We introduce a new criterion to determine the order of an autoregressive model fitted to time series data. It has the benefits of the two well-known model selection techniques, the Akaike information criterion and the Bayesian information criterion. When the data is generated from a finite order autoregression, the Bayesian information criterion is known to be consistent, and so is the new criterion. When the true order is infinity or suitably high with respect to the sample size, the Akaike information criterion is known to be efficient in the sense that its prediction performance is asymptotically equivalent to the best offered by the candidate models; in this case, the new criterion behaves in a similar manner. Different from the two classical criteria, the proposed criterion adaptively achieves either consistency or efficiency depending on the underlying true model. In practice where the observed time series is given without any prior information about the model specification, the proposed order selection criterion is more flexible and robust compared with classical approaches. Numerical results are presented demonstrating the adaptivity of the proposed technique when applied to various datasets.
This paper derives a Minimum Message Length (MML) criterion for the model selection of the Autoregressive Moving Average (ARMA) time series model. The MML87 performances on the ARMA model compared with other well known model selection criteria, Akaike Information Criterion (AIC), Corrected AIC (AICc), Bayesian Information Criterion (BIC), and Hannan Quinn (HQ). The experimental results show that the MML87 is outperformed the other model selection criteria as it select most of the models with lower prediction errors and the models selected by MML87 to have a lower mean squared error in differen…
This paper studies the model selection problem in a large class of causal time series models, which includes both the ARMA or AR($\infty$) processes, as well as the GARCH or ARCH($\infty$), APARCH, ARMA-GARCH and many others processes. We first study the asymptotic behavior of the ideal penalty that minimizes the risk induced by a quasi-likelihood estimation among a finite family of models containing the true model. Then, we provide general conditions on the penalty term for obtaining the consistency and efficiency properties. We notably prove that consistent model selection criteria outperfor…
Variable selection is essential for improving inference and interpretation in multivariate linear regression. Although a number of alternative regressor selection criteria have been suggested, the most prominent and widely used are the Akaike information criterion (AIC), Bayesian information criterion (BIC), Mallow's $C_p$, and their modifications. However, for high-dimensional data, experience has shown that the performance of these classical criteria is not always satisfactory. In the present article, we begin by presenting the necessary and sufficient conditions (NSC) for the strong consist…
Centrum voor Wiskunde en Informatica (CWI)（国家数学与计算机科学研究所（CWI））
Bayesian model averaging, model selection and its approximations such as BIC are generally statistically consistent, but sometimes achieve slower rates og convergence than other methods such as AIC and leave-one-out cross-validation. On the other hand, these other methods can br inconsistent. We identify the "catch-up phenomenon" as a novel explanation for the slow convergence of Bayesian methods. Based on this analysis we define the switch distribution, a modification of the Bayesian marginal distribution. We show that, under broad conditions,model selection and prediction based on the switch…
The vector autoregressive (VAR) model is a powerful tool in modeling complex time series and has been exploited in many fields. However, fitting high dimensional VAR model poses some unique challenges: On one hand, the dimensionality, caused by modeling a large number of time series and higher order autoregressive processes, is usually much higher than the time series length; On the other hand, the temporal dependence structure in the VAR model gives rise to extra theoretical challenges. In high dimensions, one popular approach is to assume the transition matrix is sparse and fit the VAR model…
One of the main challenges in identifying structural changes in stochastic processes is to carry out analysis for time series with dependency structure in a computationally tractable way. Another challenge is that the number of true change points is usually unknown, requiring a suitable model selection criterion to arrive at informative conclusions. To address the first challenge, we model the data generating process as a segment-wise autoregression, which is composed of several segments (time epochs), each of which modeled by an autoregressive model. We propose a multi-window method that is b…
Information criteria are an appropriate and widely used tool for solving model selection problems. However, different ways to use them exist, each leading to a more or less precise approximation of the sought model. In this paper, we mainly present two methods of utilisation of information criteria : the classical one which is generally used and an alternative one, more precise but requiring a little more calculations. Those methods are compared on 1-D and 2-D autoregressive models ; we use a synthetized process for the 1-D case and texture images for the 2-D case. We also work with the origin…
The Misspecification-Resistant Information Criterion (MRIC) proposed in [H.-L. Hsu, C.-K. Ing, H. Tong: On model selection from a finite family of possibly misspecified time series models. The Annals of Statistics. 47 (2), 1061--1087 (2019)] is a model selection criterion for univariate parametric time series that enjoys both the property of consistency and asymptotic efficiency. In this article we extend the MRIC to the case where the response is a multivariate time series and the predictor is univariate. The extension requires novel derivations based upon random matrix theory. We obtain an a…
The paper obtains analytical results for the asymptotic properties of Model Selection Criteria -- widely used in practice -- for a general family of hidden Markov models (HMMs), thereby substantially extending the related theory beyond typical i.i.d.-like model structures and filling in an important gap in the relevant literature. In particular, we look at the Bayesian and Akaike Information Criteria (BIC and AIC) and the model evidence. In the setting of nested classes of models, we prove that BIC and the evidence are strongly consistent for HMMs (under regularity conditions), whereas AIC is …
This paper examines the limit properties of information criteria (such as AIC, BIC, HQIC) for distinguishing between the unit root model and the various kinds of explosive models. The explosive models include the local-to-unit-root model, the mildly explosive model and the regular explosive model. Initial conditions with different order of magnitude are considered. Both the OLS estimator and the indirect inference estimator are studied. It is found that BIC and HQIC, but not AIC, consistently select the unit root model when data come from the unit root model. When data come from the local-to-u…
In model selection literature, two classes of criteria perform well asymptotically in different situations: Bayesian information criterion (BIC) (as a representative) is consistent in selection when the true model is finite dimensional (parametric scenario); Akaike's information criterion (AIC) performs well in an asymptotic efficiency when the true model is infinite dimensional (nonparametric scenario). But there is little work that addresses if it is possible and how to detect the situation that a specific model selection problem is in. In this work, we differentiate the two scenarios theore…
We test three common information criteria (IC) for selecting the order of a Hawkes process with an intensity kernel that can be expressed as a mixture of exponential terms. These processes find application in high-frequency financial data modelling. The information criteria are Akaike's information criterion (AIC), the Bayesian information criterion (BIC) and the Hannan-Quinn criterion (HQ). Since we work with simulated data, we are able to measure the performance of model selection by the success rate of the IC in selecting the model that was used to generate the data. In particular, we are i…
We consider the problem of change-points estimation in the mean of an AR(p) process. Taking into account the dependence structure does not allow us to use the approach of the independent case. Especially, the dynamic programming algorithm giving the optimal solution in the independent case cannot be used anymore. We propose a two-step method, based on the preliminary robust (to the change-points) estimation of the autoregression parameters. Then, we propose to follow the classical approach, by plugging this estimator in the criterion used for change-point estimation, which is equivalent to dec…
In what follows we study non asymptotic behavior of different well known estimators AIC(\cite{Tong}), BIC(\cite{Schwarz}) and EDC(\cite{Zhao,Dorea}) in contrast with the Markov chain order estimator, named as Global Depency Level - GDL(\cite{Baigorri}). The estimator GDL, is based on a different principle which makes it behave in a quite different form. It is strongly consistent and more efficient than AIC(inconsistent), outperforming the well established and consistent BIC and EDC, mainly on relatively small samples. The estimators mentioned above mainly consist in the evaluation of the Marko…
A system identification approach to clustering vector autoregressive time series
School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, China（华中科技大学人工智能与自动化学院）；School of Electrical Engineering and Telecommunications, UNSW Sydney, NSW, Australia（新南威尔士大学悉尼分校电子与电信工程学院）
In this article we propose a general class of risk measures which can be used for data based evaluation of parametric models. The loss function is defined as generalized quadratic distance between the true density and the proposed model. These distances are characterized by a simple quadratic form structure that is adaptable through the choice of a nonnegative definite kernel and a bandwidth parameter. Using asymptotic results for the quadratic distances we build a quick-to-compute approximation for the risk function. Its derivation is analogous to the Akaike Information Criterion (AIC), but u…
In the realm of the model selection context, Akaike's and Schwarz's information criteria, AIC and BIC, have been applied successfully for decades for model order identification. The Efficient Determination Criterion (EDC) is a generalization of these criteria, proposed originally to define a strongly consistent class of estimators for the dependency order of a multiple Markov chain. In this work, the EDC is generalized to partially nested models, which encompass many other order identification problems. Based on some assumptions, a class of strongly consistent estimators is established in this…
In this paper we have introduced a generalized version of alpha beta skew normal distribution in the same line of Sharafi et al. (2017) and investigated some of its basic properties. The extensions of the proposed distribution have also been studied. The appropriateness of the proposed distribution has been tested by comparing the values of Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) with the values of some other known related distributions for better model selection. Likelihood ratio test has been used for discriminating between nested models.
This paper presents a new class of probability distributions generated from the gamma distribution. For the new class proposed, we present several statistical properties, such as the risk function, the density expansions, Moment-generating function, characteristic function, the moments of order m, central moments of order m, the log likelihood and its partial derivatives and also entropy, kurtosis, symmetry and variance. These same properties are determined for a particular distribution within this new class that is used to illustrate the capability of the proposed new class through an applica…
In this study by considering Balakrishnan mechanism a new form of alpha skew distribution is proposed and properties are investigated. The suitability of the proposed distribution has tested at the end with appropriate data fitting experiment and then comparing the values of Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) with the values of some other related distributions. Likelihood ratio test is carried out for nested models, that is, for Laplace and the proposed distributions. Keywords: Skew Distribution, Alpha-Skew Distribution, Bimodal Distribution, Alpha-Ske…
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 th…
Claeskens and Hjort (2003) constructed the focused information criterion (FIC) and developed frequentist model averaging methods using maximum likelihood estimators assuming the observations to be independent and identically distributed. Towards the immediate extensions and generalizations of these results, the present article is aimed at providing the focused model selection and model averaging methods using general maximum likelihood type estimators, popularly known as $M$-estimators. The necessary asymptotic theory is derived in a setup of stationary and strong mixing stochastic processes e…
Consider $n$ independent and identically distributed $p$-dimensional Gaussian random vectors with covariance matrix $Σ.$ The problem of estimating $Σ$ when $p$ is much larger than $n$ has received a lot of attention in recent years. Yet little is known about the information criterion for covariance matrix estimation. How to properly define such a criterion and what are the statistical properties? We attempt to answer these questions in the present paper by focusing on the estimation of bandable covariance matrices when $p>n$ but $\log(p)=o(n)$. Motivated by the deep connection between Stein's …
This article investigates unsupervised classification techniques for categorical multivariate data. The study employs multivariate multinomial mixture modeling, which is a type of model particularly applicable to multilocus genotypic data. A model selection procedure is used to simultaneously select the number of components and the relevant variables. A non-asymptotic oracle inequality is obtained, leading to the proposal of a new penalized maximum likelihood criterion. The selected model proves to be asymptotically consistent under weak assumptions on the true probability underlying the obser…
The problem of skewness is common among clinical trials and survival data which has being the research focus derivation and proposition of different flexible distributions. Thus, a new distribution called Extended Rayleigh Lomax distribution is constructed from Rayleigh Lomax distribution to capture the excessiveness of some survival data. We derive the new distribution by using beta logit function proposed by Jones (2004). Some statistical properties of the distribution such as probability density function, cumulative density function, reliability rate, hazard rate, reverse hazard rate, momen…
