect to consistency and pointwise asymptotic loss efficiency in the parametric and nonparametric cases.     

Probit and Logit Model Selection

Monte Carlo experiments are conducted to compare the Bayesian and sample theory model selection criteria in choosing the univariate probit and logit models. We use five criteria: the deviance information criterion (DIC), predictive deviance information criterion (PDIC), Akaike information criterion (AIC), weighted, and unweighted sums of squared errors. The first two criteria are Bayesian while the others are sample theory criteria. The results show that if data are balanced none of the model selection criteria considered in this article can distinguish the probit and logit models. If data are unbalanced and the sample size is large the DIC and AIC choose the correct models better than the other criteria. We show that if unbalanced binary data are generated by a leptokurtic distribution the logit model is preferred over the probit model. The probit model is preferred if unbalanced data are generated by a platykurtic distribution. We apply the model selection criteria to the probit and logit models that link the ups and downs of the returns on S&P500 to the crude oil price.