Influential subsets on the variable selection

When one or few observations are deleted in the multiple linear regression model, they can affect the variable selection. In this paper we derived the formula for the Mallows C_p criterion when k observations are deleted and express it as a functionn of basic building blocks such as residuals and leverages. Also, two real date sets are used to see how the selected model changes as few observations re deleted.     

Weighted-Average Least Squares Prediction

Prediction under model uncertainty is an important and difficult issue. Traditional prediction methods (such as pretesting) are based on model selection followed by prediction in the selected model, but the reported prediction and the reported prediction variance ignore the uncertainty from the selection procedure. This article proposes a weighted-average least squares (WALS) prediction procedure that is not conditional on the selected model. Taking both model and error uncertainty into account, we also propose an appropriate estimate of the variance of the WALS predictor. Correlations among the random errors are explicitly allowed. Compared to other prediction averaging methods, the WALS predictor has important advantages both theoretically and computationally. Simulation studies show that the WALS predictor generally produces lower mean squared prediction errors than its competitors, and that the proposed estimator for the prediction variance performs particularly well when model uncertainty increases.     

Interval Estimation by Frequentist Model Averaging

An important contribution to the literature on frequentist model averaging (FMA) is the work of Hjort and Claeskens (2003 Hjort , N. L. , Claeskens , G. ( 2003 ). Frequestist model average estimators . J. Amer. Statist. Assoc. 98 : 879 – 899 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), who developed an asymptotic theory for frequentist model averaging in parametric models based on a local mis-specification framework. They also proposed a simple method for constructing confidence intervals of the unknown parameters. This article shows that the confidence intervals based on the FMA estimator suggested by Hjort and Claeskens (2003 Hjort , N. L. , Claeskens , G. ( 2003 ). Frequestist model average estimators . J. Amer. Statist. Assoc. 98 : 879 – 899 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) are asymptotically equivalent to that obtained from the full model under both parametric and the varying-coefficient partially linear models. Thus, as long as interval estimation rather than point estimation is concerned, the confidence interval based on the full model already fulfills the objective and model averaging provides no additional useful information.     

Multiple Regression Model Averaging and the Focused Information Criterion With an Application to Portfolio Choice

ABSTRACT

We consider multiple regression (MR) model averaging using the focused information criterion (FIC). Our approach is motivated by the problem of implementing a mean-variance portfolio choice rule. The usual approach is to estimate parameters ignoring the intention to use them in portfolio choice. We develop an estimation method that focuses on the trading rule of interest. Asymptotic distributions of submodel estimators in the MR case are derived using a localization framework. The localization is of both regression coefficients and error covariances. Distributions of submodel estimators are used for model selection with the FIC. This allows comparison of submodels using the risk of portfolio rule estimators. FIC model averaging estimators are then characterized. This extension further improves risk properties. We show in simulations that applying these methods in the portfolio choice case results in improved estimates compared with several competitors. An application to futures data shows superior performance as well.     

Bayesian Model Averaging in Proportional Hazard Models: Assessing the Risk of a Stroke

In the context of the Cardiovascular Health Study, a comprehensive investigation into the risk factors for strokes, we apply Bayesian model averaging to the selection of variables in Cox proportional hazard models. We use an extension of the leaps-and-bounds algorithm for locating the models that are to be averaged over and make available S-PLUS software to implement the methods. Bayesian model averaging provides a posterior probability that each variable belongs in the model, a more directly interpretable measure of variable importance than a P -value. P -values from models preferred by stepwise methods tend to overstate the evidence for the predictive value of a variable and do not account for model uncertainty. We introduce the partial predictive score to evaluate predictive performance. For the Cardiovascular Health Study, Bayesian model averaging predictively outperforms standard model selection and does a better job of assessing who is at high risk for a stroke.     

Bandwidth selection for kernel regression with correlated errors

In this paper, we propose bandwidth selectors for nonparametric regression with dependent errors. The methods are based on criteria that approximate the average squared error. We show that these approximations are uniform over the bandwidth sequence. The criteria involve some constants that depend on the unknown error correlations. We propose a novel way of estimating these constants. Our numerical study shows that the method is quite efficient in a variety of error models.     

A modified Cp statistic in a system-of-equations model

A new statistic, SΓ(p), is developed for variable selection in a system-of-equations model. The standardized total mean square error in the SΓ(p)statistic is weighted by the covariance matrix of dependent variables instead of the error covariance matrix of the true model as in the original definition. The new statistic can be also used for model selection in the non-nested models. The estimate of SΓ(p), SC(p), is derived and shown to become SC_ε(p) in the similar form of C_p in a single-equation model when the covariance matrix of sampled dependent variables is replaced by the error covariance matrix under the full model.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

A MORE GENERAL CRITERION FOR SUBSET SELECTION IN MULTIPLE LINEAR REGRESSION

ABSTRACT

In this article, we propose a more general criterion called S_p -criterion, for subset selection in the multiple linear regression Model. Many subset selection methods are based on the Least Squares (LS) estimator of β, but whenever the data contain an influential observation or the distribution of the error variable deviates from normality, the LS estimator performs ‘poorly’ and hence a method based on this estimator (for example, Mallows’ C_p -criterion) tends to select a ‘wrong’ subset. The proposed method overcomes this drawback and its main feature is that it can be used with any type of estimator (either the LS estimator or any robust estimator) of β without any need for modification of the proposed criterion. Moreover, this technique is operationally simple to implement as compared to other existing criteria. The method is illustrated with examples.     

A Statistical Model for Social Network Labeling

We consider a social network from which one observes not only network structure (i.e., nodes and edges) but also a set of labels (or tags, keywords) for each node (or user). These labels are self-created and closely related to the user’s career status, life style, personal interests, and many others. Thus, they are of great interest for online marketing. To model their joint behavior with network structure, a complete data model is developed. The model is based on the classical p₁ model but allows the reciprocation parameter to be label-dependent. By focusing on connected pairs only, the complete data model can be generalized into a conditional model. Compared with the complete data model, the conditional model specifies only the conditional likelihood for the connected pairs. As a result, it suffers less risk from model misspecification. Furthermore, because the conditional model involves connected pairs only, the computational cost is much lower. The resulting estimator is consistent and asymptotically normal. Depending on the network sparsity level, the convergence rate could be different. To demonstrate its finite sample performance, numerical studies (based on both simulated and real datasets) are presented.     

Goodness of fit for thei ordered categories discrete uniform distribution

Goodness of fit for thei ordered categories discrete uniform distribution can be carried out using Pearson's X² _pstatistic and its components. Applications of this technique are considered and comparisons made with recently suggested empirical uniform distribution     

Resistant selection of the smoothing parameter for smoothing splines

Robust automatic selection techniques for the smoothing parameter of a smoothing spline are introduced. They are based on a robust predictive error criterion and can be viewed as robust versions of C _p and cross-validation. They lead to smoothing splines which are stable and reliable in terms of mean squared error over a large spectrum of model distributions.     

Statistical analysis of process capability indices with measurement errors: The case ofC p

Process capability indices (PCIs) have been widely used in manufacturing industries to previde a quantitative measure of process potential and performance. While some efforts have been dedicated in the literature to the statistical properties of PCIs estimators, scarce attention has been given to the evaluation of these properties when sample data are affected by measurement errors. In this work we deal with the problem of measurement errors effects on the performance of PCIs. The analysis is illustrated with reference toC _p , i.e. the simplest and most common measure suggested to evaluate process capability. The authors would like to thank two anonymous referees for their comments and suggestion that were useful in the preparation and improvement of this paper. This work was partially supported by a MURST research grant.     

On the Mixture Proportional Mean Residual Life Model

In this article, we propose a new mixture model induced by the model of proportional mean residual life. Under some appropriate assumptions, it is shown that the mixing and overall variables in the model admit the positive likelihood ratio dependence structure. To see how the overall variable is affected by the stochastic variation of the mixing variable, we study some stochastic comparisons using these variables. Finally, some useful bounds for tail probability of the overall variable for large values of the mixing variable are derived.     

Pilot–pivotal trials for average bioequivalence

Before carrying out a full scale bioequivalence trial, it is desirable to conduct a pilot trial to decide if a generic drug product shows promise of bioequivalence. The purpose of a pilot trial is to screen test formulations, and hence small sample sizes can be used. Based on the outcome of the pilot trial, one can decide whether or not a full scale pivotal trial should be carried out to assess bioequivalence. This article deals with the design of a pivotal trial, based on the evidence from the pilot trial. A two-stage adaptive procedure is developed in order to determine the sample size and the decision rule for the pivotal trial, for testing average bioequivalence using the two one-sided test (TOST). Numerical implementation of the procedure is discussed in detail, and the required tables are provided. Numerical results indicate that the required sample sizes could be smaller than that recommended by the FDA for a single trial, especially when the pilot study provides strong evidence in favor of bioequivalence.     

Alternative beta estimation for the market model using partially adaptive techniques

The leptokurtosls of many security market return distributions can contaminate ordinary least squares estimates of the β coefficient of the market model. Partially adaptive estimation techniques accommodate the possibility of fat tailed distributions. this methodology limits the influence of extremely large residuals and yields estimates which are both statistically and practically different from ordinary least squares.