Abstract
Estimation of prediction accuracy is important when our aim is prediction. The training error is an easy estimate of prediction error, but it has a downward bias. On the other hand, K-fold cross-validation has an upward bias. The upward bias may be negligible in leave-one-out cross-validation, but it sometimes cannot be neglected in 5-fold or 10-fold cross-validation, which are favored from a computational standpoint. Since the training error has a downward bias and K-fold cross-validation has an upward bias, there will be an appropriate estimate in a family that connects the two estimates. In this paper, we investigate two families that connect the training error and K-fold cross-validation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Burman, P.: A comparative study of ordinary cross-validation, v-fold cross-validation and the repeated learning-testing methods. Biometrika 76, 503–514 (1989)
Davison, A.C., Hinkley, D.V.: Bootstrap Methods and Their Application. Cambridge University Press, Cambridge (1997)
Efron, B.: The estimation of prediction error: covariance penalties and cross-validation (with discussion). J. Am. Stat. Assoc. 99, 619–642 (2004)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer, New York (2001)
Konishi, S., Kitagawa, G.: Information Criteria and Statistical Modeling. Springer, New York (2007)
Li, K.-C.: Asymptotic optimality for C p , C L , cross-validation and generalized cross-validation: discrete index set. Ann. Stat. 15(3), 958–975 (1987)
Shao, J.: Linear model selection by cross-validation. J. Am. Stat. Assoc. 88, 486–494 (1993)
Stone, M.: Cross-validatory choice and assessment of statistical predictions (with discussion). J. R. Stat. Soc., Ser. B 36, 111–147 (1974)
Stone, M.: An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. J. R. Stat. Soc., Ser. B 39, 44–47 (1977)
van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998)
Yanagihara, H., Tonda, T., Matsumoto, C.: Bias correction of cross-validation criterion based on Kullback-Leibler information under a general condition. J. Multivar. Anal. 97, 1965–1975 (2006)
Yang, Y.: Consistency of cross validation for comparing regression procedures. Ann. Stat. 35, 2450–2473 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fushiki, T. Estimation of prediction error by using K-fold cross-validation. Stat Comput 21, 137–146 (2011). https://doi.org/10.1007/s11222-009-9153-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-009-9153-8