Hiroshima Mathematical Journal

Statistical inference for functional relationship between the specified and the remainder populations

Yasutomo Maeda

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Abstract

This paper is concerned with discovering linear functional relationships among $k$ $p$-variate populations with mean vectors $\vmu_{i}$, $i=1,\ldots ,k$ and a common covariance matrix $\Sigma$. We consider a linear functional relationship to be one in which each of the specified $r$ mean vectors, for example, $\vmu_{1}, \ldots, \vmu_{r}$ are expressed as linear functions of the remainder mean vectors $\vmu_{r+1}, \ldots, \vmu_{k}$. This definition differs from the classical linear functional relationship, originally studied by Anderson [1], Fujikoshi [8] and others, in that there are $r$ linear relationships among $k$ mean vectors without any specification of $k$ populations. To derive our linear functional relationship, we first obtain a likelihood test statistic when the covariance matrix $\Sigma$ is known. Second, the asymptotic distribution of the test statistic is studied in a high-dimensional framework. Its accuracy is examined by simulation.

Article information

Source
Hiroshima Math. J., Volume 40, Number 2 (2010), 215-228.

Dates
First available in Project Euclid: 2 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1280754422

Digital Object Identifier
doi:10.32917/hmj/1280754422

Mathematical Reviews number (MathSciNet)
MR2680657

Zentralblatt MATH identifier
1284.62140

Subjects
Primary: 12A34 98B76 23C57

Keywords
asymptotic distribution high-dimensional framework likelihood ratio test statistics (LR test statistics) linear functional relationship maximum likelihood estimators (MLE)

Citation

Maeda, Yasutomo. Statistical inference for functional relationship between the specified and the remainder populations. Hiroshima Math. J. 40 (2010), no. 2, 215--228. doi:10.32917/hmj/1280754422. https://projecteuclid.org/euclid.hmj/1280754422


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References

  • Anderson, T. W., Estimating linear restrictions on regression coefficients for multivariate normal distributions, Ann. Math. Statist., 22 (1951), 327--351.
  • Anderson, T. W., Asymptotic theory for principal component analysis, Ann. Math. Statist., 34 (1963), 122--148.
  • Anderson, T. W., Estimation of linear functional Relationships: Approximate distributions and communications with simultaneous equations in econometrics, J. Roy. Statist. Soc. Ser. B, 38 (1976), 1--20.
  • Anderson, T. W., Estmating linear statistical relationships, Ann. Statist., 12 (1984), 1--45.
  • Arellano-Valle, R. B., Bolfarine, H. and Gasco, L., Measurement error models with nonconstant covariance matrices, J. Multivariate Anal., 82 (2002), 395--415.
  • Bai, Z. D., Methodologies in spectral analysis of large dimensional random matrices, a review, Statist. Sinica, 9 (1999), 611--677.
  • Chi-Lun Cheng and John W. Van Ness, Statistical Regression with Measurement Error, Kendall's Library of Statistics 6, 1999.
  • Fujikoshi, Y., The likelihood ratio tests for the dimensionality of regression coefficients, J. Multivariate Anal., 4 (1974), 327--340.
  • Fujikoshi, Y., Asymptotic expansions for the distributions of some multivariate tests, Multivariate Analysis-IV, (P. R. Krishnaiah, Ed.), North-Holland Publishing Company, Amsterdam., (1977), 55--71.
  • Fujikoshi, Y., Himeno, T. and Wakaki, H., Asymptotic results in canonical discriminant analysis when the dimension is large compared to the sample size, J. Statist. Plann. Inference, 138 (2008), 3457--3466.
  • Fuller, Wayne A., Properties of some estimators for the errors-in-variables model, Ann. Statist., 8 (1980), 407--422.
  • Fuller, Wayne A., Measurement Error Models, Wiley, New York, 1987.
  • Gleser, Leon Jay, Estimation in a multivariate `errors in variables' regression model: large sample results, Ann. Statist., (1981), 9 24--44.
  • Gupta, A. K. and Nagar D. K., Matrix Variate Distributions, Chapman & Hall, 2000.
  • Kraft, C. H., Olkin, I. and van Eeden, C., Estimation and testing for differences in magnitude or displacement in the mean vectors of two multivariate normal populations., Ann. Math. Statist., 43 (1972), 455--467.
  • Johnstone, I. M., On the distribution of the largest eigenvalue in principal component analysis, Ann. Statist., 29 (2001), 295--327.
  • Lawley, D. N., Tests of significance for the latent roots of covariance and correlation matrices, Biometrika, 43 (1956), 128--136.
  • Lawley, D. N., Tests of significance in canonical analysis, Biometrika, 46 (1959), 59--66.
  • Ledoit, O. Wolf, M., Some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size, Ann. Statist., 30 (2002), 1081--1102.
  • Raudys, S. and Young D. M., Results in statistical discriminant analysis: A review of the former Soviet Union literature, J. Multivariate Anal., 89 (2004), 1--35.
  • Seber, George A. F., A matrix handbook for statisticians, John Wiley & Sons, 2008.
  • Wakaki, H., Edgeworth expansion of Wilks' lambda statistic, J. Multivariate Anal., 97 (2006), 1958--1964.