, Volume 44, Issue 1, pp 25–56 | Cite as

Distribution-free properties of some asymptotic cumulants for the Mallows C p and its modifications in usual and ridge regression

Original Paper


In multivariate multiple linear regression with a non-negative ridge parameter, when a model is underspecified, the asymptotic biases of the Mallows C p and its modifications are derived up to order O(1) under non-normality. For a not underspecified model, the asymptotic biases are of smaller order and are shown to be distribution free. Similarly, under the latter condition, the common asymptotic variance of order O(1) for the statistics is distribution free. It is shown that the above results hold irrespective of the ridge parameter. Numerical illustrations with simulations under normality and non-normality give similar simulated results. These results justify the robust correct model selection under non-normality shown by simulations.


Multivariate regression Asymptotic bias Asymptotic variance Underspecified model Non-normality Ridge regression 



The author is indebted to the comments of the reviewers for the improvement of earlier versions of this paper. This work was partially supported by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology (JSPS KAKENHI, Grant No. 26330031).


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Copyright information

© The Behaviormetric Society 2016

Authors and Affiliations

  1. 1.Department of Information and Management ScienceOtaru University of CommerceOtaruJapan

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