# Near-exact distributions for the generalized Wilks Lambda statistic

• Volume: 30, Issue: 1, page 53-86
• ISSN: 1509-9423

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## Abstract

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Two near-exact distributions for the generalized Wilks Lambda statistic, used to test the independence of several sets of variables with a multivariate normal distribution, are developed for the case where two or more of these sets have an odd number of variables. Using the concept of near-exact distribution and based on a factorization of the exact characteristic function we obtain two approximations, which are very close to the exact distribution but far more manageable. These near-exact distributions equate, by construction, some of the first exact moments and correspond to cumulative distribution functions which are practical to use, allowing for an easy computation of quantiles. We also develop three asymptotic distributions which also equate some of the first exact moments. We assess the proximity of the asymptotic and near-exact distributions obtained to the exact distribution using two measures based on the Berry-Esseen bounds. In our comparative numerical study we consider different numbers of sets of variables, different numbers of variables per set and different sample sizes.

## How to cite

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Luís M. Grilo, and Carlos A. Coelho. "Near-exact distributions for the generalized Wilks Lambda statistic." Discussiones Mathematicae Probability and Statistics 30.1 (2010): 53-86. <http://eudml.org/doc/277037>.

@article{LuísM2010,
abstract = {Two near-exact distributions for the generalized Wilks Lambda statistic, used to test the independence of several sets of variables with a multivariate normal distribution, are developed for the case where two or more of these sets have an odd number of variables. Using the concept of near-exact distribution and based on a factorization of the exact characteristic function we obtain two approximations, which are very close to the exact distribution but far more manageable. These near-exact distributions equate, by construction, some of the first exact moments and correspond to cumulative distribution functions which are practical to use, allowing for an easy computation of quantiles. We also develop three asymptotic distributions which also equate some of the first exact moments. We assess the proximity of the asymptotic and near-exact distributions obtained to the exact distribution using two measures based on the Berry-Esseen bounds. In our comparative numerical study we consider different numbers of sets of variables, different numbers of variables per set and different sample sizes.},
author = {Luís M. Grilo, Carlos A. Coelho},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {independent Beta random variables; characteristic function; sum of Gamma random variables; likelihood ratio test statistic; proximity measures},
language = {eng},
number = {1},
pages = {53-86},
title = {Near-exact distributions for the generalized Wilks Lambda statistic},
url = {http://eudml.org/doc/277037},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Luís M. Grilo
AU - Carlos A. Coelho
TI - Near-exact distributions for the generalized Wilks Lambda statistic
JO - Discussiones Mathematicae Probability and Statistics
PY - 2010
VL - 30
IS - 1
SP - 53
EP - 86
AB - Two near-exact distributions for the generalized Wilks Lambda statistic, used to test the independence of several sets of variables with a multivariate normal distribution, are developed for the case where two or more of these sets have an odd number of variables. Using the concept of near-exact distribution and based on a factorization of the exact characteristic function we obtain two approximations, which are very close to the exact distribution but far more manageable. These near-exact distributions equate, by construction, some of the first exact moments and correspond to cumulative distribution functions which are practical to use, allowing for an easy computation of quantiles. We also develop three asymptotic distributions which also equate some of the first exact moments. We assess the proximity of the asymptotic and near-exact distributions obtained to the exact distribution using two measures based on the Berry-Esseen bounds. In our comparative numerical study we consider different numbers of sets of variables, different numbers of variables per set and different sample sizes.
LA - eng
KW - independent Beta random variables; characteristic function; sum of Gamma random variables; likelihood ratio test statistic; proximity measures
UR - http://eudml.org/doc/277037
ER -

## References

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1. [1] M.Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, (eds.), 9-th printing, Dover, New York 1974.
2. [2] T.W. Anderson, An Introduction to Multivariate Statistical Analysis, 3.rd ed., J. Wiley & Sons, New York 2003. Zbl1039.62044
3. [3] G.E.P. Box, A general distribution theory for a class of likelihood criteria, Biometrika 36 (1949), 317-346. Zbl0035.09101
4. [4] C.A. Coelho, The generalized integer Gamma distribution - a basis for distributions in Multivariate Statistics, Journal of Multivariate Analysis 64 (1998), 86-102. Zbl0893.62046
5. [5] C.A. Coelho, A generalized Integer Gamma distribution as an asymptotic replacement for a Logbeta distribution & applications, American Journal of Mathematical Management Sciences 23 (2003), 383-399.
6. [6] C.A. Coelho, The generalized near-integer Gamma distribution: a basis for 'near-exact' approximations to the distribution of statistics which are the product of an odd number of independent Beta random variables, Journal of Multivariate Analysis 89 (2004), 191-218. Zbl1047.62014
7. [7] L.M. Grilo, Development of near-exact distributions for different scenarios of application of the Wilks Lambda statistic (in Portuguese), Ph.D. thesis, Technical University of Lisbon, Portugal 2005.
8. [8] L.M. Grilo and C.A. Coelho, Development and Comparative Study of two Near-exact Approximations to the Distribution of the Product of an Odd Number of Independent Beta Random Variables, Journal of Statistical Planning and Inference 137 (2007), 1560-1575. Zbl1110.62020
9. [9] L.M. Grilo and C.A. Coelho, The exact and near-exact distribution for the Wilks Lambda statistic used in the test of independence of two sets of variables, American Journal of Mathematical and Management Sciences (2010), (in print). Zbl1228.62019
10. [10] N.L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, vol. 2, J. Wiley & Sons, New York 1995.
11. [11] S.S. Wilks, Certain generalizations in the analysis of variance, Biometrika 24 (1932), 471-494. Zbl0006.02301
12. [12] S.S. Wilks, On the independence of k sets of normally distributed statistical variables, Econometrika 3 (1935), 309-326. Zbl0012.02903

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