Tests of independence of normal random variables with known and unknown variance ratio

Edward Gąsiorek; Andrzej Michalski; Roman Zmyślony

Discussiones Mathematicae Probability and Statistics (2000)

  • Volume: 20, Issue: 2, page 233-247
  • ISSN: 1509-9423

Abstract

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In the paper, a new approach to construction test for independenceof two-dimensional normally distributed random vectors is given under the assumption that the ratio of the variances is known. This test is uniformly better than the t-Student test. A comparison of the power of these two tests is given. A behaviour of this test forsome ε-contamination of the original model is also shown. In the general case when the variance ratio is unknown, an adaptive test is presented. The equivalence between this test and the classical t-test for independence of normal variables is shown. Moreover, the confidence interval for correlation coefficient is given. The results follow from the unified theory of testing hypotheses both for fixed effects and variance components presented in papers [6] and [7].

How to cite

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Edward Gąsiorek, Andrzej Michalski, and Roman Zmyślony. "Tests of independence of normal random variables with known and unknown variance ratio." Discussiones Mathematicae Probability and Statistics 20.2 (2000): 233-247. <http://eudml.org/doc/287716>.

@article{EdwardGąsiorek2000,
abstract = {In the paper, a new approach to construction test for independenceof two-dimensional normally distributed random vectors is given under the assumption that the ratio of the variances is known. This test is uniformly better than the t-Student test. A comparison of the power of these two tests is given. A behaviour of this test forsome ε-contamination of the original model is also shown. In the general case when the variance ratio is unknown, an adaptive test is presented. The equivalence between this test and the classical t-test for independence of normal variables is shown. Moreover, the confidence interval for correlation coefficient is given. The results follow from the unified theory of testing hypotheses both for fixed effects and variance components presented in papers [6] and [7].},
author = {Edward Gąsiorek, Andrzej Michalski, Roman Zmyślony},
journal = {Discussiones Mathematicae Probability and Statistics},
keywords = {mixed linear models; variance components; correlation; quadratic unbiased estimation; testing hypotheses; confidence intervals},
language = {eng},
number = {2},
pages = {233-247},
title = {Tests of independence of normal random variables with known and unknown variance ratio},
url = {http://eudml.org/doc/287716},
volume = {20},
year = {2000},
}

TY - JOUR
AU - Edward Gąsiorek
AU - Andrzej Michalski
AU - Roman Zmyślony
TI - Tests of independence of normal random variables with known and unknown variance ratio
JO - Discussiones Mathematicae Probability and Statistics
PY - 2000
VL - 20
IS - 2
SP - 233
EP - 247
AB - In the paper, a new approach to construction test for independenceof two-dimensional normally distributed random vectors is given under the assumption that the ratio of the variances is known. This test is uniformly better than the t-Student test. A comparison of the power of these two tests is given. A behaviour of this test forsome ε-contamination of the original model is also shown. In the general case when the variance ratio is unknown, an adaptive test is presented. The equivalence between this test and the classical t-test for independence of normal variables is shown. Moreover, the confidence interval for correlation coefficient is given. The results follow from the unified theory of testing hypotheses both for fixed effects and variance components presented in papers [6] and [7].
LA - eng
KW - mixed linear models; variance components; correlation; quadratic unbiased estimation; testing hypotheses; confidence intervals
UR - http://eudml.org/doc/287716
ER -

References

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  1. [1] S. Geisser, Estimation in the uniform covariance case, JASA 59 (1964), 477-483. Zbl0192.25902
  2. [2] S. Gnot and A. Michalski, Tests based on admissible estimators in two variance components models, Statistics 25 (1994), 213-223. Zbl0816.62019
  3. [3] N.L. Johnson and S. Kotz, Distribution in Statistics: continuous univariate distributions - 2, Houghton Mifflin, New York 1970. 
  4. [4] J.M. Kinderman and J.F. Monahan, Computer generation of random variables using the ratio of uniform deviates, ACH Trans. Math. Soft. 3 (1977), 257-260. Zbl0387.65006
  5. [5] E.L. Lehmann, Testing Statistical Hypotheses, Wiley, New York 1986. 
  6. [6] A. Michalski and R. Zmyślony, Testing hypotheses for variance components in mixed linear models, Statistics 27 (1996), 297-310. Zbl0842.62059
  7. [7] A. Michalski and R. Zmyślony, Testing hypotheses for linear functions of parameters in mixed linear models, Tatra Mountains Math. Publ. 17 (1999), 103-110. Zbl0987.62012
  8. [8] J. Seely, Quadratic subspaces and completeness, Ann. Math. Statist. 42 (1971), 710-721. Zbl0249.62067
  9. [9] R. Zmyślony, On estimation of parameters in linear models, Zastosowania Matematyki 15 (1976), 271-276. Zbl0401.62049
  10. [10] R. Zmyślony, Completeness for a family of normal distributions, Banach Center Publications 6 (1980), 355-357. Zbl0464.62003

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