[1] Aoshima, M., Shen, D., Shen, H., Yata, K., Zhou, Y. H. & Marron, J. (2018). A survey
of high dimension low sample size asymptotics. Aust. Nz. J. Stat., 60, 4–19.
[2] Aoshima, M. & Yata, K. (2018). Two-sample tests for high-dimension, strongly spiked
eigenvalue models. Stat. Sinica, 28, 43–62.
[3] Aoshima, M. & Yata, K. (2019). Distance-based classifier by data transformation for
high-dimension, strongly spiked eigenvalue models. Ann. I. Stat. Math., 71, 473–503.
[4] Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In 2nd International Symposium on Information Theory (eds. B. N. Petrov & F.
Cs´
aki), pp. 995–1010. Akad´
emiai Kiad´
o, Budapest.
[5] Akaike, H. (1974). A new look at the statistical model identification. Institute of Electrical
and Electronics Engineers. Transactions on Automatic Control AC − 19, 716–723.
[6] Bozdogan, H. (1987). Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika, 52, 345–370.
[7] Dempster, A. P. (1958). A high dimensional two sample significance test. Ann. Math.
Statist., 29, 995–1010.
[8] Dempster, A. P. (1960). A significance test for the separation of two highly multivariate
small samples. Biometrics, 16, 41–50.
[9] Fujikoshi, Y., Kan, T., Takahashi, S. & Sakurai, T. (2011). Prediction error criterion for
selecting variables in a linear regression model. Ann. I. Stat. Math., 63, 387–403.
[10] Fujikoshi, Y., Himeno, T. & Wakaki, H. (2004). Asymptotic results of a high dimensional
MANOVA test and power comparison when the dimension is large compared to the sample
size. J. Japan Statist. Soc., 34, 19–26.
[11] Fujikoshi, Y., Sakurai, T. & Yanagihara, H. (2014). Consistency of high-dimensional AICtype and Cp -type criteria in multivariate linear regression. J. Multivariate Anal., 123,
184–200.
[12] Fujikoshi, Y. & Satoh, K. (1997). Modified AIC and Cp in multivariate linear regression.
Biometrika, 84, 707–716.
[13] Hannan, E. J. & Quinn, B. G. (1979). The determination of the order of an autoregression.
J. Roy. Statist. Soc. Ser. B, 26, 270–273.
[14] Harville, D. A. (1997). Matrix Algebra from a Statistician’s Perspective. Springer-Verlag,
New York.
32
Ryoya Oda
[15] Himeno, T. & Yamada, T. (2014). Estimations for some functions of covariance matrix
in high dimension under non-normality and its applications. J. Multivariate Anal., 130,
27–44.
[16] Katayama, S. & Imori, S. (2014). Lasso penalized model selection criteria for highdimensional multivariate linear regression analysis. J. Multivariate Anal., 132, 138–150.
[17] Kubokawa, T. & Srivastava, M. S. (2012). Selection of variables in multivariate regression
models for large dimensions. Comm. Statist. A-Theory Methods, 41, 2465–2489.
[18] Magnus, J. R. & Neudecker, H. (1979). The commutation matrix: some properties and
applications. Ann. Statist., 7, 381–894.
[19] Mallows, C. L. (1973). Some comments on Cp . Technometrics, 15, 661–675.
[20] Mallows, C. L. (1995). More comments on Cp . Technometrics, 37, 362–372.
[21] Nagai, I., Yanagihara, H. & Satoh, K. (2012). Optimization of ridge parameters in multivariate generalized ridge regression by plug-in methods. Hiroshima Math. J., 42, 301–324.
[22] Nishii, R., Bai, Z. D. & Krishnaiah, P. R. (1988). Strong consistency information criterion
for model selection in multivariate analysis. Hiroshima Math. J., 18, 451–462.
[23] Sparks, R. S., Coutsourides, D. & Troskie, L. (1983). The multivariate Cp . Comm. Statist.
A-Theory Methods, 12, 1775–1793.
[24] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist., 6, 461–464.
[25] Srivastava, M. S. (2002). Methods of Multivariate Statistics. John Wiley & Sons, New
York.
[26] Timm, N. H. (2002). Applied Multivariate Analysis. Springer-Verlag, New York.
[27] Wille, A., Zimmermann, P., Vranova, E., F¨
urholz, A., Laule, O., Bleuler, S., Hennig,
L., Prelic, A., von Rohr, P., Thiele, L., Zitzler, E., Gruissem, W. & B¨
uhlmenn, P.
(2004). Sparse graphical Gaussian modeling of the isoprenoid gene network in Arabidopsis
thaliana. Genome Biol., 5, 1–13.
[28] Yamamura, M., Yanagihara, H. & Srivastava, M. S. (2010). Variable selection in multivariate linear regression models with fewer observations than the dimension. Japan. J.
Appl. Stat., 39, 1–19.
[29] Yanagihara, H. (2015). Conditions for consistency of a log-likelihood-based information
criterion in normal multivariate linear regression models under the violation of the normality assumption. J. Japan Statist. Soc., 45, 21–56.
[30] Yanagihara, H. (2016). A high-dimensionality-adjusted consistent Cp -type statistic for
selecting variables in a normality-assumed linear regression with multiple responses. Procedia Comput. Sci., 96, 1096–1105.
[31] Yanagihara, H. (2019). Evaluation of consistency of model selection criteria in multivariate
linear regression models by large-sample and high-dimensional asymptotic theory under
nonnormality. J. Jpn. Stat. Soc. Jpn. Issue, 48, 1–13.
[32] Yanagihara, H., Wakaki, H. & Fujikoshi, Y. (2015). A consistency property of the AIC for
multivariate linear models when the dimension and the sample size are large. Electron. J.
Statist., 9, 869–897.
[33] Zhao, L. C., Krishnaiah, P. R. & Bai, Z. D. (1986). On detection of the number of signals
in presence of white noise. J. Multivariate Anal., 20, 1–25.
Ryoya Oda
Department of Mathematics
Graduate School of Science
Hiroshima University
Higashi-Hiroshima 739-8526, JAPAN
E-mail : ryoya-oda@hiroshima-u.ac.jp
...