Selecting the regularization parameters in high-dimensional panel data models: Consistency and efficiency

This article considers panel data models in the presence of a large number of potential predictors and unobservable common factors. The model is estimated by the regularization method together with the principal components procedure. We propose a panel information criterion for selecting the regularization parameter and the number of common factors under a diverging number of predictors. Under the correct model specification, we show that the proposed criterion consistently identifies the true model. If the model is instead misspecified, the proposed criterion achieves asymptotically efficient model selection. Simulation results confirm these theoretical arguments.     

Efficient Regularization Parameter Selection Via Information Criteria

In a nonlinear regression model based on a regularization method, selection of appropriate regularization parameters is crucial. Information criteria such as generalized information criterion (GIC) and generalized Bayesian information criterion (GBIC) are useful for selecting the optimal regularization parameters. However, the optimal parameter is often determined by calculating information criterion for all candidate regularization parameters, and so the computational cost is high. One simple method by which to accomplish this is to regard GIC or GBIC as a function of the regularization parameters and to find a value minimizing GIC or GBIC. However, it is unclear how to solve the optimization problem. In the present article, we propose an efficient Newton–Raphson type iterative method for selecting optimal regularization parameters with respect to GIC or GBIC in a nonlinear regression model based on basis expansions. This method reduces the computational time remarkably compared to the grid search and can select more suitable regularization parameters. The effectiveness of the method is illustrated through real data examples.     

Some computational aspects of the generalized von Mises distribution

This article deals with some important computational aspects of the generalized von Mises distribution in relation with parameter estimation, model selection and simulation. The generalized von Mises distribution provides a flexible model for circular data allowing for symmetry, asymmetry, unimodality and bimodality. For this model, we show the equivalence between the trigonometric method of moments and the maximum likelihood estimators, we give their asymptotic distribution, we provide bias-corrected estimators of the entropy, the Akaike information criterion and the measured entropy for model selection, and we implement the ratio-of-uniforms method of simulation.     

Variable selection for model-based clustering using the integrated complete-data likelihood

Variable selection in cluster analysis is important yet challenging. It can be achieved by regularization methods, which realize a trade-off between the clustering accuracy and the number of selected variables by using a lasso-type penalty. However, the calibration of the penalty term can suffer from criticisms. Model selection methods are an efficient alternative, yet they require a difficult optimization of an information criterion which involves combinatorial problems. First, most of these optimization algorithms are based on a suboptimal procedure (e.g. stepwise method). Second, the algorithms are often computationally expensive because they need multiple calls of EM algorithms. Here we propose to use a new information criterion based on the integrated complete-data likelihood. It does not require the maximum likelihood estimate and its maximization appears to be simple and computationally efficient. The original contribution of our approach is to perform the model selection without requiring any parameter estimation. Then, parameter inference is needed only for the unique selected model. This approach is used for the variable selection of a Gaussian mixture model with conditional independence assumed. The numerical experiments on simulated and benchmark datasets show that the proposed method often outperforms two classical approaches for variable selection. The proposed approach is implemented in the R package VarSelLCM available on CRAN.     

Tuning Parameter Selection in Cox Proportional Hazards Model with a Diverging Number of Parameters

Regularized variable selection is a powerful tool for identifying the true regression model from a large number of candidates by applying penalties to the objective functions. The penalty functions typically involve a tuning parameter that controls the complexity of the selected model. The ability of the regularized variable selection methods to identify the true model critically depends on the correct choice of the tuning parameter. In this study, we develop a consistent tuning parameter selection method for regularized Cox's proportional hazards model with a diverging number of parameters. The tuning parameter is selected by minimizing the generalized information criterion. We prove that, for any penalty that possesses the oracle property, the proposed tuning parameter selection method identifies the true model with probability approaching one as sample size increases. Its finite sample performance is evaluated by simulations. Its practical use is demonstrated in The Cancer Genome Atlas breast cancer data.     

Adaptive order selection for autoregressive models

Autoregressive model is a popular method for analysing the time dependent data, where selection of order parameter is imperative. Two commonly used selection criteria are the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), which are known to suffer the potential problems regarding overfit and underfit, respectively. To our knowledge, there does not exist a criterion in the literature that can satisfactorily perform under various situations. Therefore, in this paper, we focus on forecasting the future values of an observed time series and propose an adaptive idea to combine the advantages of AIC and BIC but to mitigate their weaknesses based on the concept of generalized degrees of freedom. Instead of applying a fixed criterion to select the order parameter, we propose an approximately unbiased estimator of mean squared prediction errors based on a data perturbation technique for fairly comparing between AIC and BIC. Then use the selected criterion to determine the final order parameter. Some numerical experiments are performed to show the superiority of the proposed method and a real data set of the retail price index of China from 1952 to 2008 is also applied for illustration.     

Model Selection,Transformations and Variance Estimation in Nonlinear Regression

The results of analyzing experimental data using a parametric model may heavily depend on the chosen model for regression and variance functions, moreover also on a possibly underlying preliminary transformation of the variables. In this paper we propose and discuss a complex procedure which consists in a simultaneous selection of parametric regression and variance models from a relatively rich model class and of Box-Cox variable transformations by minimization of a cross-validation criterion. For this it is essential to introduce modifications of the standard cross-validation criterion adapted to each of the following objectives: 1. estimation of the unknown regression function, 2. prediction of future values of the response variable, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Our idea of a criterion oriented combination of procedures (which usually if applied, then in an independent or sequential way) is expected to lead to more accurate results. We show how the accuracy of the parameter estimators can be assessed by a “moment oriented bootstrap procedure", which is an essential modification of the “wild bootstrap” of Härdle and Mammen by use of more accurate variance estimates. This new procedure and its refinement by a bootstrap based pivot (“double bootstrap”) is also used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated, e.g., by an application in radioimmunological assay.     

Shrinkage tuning parameter selection in precision matrices estimation

Recent literature provides many computational and modeling approaches for covariance matrices estimation in a penalized Gaussian graphical models but relatively little study has been carried out on the choice of the tuning parameter. This paper tries to fill this gap by focusing on the problem of shrinkage parameter selection when estimating sparse precision matrices using the penalized likelihood approach. Previous approaches typically used K-fold cross-validation in this regard. In this paper, we first derived the generalized approximate cross-validation for tuning parameter selection which is not only a more computationally efficient alternative, but also achieves smaller error rate for model fitting compared to leave-one-out cross-validation. For consistency in the selection of nonzero entries in the precision matrix, we employ a Bayesian information criterion which provably can identify the nonzero conditional correlations in the Gaussian model. Our simulations demonstrate the general superiority of the two proposed selectors in comparison with leave-one-out cross-validation, 10-fold cross-validation and Akaike information criterion.     

The Mean Squared Error Optimum Design Criterion for Parameter Estimation in Nonlinear Regression Models

In the context of nonlinear regression models, we propose an optimal experimental design criterion for estimating the parameters that account for the intrinsic and parameter-effects nonlinearity. The optimal design criterion proposed in this article minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated using the curvature array. The design criterion reduces to the D-optimal design criterion if there are no intrinsic and parameter-effects nonlinearity in the model, and depends on the scale parameter estimator and on the reparameterization used. Some examples, using a well known nonlinear kinetics model, demonstrate the application of the proposed criterion to nonsequential design of experiments as compared with the D-optimal criterion.     

Statistical analysis of controlled calibration model with replicates

In this work, we generalize the controlled calibration model by assuming replication on both variables. Likelihood-based methodology is used to estimate the model parameters and the Fisher information matrix is used to construct confidence intervals for the unknown value of the regressor variable. Further, we study the local influence diagnostic method which is based on the conditional expectation of the complete-data log-likelihood function related to the EM algorithm. Some useful perturbation schemes are discussed. A simulation study is carried out to assess the effect of the measurement error on the estimation of the parameter of interest. This new approach is illustrated with a real data set.     

Penalized semiparametric density estimation

In this article we propose a penalized likelihood approach for the semiparametric density model with parametric and nonparametric components. An efficient iterative procedure is proposed for estimation. Approximate generalized maximum likelihood criterion from Bayesian point of view is derived for selecting the smoothing parameter. The finite sample performance of the proposed estimation approach is evaluated through simulation. Two real data examples, suicide study data and Old Faithful geyser data, are analyzed to demonstrate use of the proposed method.     

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.     

Focused Information Criterion and Model Averaging in Quantile Regression

In this article, we study model selection and model averaging in quantile regression. Under general conditions, we develop a focused information criterion and a frequentist model average estimator for the parameters in quantile regression model, and examine their theoretical properties. The new procedures provide a robust alternative to the least squares method or likelihood method, and a major advantage of the proposed procedures is that when the variance of random error is infinite, the proposed procedure works beautifully while the least squares method breaks down. A simulation study and a real data example are presented to show that the proposed method performs well with a finite sample and is easy to use in practice.     

Identifiability and Comparison of Estimation Methods on Weibull Mixture Models

In this article, the finite mixture model of Weibull distributions is studied, the identifiability of the model with m components is proven, and the parameter estimators for the case of two components resulted by several algorithms are compared. The parameter estimators are obtained with maximum likelihood performing calculations with different algorithms: expectation-maximization (EM), Fisher scoring, backfitting, optimization of k-nearest neighbor approach, and random walk algorithm using Monte Carlo simulation. The Akaike information criterion and the log-likelihood value are used to compare models. In general, the proposed random walk algorithm shows better performance in mean square error and bias. Finally, the results are applied to electronic component lifetime data.     

A non-iterative optimization method for smoothness in penalized spline regression

Typically, an optimal smoothing parameter in a penalized spline regression is determined by minimizing an information criterion, such as one of the C _p , CV and GCV criteria. Since an explicit solution to the minimization problem for an information criterion cannot be obtained, it is necessary to carry out an iterative procedure to search for the optimal smoothing parameter. In order to avoid such extra calculation, a non-iterative optimization method for smoothness in penalized spline regression is proposed using the formulation of generalized ridge regression. By conducting numerical simulations, we verify that our method has better performance than other methods which optimize the number of basis functions and the single smoothing parameter by means of the CV or GCV criteria.     

The Calibration for Two-Phase Randomized Response Estimator

This article presents the calibration procedure of the two-phase randomized response (RR) technique for surveying the sensitive characteristic. When the sampling scheme is two-phase or double sampling, auxiliary information known from the entire population can be used, but the auxiliary information should be information available from both the first and second phases of the sample. If there is auxiliary information available from both the first and second phases, then we can improve the ordinary two-phase RR estimator by incorporating this information in the estimation procedure. In this article, we used the new two-step Newton's method for computing unknown constants in the calibration procedure and compared the efficiency of the proposed estimator through some numerical study.     

Focused information criteria for copulas

In this paper, we extend the focused information criterion (FIC) to copula models. Copulas are often used for applications where the joint tail behavior of the variables is of particular interest, and selecting a copula that captures this well is then essential. Traditional model selection methods such as the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) aim at finding the overall best‐fitting model, which is not necessarily the one best suited for the application at hand. The FIC, on the other hand, evaluates and ranks candidate models based on the precision of their point estimates of a context‐given focus parameter. This could be any quantity of particular interest, for example, the mean, a correlation, conditional probabilities, or measures of tail dependence. We derive FIC formulae for the maximum likelihood estimator, the two‐stage maximum likelihood estimator, and the so‐called pseudo‐maximum‐likelihood (PML) estimator combined with parametric margins. Furthermore, we confirm the validity of the AIC formula for the PML estimator combined with parametric margins. To study the numerical behavior of FIC, we have carried out a simulation study, and we have also analyzed a multivariate data set pertaining to abalones. The results from the study show that the FIC successfully ranks candidate models in terms of their performance, defined as how well they estimate the focus parameter. In terms of estimation precision, FIC clearly outperforms AIC, especially when the focus parameter relates to only a specific part of the model, such as the conditional upper‐tail probability.     

On Variational Bayes Estimation and Variational Information Criteria for Linear Regression Models

Variational Bayes (VB) estimation is a fast alternative to Markov Chain Monte Carlo for performing approximate Baesian inference. This procedure can be an efficient and effective means of analyzing large datasets. However, VB estimation is often criticised, typically on empirical grounds, for being unable to produce valid statistical inferences. In this article we refute this criticism for one of the simplest models where Bayesian inference is not analytically tractable, that is, the Bayesian linear model (for a particular choice of priors). We prove that under mild regularity conditions, VB based estimators enjoy some desirable frequentist properties such as consistency and can be used to obtain asymptotically valid standard errors. In addition to these results we introduce two VB information criteria: the variational Akaike information criterion and the variational Bayesian information criterion. We show that variational Akaike information criterion is asymptotically equivalent to the frequentist Akaike information criterion and that the variational Bayesian information criterion is first order equivalent to the Bayesian information criterion in linear regression. These results motivate the potential use of the variational information criteria for more complex models. We support our theoretical results with numerical examples.     

Analysing nonlinear time series with central subspace

Traditionally, time series analysis involves building an appropriate model and using either parametric or nonparametric methods to make inference about the model parameters. Motivated by recent developments for dimension reduction in time series, an empirical application of sufficient dimension reduction (SDR) to nonlinear time series modelling is shown in this article. Here, we use time series central subspace as a tool for SDR and estimate it using mutual information index. Especially, in order to reduce the computational complexity in time series, we propose an efficient estimation method of minimal dimension and lag using a modified Schwarz–Bayesian criterion, when either of the dimensions and the lags is unknown. Through simulations and real data analysis, the approach presented in this article performs well in autoregression and volatility estimation.     

Penalized calibration in survey sampling: Design-based estimation assisted by mixed models

Calibration techniques in survey sampling, such as generalized regression estimation (GREG), were formalized in the 1990s to produce efficient estimators of linear combinations of study variables, such as totals or means. They implicitly lie on the assumption of a linear regression model between the variable of interest and some auxiliary variables in order to yield estimates with lower variance if the model is true and remaining approximately design-unbiased even if the model does not hold. We propose a new class of model-assisted estimators obtained by releasing a few calibration constraints and replacing them with a penalty term. This penalization is added to the distance criterion to minimize. By introducing the concept of penalized calibration, combining usual calibration and this ‘relaxed’ calibration, we are able to adjust the weight given to the available auxiliary information. We obtain a more flexible estimation procedure giving better estimates particularly when the auxiliary information is overly abundant or not fully appropriate to be completely used. Such an approach can also be seen as a design-based alternative to the estimation procedures based on the more general class of mixed models, presenting new prospects in some scopes of application such as inference on small domains.