Summary
Many classical multivariate tests, such as the one-sample Hotelling’s T 2-test, are based in considering a multivariate normal distribution as underlying model because, in this situation, we can use the usual F (a,b)-distribution to compute p-values and critical values. In this contributed paper we obtain good analytic approximations for these elements in a close to normal situation which allow us to analyze the robustness of this test against small departures from normality.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Daniels, H.E.: Saddlepoint approximations for estimating equations. Biometrika 70, 89–96 (1983)
Everitt, B.S.: A Monte Carlo investigation of the robustness of Hotelling’s one and two-sample T 2 statistic. J. Amer. Statist. Assoc. 74, 48–51 (1979)
Field, C.A., Ronchetti, E.: A tail area influence function and its application to testing. Sequent. Anal. 4, 19–41 (1985)
Fujikoshi, Y.: An asymptotic expansion for the distribution of Hotelling’s T 2-statistic under nonnormality. J. Multivar. Anal. 61, 187–193 (1997)
García-Pérez, A.: Von Mises approximation of the critical value of a test. TEST 12, 385–411 (2003)
García-Pérez, A.: Chi-square tests under models close to the normal distribution. Metrika 63, 343–354 (2006)
García-Pérez, A.: t-tests with models close to the normal distribution. In: Balakrishnan, N., Castillo, E., Sarabia, J.M. (eds.) Advances in Distributions, Order Statistics, and Inference, pp. 363–379. Birkhäuser, Boston (2006)
García-Pérez, A.: Approximations for F-tests which are ratios of sums of squares of independent variables with a model close to the normal. TEST 17, 350–369 (2008)
Gupta, A.K., Kabe, D.G.: Distributions of Hotelling’s T 2 and multiple and partial correlation coefficients for the mixture of two multivariate Gaussian populations. Statistics 32, 331–339 (1999)
Hampel, E.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics: The Approach Based on Influence Functions. Wiley, New York (1986)
Iwashita, T.: Asymptotic null and nonnull distribution of Hotelling’s T 2-statistic under the elliptical distribution. J. Statist. Plan Infer. 61, 85–104 (1997)
Lugannani, R., Rice, S.: Saddle point approximation for the distribution of the sum of independent random variables. Adv. Appl. Probab. 12, 475–490 (1980)
Nachtsheim, C.J., Johnson, M.E.: A new family of multivariate distributions with applications to Monte Carlo studies. J. Amer. Statist. Assoc. 83, 984–989 (1988)
Rao, C.R.: Linear Statistical Inference and its Applications, 2nd edn. Wiley, New York (1973)
Rencher, A.C.: Multivariate Statistical Inference and Applications. Wiley, New York (1998)
Ronchetti, E.: Robust inference by influence functions. J. Statist. Plan. Infer. 57, 59–72 (1997)
Srivastava, M.S., Awan, H.M.: On the robustness of Hotelling’s T 2-test and distribution of linear and quadratic forms in sampling from a mixture of two multivariate normal populations. Comm. Statist.-Theor. Meth. 11, 81–107 (1982)
Staudte, R.G., Sheather, S.J.: Robust Estimation and Testing. Wiley, New York (1990)
Withers, C.S.: Expansions for the distribution and quantiles of a regular functional of the empirical distribution with applications to nonparametric confidence intervals. Ann. Statist. 11, 577–587 (1983)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
García-Pérez, A. (2011). Hotelling’s T 2-Test with Multivariate Normal Mixture Populations: Approximations and Robustness. In: Pardo, L., Balakrishnan, N., Gil, M.Á. (eds) Modern Mathematical Tools and Techniques in Capturing Complexity. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20853-9_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-20853-9_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-20852-2
Online ISBN: 978-3-642-20853-9
eBook Packages: EngineeringEngineering (R0)