A risk function based on the standardized mean squared error of prediction is widely used for measuring the goodness of a model. However, using the risk function directly for model selection is generally not possible due to its inclusion of unknown parameters. The C p criterion is an asymptotic unbiased estimator of the risk function that has been commonly used in place of the risk function for model selection. Because the C p criterion can have a non-negligible bias against the risk function when the sample size is small, a modified C p criterion that completely corrects this bias was proposed by Fujikoshi and Satoh (1997) in a multivariate linear regression model. In this paper, we propose a modified C p criterion for common univariate models that generalize previous studies, i.e., the models in which the predicted response variables are defined as the product of a hat matrix and a vector of response variables.

For multivariate models, one of the important properties of a variable selection criterion is consistency, whereby the probability of selecting the true subset of explanatory variables approaches 1 as the sample size goes to infinity. However, in cases of high-dimensionality, the BIC, which is expected to be consistent under large-sample asymptotic theory, may not be consistent, and the AIC, which is said to be inconsistent under the same large-sample asymptotic theory, may be consistent. In this paper, we propose a high-dimensionality-adjusted consistent AIC called the HCAIC, which is an adjusted version of the consistent AIC proposed by Bozdogan (1987) regarding the effect of high-dimensionality through an asymptotic theory whereby the dimension of the vector of response variables allows both an increasing and a fixed condition as sample size increases. Through a series of numerical experiments, we verify that the proposed HCAIC is a high-performance information criterion in the sense that it can select the true subset of explanatory variables with high probability for any dimensionality.

In this study, we use the log-linear link function and propose a generalized fused Lasso (GFL) Poisson regression model in which the nonlinear trend is discretely represented by categorical covariates in the additive model. We use the coordinate descent algorithm for the estimation and show that the optimal solution in a coordinate axis can be found explicitly. To demonstrate the proposed approach, we analyze Japanese crime data. Simulation results showed a fitness ratio for true fusion to be more than 90% in total, demonstrating the reliability of the estimates.

This paper proposes the optimization of ridge parameters in a multivariate generalized ridge regression (MGRR) by minimizing a modified Cp (MCp) statistic, which is defined by completely removing the bias against the mean square error of prediction from Mallows’ Cp statistic. The model considered is the generalized multivariate analysis of the variance model, referred to as the GMANOVA model. In the multivariate linear regression model, which is a special case of the GMANOVA model, the ridge parameters in the GMRR that minimize the MCp statistic can be obtained in closed form. In this paper, we show that a solution to the minimization problem involving the MCp statistic can be obtained explicitly even for the GMANOVA model. Through numerical studies, we verify that an MGRR based on ridge parameters optimized by minimizing the MCp statistic is superior to that produced by minimizing the Cp statistic in the sense of a smaller mean squared error of the predictor.

Data containing information about observed location and time are called geographical and chronological data. The purpose of this paper is to propose how we can analyze geographical and chronological data with multiple response variables by innovating the varying coefficient model in canonical correlation analysis. In addition, the variable selection proposed by Hashiyama et al . (2014) is applied to our model. As numerical background, we propose to apply an approach where we use a body condition data set from common minke whales ( Balaenoptera acutorostrata acutorostrata ) in the Barents Sea (Solvang et al . (2016)). From the estimation results, minke whale body condition is affected by geography in females and by chronology in males, however the geographical effect seems not so strong, and male and female whales gain their body condition as fall approaches, which is the well known as their general habits in the Barents Sea.

In this paper, we consider the consistency of Cp-type statistics for selecting variables in a normality-assumed linear regression with multiple responses when the dimension of the vector of the response variables may be large. We propose a new consistent Cp-type statistic for which consistency can be achieved whenever the dimension of the response variables vector is fixed or goes to infinity. A high probability of selecting the true subset of explanatory variables can be expected under a moderate sample size when the proposed Cp-type statistic is used to select variables, even when there is a high-dimensional response variables vector.