Asymptotic biases of information and cross-validation criteria under canonical parametrization

ABSTRACT

An asymptotic expansion of the cross-validation criterion (CVC) using the Kullback-Leibler distance is derived when the leave-k-out method is used and when parameters are estimated by the weighted score method. By this expansion, the asymptotic bias of the Takeuchi information criterion (TIC) is derived as well as that of the CVC. Under canonical parametrization in the exponential family of distributions when maximum likelihood estimation is used, the magnitudes of the asymptotic biases of the Akaike information criterion (AIC) and CVC are shown to be smaller than that of the TIC. Examples in typical statistical distributions are shown.

KEYWORDS:

• Takeuchi information criterion
• Akaike information criterion
• leave-k-out method
• exponential family of distributions
• infinitely divisible.

MATHEMATICS SUBJECT CLASSIFICATION:

• 62F12
• 62H10

Additional information

Funding

Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, Culture, Sports, Science and Technology [JSPS KAKENHI, Grant No.17K00042].