An information criterion for auxiliary variable selection in incomplete data analysis

Statistical inference is considered for variables of interest, called primary variables, when auxiliary variables are observed along with the primary variables. We consider the setting of incomplete data analysis, where some primary variables are not observed. Utilizing a parametric model of joint distribution of primary and auxiliary variables, it is possible to improve the estimation of parametric model for the primary variables when the auxiliary variables are closely related to the primary variables. However, the estimation accuracy reduces when the auxiliary variables are irrelevant to the primary variables. For selecting useful auxiliary variables, we formulate the problem as model selection, and propose an information criterion for predicting primary variables by leveraging auxiliary variables. The proposed information criterion is an asymptotically unbiased estimator of the Kullback-Leibler divergence for complete data of primary variables under some reasonable conditions. We also clarify an asymptotic equivalence between the proposed information criterion and a variant of leave-one-out cross validation. Performance of our method is demonstrated via a simulation study and a real data example

一般化線形モデル及びその拡張におけるモデル選択規準

広島大学(Hiroshima University)博士(理学)Doctor of Sciencedoctora

A Study on the Bias-Correction Effect of the AIC for Selecting Variables in Normal Multivariate Linear Regression Models under Model Misspecification

Abstract By numerically comparing a variable-selection method using the crude AIC with those using the bias-corrected AICs, we find out knowledge about what kind of bias correction gives a positive effect to variable selection under model misspecification. Actually, since it can be proved theoretically that all the variable-selection methods considered in this paper asymptotically choose the same model as the best model, we conduct numerical examinations using small and moderate sample sizes. Our results show that bias correction under assumption that the mean structure is misspecified has a better effect on variable selection than that under the assumption that the distribution of the model is misspecified

On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. The article complements the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart matrix are close to the degrees of freedom of the distribution

On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. We complement the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart  matrix are close to the degrees of freedom of the distribution

On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. We complement the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart  matrix are close to the degrees of freedom of the distribution

On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. We complement the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart  matrix are close to the degrees of freedom of the distribution