Model averaging for M-estimation

M-estimation is a widely used technique for robust statistical inference. In this paper, we study model selection and model averaging for M-estimation to simultaneously improve the coverage probability of confidence intervals of the parameters of interest and reduce the impact of heavy-tailed errors or outliers in the response. Under general conditions, we develop robust versions of the focused information criterion and a frequentist model average estimator for M-estimation, and we examine their theoretical properties. In addition, we carry out extensive simulation studies as well as two real examples to assess the performance of our new procedure, and find that the proposed method produces satisfactory results.     

Cointegration Detection Using Dynamic Factor Models

A method of information-criterion-based cointegration detection using dynamic factor models is proposed. The results of the data-based and non data-based Monte Carlo simulations suggest that this method is as effective as conventional hypothesis-testing methods. In the proposed method, an observed multivariate time series is described in terms of common stochastic trends plus stationary autoregressive cycles. Then the best model is selected from among alternative models obtained by changing the number of common stochastic trends, on the basis of information criteria. Consequently, the cointegration rank is determined on the basis of the selected model. Two advantages of the proposed method are also discussed.     

An Application of Matrix Power Series to Linear Models

In the system of two seemingly unrelated regressions, employing a matrix power series, we show that the two-stage estimator is better than the ordinary least square estimator (OLSE) in terms of the mean square error matrix (MSEM) criterion. The result enriches the existing literature and can be applied to many fields of applications related to economics and statistics.     

Akaike Information Criterion for Selecting Variables in the Nested Error Regression Model

The Akaike Information Criterion (AIC) is developed for selecting the variables of the nested error regression model where an unobservable random effect is present. Using the idea of decomposing the likelihood into two parts of “within” and “between” analysis of variance, we derive the AIC when the number of groups is large and the ratio of the variances of the random effects and the random errors is an unknown parameter. The proposed AIC is compared, using simulation, with Mallows' C _p , Akaike's AIC, and Sugiura's exact AIC. Based on the rates of selecting the true model, it is shown that the proposed AIC performs better.     

The best subset of parameters in min-max linear regression

These Fortran-77 subroutines provide building blocks for Generalized Cross-Validation (GCV) (Craven and Wahba, 1979) calculations in data analysis and data smoothing including ridge regression (Golub, Heath, and Wahba, 1979), thin plate smoothing splines (Wahba and Wendelberger, 1980), deconvolution (Wahba, 1982d), smoothing of generalized linear models (O'sullivan, Yandell and Raynor 1986, Green 1984 and Green and Yandell 1985), and ill-posed problems (Nychka et al., 1984, O'sullivan and Wahba, 1985). We present some of the types of problems for which GCV is a useful method of choosing a smoothing or regularization parameter and we describe the structure of the subroutines.Ridge Regression: A familiar example of a smoothing parameter is the ridge parameter X in the ridge regression problem which we write.     

Ranking paired contestants

The ranking of paired contestants (players) after a series of contests is difficult when every player does not play every other player. In the 1975 JASA Mark Thompson presented a maximum likelihood solution based on the assumption that the probability of any one player defeating any other is a function only of the difference in their ranks. Here the linear approximation to that likelihood is shown to lead to a nonparametric measure of the efficacy of the ranking, called the net difference in ranks (NDR) , which is the sum of the differences in ranks of the paired players in the observed contests that agree with the ranking minus the sum of the differences in ranks in the observed contests that disagree with the ranking (upsets) . The subject is part of a large literature that has been consolidated by H.A. David in The Method of Paired Comparisons (1963, 1988). The method was introduced by the psychophysicist Fechner in 1860 and has been widely applied to sensory testing,     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

Likelihood-based cross-validation score for selecting the smoothing parameter in maximum penalized likelihood estimation

Maximum penalized likelihood estimation is applied in non(semi)-para-metric regression problems, and enables us exploratory identification and diagnostics of nonlinear regression relationships. The smoothing parameter A controls trade-off between the smoothness and the goodness-of-fit of a function. The method of cross-validation is used for selecting A, but the generalized cross-validation, which is based on the squared error criterion, shows bad be¬havior in non-normal distribution and can not often select reasonable A. The purpose of this study is to propose a method which gives more suitable A and to evaluate the performance of it.

A method of simple calculation for the delete-one estimates in the likeli¬hood-based cross-validation (LCV) score is described. A score of similar form to the Akaike information criterion (AIC) is also derived. The proposed scores are compared with the ones of standard procedures by using data sets in liter¬atures. Simulations are performed to compare the patterns of selecting A and overall goodness-of-fit and to evaluate the effects of some factors. The LCV-scores by the simple calculation provide good approximation to the exact one if λ is not extremeiy smaii Furthermore the LCV scores by the simple size it possible to select X adaptively They have the effect, of reducing the bias of estimates and provide better performance in the sense of overall goodness-of fit. These scores are useful especially in the case of small sample size and in the case of binary logistic regression.     

Smoothed Histograms for Frequency Data on Irregular Intervals

Frequency tables are often constructed on intervals of irregular width. When plotted as bar charts, the underlying true density information may be quite distorted. The majority of introductory statistics texts recommend tabulating data into intervals of equal width, but seldom caution the consequences of failing to do so. An occasional introductory text correctly emphasizes that area rather than frequency should be plotted. Nevertheless, the correctly scaled density figure is often visually less informative than one might expect, with wide bins at constant height. In many cases, the right most bin interval has no well-defined end point, making its depiction some what arbitrary. In this note, we introduce a regular histogram approximation that matches the frequencies and also minimizes a roughness criterion for visual and exploratory appeal. The resulting estimate can reveal the density structure much more clearly. We also formulate an alternative criterion that explicitly takes account of the uncertainty in the bin frequencies.