Accurate Approximation of Correlation Coefficients by Short Edgeworth-Chebyshev Expansion and Its Statistical Applicatio
In Christoph, Prokhorov and Ulyanov (Theory Probab Appl 40(2):250–260, 1996) we studied properties of high-dimensional Gaussian random vectors. Yuri Vasil’evich Prokhorov initiated these investigations. In the present paper we continue these investigation
- PDF / 248,241 Bytes
- 22 Pages / 439.36 x 666.15 pts Page_size
- 68 Downloads / 177 Views
Abstract In Christoph, Prokhorov and Ulyanov (Theory Probab Appl 40(2):250– 260, 1996) we studied properties of high-dimensional Gaussian random vectors. Yuri Vasil’evich Prokhorov initiated these investigations. In the present paper we continue these investigations. Computable error bounds of order O.n3 / or O.n2 / for the approximations of sample correlation coefficients and the angle between high-dimensional Gaussian vectors by the standard normal law are obtained. We give some numerical results as well. Moreover, different types of Bartlett corrections are suggested. Keywords High-dimensional Gaussian random vectors • Sample correlation coefficient • Short Edgeworth-Chebyshev expansions • Computable error bound • Bartlett correction • Fisher transform
Mathematics Subject Classification (2010): Primary 62H10; Secondary 62E20
G. Christoph () Department of Mathematics, University of Magdeburg, Postfach 4120, D-39016, Magdeburg, Germany e-mail: [email protected] V.V. Ulyanov Faculty of Computational Mathematics and Cybernetics, Moscow State University, Vorobyevy Gori, 119899, Moscow, Russia e-mail: [email protected] Y. Fujikoshi Emeritus Professor, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, 739–8526, Japan e-mail: fujikoshi [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 13, © Springer-Verlag Berlin Heidelberg 2013
239
240
G. Christoph et al.
1 Introduction In the present paper we continue to study properties of high-dimensional Gaussian random vectors. We get new results for basic statistics connected with highdimensional vectors. In Christoph, Prokhorov and Ulyanov [2] two-sided bounds were constructed for a probability density function p.u; a/ of a random variable jY aj2 ; where Y is a Gaussian random element with zero mean in a Hilbert space H . The constructed bounds are sharp in the sense that starting from large enough u a ratio of upper bound to lower one equals 8 and does not depend on any parameters of a distribution of jY aj2 . The results hold for finite-dimensional space H D Rd as well provided that its dimension d 3. In Kawaguchi, Ulyanov and Fujikoshi [8] geometric representation of N observations on n variables were studied. It is useful to describe asymptotic behavior of the following statistics: • Length of n-dimensional observation vector, • Distance between two independent observation vectors and • Angle between these vectors. In Hall, Marron and Neeman [6] the asymptotic distributions of these statistics were pointed out in a high-dimensional framework when the dimension n tends to infinity while the sample size N is fixed. In Kawaguchi, Ulyanov and Fujikoshi [8] we obtained the computable error bounds for approximations of the length and the distance. The aim of the present paper is to get a computable error bounds for the angle. Moreover, in order to construct the bounds we study approximations for the sample correlation coe
Data Loading...
