Abstract
This chapter is concerned with theoretical accuracies for asymptotic approximations of the expected probabilities of misclassification (EPMC) when the linear discriminant function and the quadratic discriminant function are used. The method in this chapter is based on asymptotic bounds for asymptotic approximations of a location and scale mixture. The asymptotic approximations considered in detail are those in which both the sample size and the dimension are large, and the sample size is large.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Anderson, T. W. (1973). An asymptotic expansion of the distribution of the studentized classification statistic \(W\). The Annals of Statistics, 1, 964–972.
Fujikoshi, Y. (2000). Error bounds for asymptotic approximations of the linear discriminant function when the sample size and dimensionality are large. Journal of Multivariate Analysis, 73, 1–17.
Fujikoshi, Y. (2002). Selection of variables for discriminant analysis in a high-dimensional case. Sankhyā Series A, 64, 256–257.
Fujikoshi, Y. (2020). Computable error bounds for asymptotic approximations of the quadratic discriminant function when the sample sizes and dimensionality are large. Hiroshima Mathematical Journal, 50.
Fujikoshi, Y., & Seo, T. (1998). Asymptotic approximations for EPMC’s of the linear and the quadratic discriminant functions when the samples sizes and the dimension are large. Statistics Analysis Random Arrays, 6, 269–280.
Fujikoshi, Y., Ulyanov, V. V., & Shimizu, R. (2010). Multivariate analysis: High-dimensional and large-sample approximations. Hoboken: Wiley.
Lachenbruch, P. A. (1968). On expected probabilities of misclassification in discriminant analysis, necessary sample size, and a relation with the multiple correlation coefficients. Biometrics, 24, 823–834.
Okamoto, M. (1963). An asymptotic expansion for the distribution of the linear discriminant function. The Annals of Mathematical Statistics, 34, 1286–1301.
Raudys, S. (1972). On the amount of priori information in designing the classification algorithm. Technical. Cybernetics, 4, 168–174. (in Russian).
Shimizu, R., & Fujikoshi, Y. (1997). Sharp error bounds for asymptotic expansions of the distribution functions of scale mixtures. Annals of the Institute of Statistical Mathematics, 49, 285–297.
Siotani, M. (1982). Large sample approximations and asymptotic expansions of classification statistic. In P. R. Krishnaiah & L. N. Kanal (Eds.), Handbook of statistics (Vol. 2, pp. 47–60). Amsterdam: North-Holland Publishing Company.
Yamada, T., Sakurai, T., & Fujikoshi, Y. (2017). High-dimensional asymptotic results for EPMCs of W- and Z-rules. Hiroshima Statistical Research Group, TR, 17, 12.
Wyman, F. J., Young, D. M., & Turner, D. W. (1990). A comparison of asymptotic error rate expansions for the sample linear discriminant function. Pattern Recognition, 23, 775–783.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this chapter
Cite this chapter
Fujikoshi, Y., Ulyanov, V.V. (2020). Linear and Quadratic Discriminant Functions. In: Non-Asymptotic Analysis of Approximations for Multivariate Statistics. SpringerBriefs in Statistics(). Springer, Singapore. https://doi.org/10.1007/978-981-13-2616-5_4
Download citation
DOI: https://doi.org/10.1007/978-981-13-2616-5_4
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-13-2615-8
Online ISBN: 978-981-13-2616-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)