Computable error bounds for asymptotic approximations of the quadratic discriminant function. (English) Zbl 1469.62286
Summary: This paper is concerned with computable error bounds for asymptotic approximations of the expected probabilities of misclassification (EPMC) of the quadratic discriminant function \(Q\). A location and scale mixture expression for \(Q\) is given as a special case of a general discriminant function including the linear and quadratic discriminant functions. Using the result, we provide computable error bounds for asymptotic approximations of the EPMC of \(Q\) when both the sample size and the dimensionality are large. The bounds are numerically explored. Similar results are given for a quadratic discriminant function \(Q_0\) when the covariance matrix is known.
MSC:
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
| 62-08 | Computational methods for problems pertaining to statistics |
Keywords:
asymptotic approximations; error bounds; expected probability of misclassification; high-dimension; large-sample; linear discriminant function; quadratic discriminant functionReferences:
| [1] | Fujikoshi, Y. (2000). Error bounds for asymptotic approximations of the linear discriminant function when the sample size and dimensionality are large. J. Multivariate Anal., 73, 1-17. · Zbl 0948.62044 |
| [2] | Fujikoshi, Y. (2002). Selection of variables for discriminant analysis in a high-dimensional case. Sankhyā Ser. A, 64, 256-257. · Zbl 1192.62162 |
| [3] | Fujikoshi, Y. and 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. Statist. Anal. Random Arrays, 6, 269-280. · Zbl 0924.62070 |
| [4] | Fujikoshi, Y. and Ulyanov, V. V. (2006). On accuracy of approximations for location and scale mixture. J. Math. Sci., 138, 5390-5395. · Zbl 1113.62014 |
| [5] | Fujikoshi, Y., Ulyanov, V. V. and Shimizu, R. (2010). Multivariate Analysis: High-Dimensional and Large-Sample Approximations. Wiley, Hoboken, New Jersey. · Zbl 1304.62016 |
| [6] | Hyodo, M. and Kubokawa, T. (2014). A variable selection criterion for linear discriminant rule and its optimality in high dimensional and large sample data. J. Multivariate Anal., 123, 364-379. · Zbl 1278.62096 |
| [7] | Matsumoto, C. (2004). An optimal discriminant rule in the class of linear and quadratic discriminant functions for large dimension and samples. Hiroshima Math. J., 34, 231-250. · Zbl 1056.62078 |
| [8] | McLachlan, G. J. (1991). Discriminant Analysis and Statistical Pattern Recognition. Wiley, New York. |
| [9] | Siotani, M. (1982). Large sample approximations and asymptotic expansions of classifica-tion statistic. Handbook of Statistics 2 (P. R. Krishnaiah and L. N. Kanal, Eds.), North-Holland Publishing Company, 47-60. · Zbl 0496.00013 |
| [10] | Tonda, T., Nakagawa, T. and Wakaki, H. (2017). EPMC estimation in discriminant analysis when the dimension and sample are large. Hiroshima Math. J., 47, 43-62. · Zbl 1365.62250 |
| [11] | Yamada, T., Himeno, T. and Sakurai, T. (2017). Asymptotic cut-o¤ point in linear discriminant rule to adjust the misclassification probability for large dimensions. Hiroshima Math. J., 47, 319-334. · Zbl 1381.62198 |
| [12] | Yamada, T., Sakurai, T. and Fujikoshi, Y. (2017). High-dimensional asymptotic results for EPMCs of W-and Z-rules. Hiroshima Statistical Research Group, TR: 17-12 |
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