The Annals of Mathematical Statistics

Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions

David Durand and J. Arthur Greenwood

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Abstract

The distribution of the sum of $n$ random coplanar unit vectors and of a given component of the sum has been discussed by many authors, who have shown that each distribution can be approximated in series that are asymptotically normal. But the difficult question of the usefulness of these approximations for finite $n$--in particular for small $n$--has not been exhaustively treated. Accordingly, this paper reexamines some analyses of Pearson's series for the vector sum, presents corresponding series for a component, and examines the accuracy of the latter series.

Article information

Source
Ann. Math. Statist., Volume 28, Number 4 (1957), 978-986.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177706798

Digital Object Identifier
doi:10.1214/aoms/1177706798

Mathematical Reviews number (MathSciNet)
MR93805

Zentralblatt MATH identifier
0083.14005

JSTOR
links.jstor.org

Citation

Durand, David; Greenwood, J. Arthur. Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions. Ann. Math. Statist. 28 (1957), no. 4, 978--986. doi:10.1214/aoms/1177706798. https://projecteuclid.org/euclid.aoms/1177706798


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See also

  • Part I: J. Arthur Greenwood, David Durand. The Distribution of Length and Components of the Sum of $n$ Random Unit Vectors. Ann. Math. Statist., Volume 26, Number 2 (1955), 233--246.