Theorem: Let X be a bivariate random variable with distribution function F. Given two nonsingular 2X2 matrices, say A and B, such that A-1B or B-1A has no zero element. Further, if the components of BX and AX are independent then F is bivariate normal distribution function.
How do we construct bivariate normality test using the above theorem ?
Hi,
To construct bivariate normality test,
Kindly see the links:
http://jtemplin.coe.uga.edu/files/multivariate/mv11ersh8350/mv11ersh8350_R3.3.pdf
http://www.iimahd.ernet.in/publications/data/2006-12-04_tdesai.pdf
http://users.jyu.fi/~slahola/files/icorsmar.pdf
http://www.math.hkbu.edu.hk/~hpeng/stat3710/Lecture_note3.pdf
http://lya.fciencias.unam.mx/rfuentes/mv_norm_tests.pdf
http://www.public.iastate.edu/~maitra/stat501/lectures/AssessingNormality.pdf
Hi,
I need to develop a test using linear combination property of normal distribution.
Choose two non-singular matrix A and B with A-B has no zero element. If you have the data from F firstly look at the person correlation coefficient between AX and BX. If it is around 0, you may say that "H0: X has bivariate normal distribution" hypothesis can not be rejected. You can construct the critical values for different significance levels by using monte carlo simulation. But I think that it is not goog idea using above theorem to test bivariate normality. Because choosing A and B is big problem.
Hi,
Yes, two pairs of linear combinations with marginal independence is enough to say about bivariate normality.
It is obvious that correlation is zero does not mean the independence. In this case, the test with above contsruction does not work well in some cases in whihc uncorrelated but dependent random variables.
As long we are testing for normality correlation zero implies independence. For normal distribution it is true.
See also Mecklin and Mundfrom (2005) who have surveyed 50 normality tests:
C. J. Mecklin and D. J. Mundfrom. A Monte Carlo Comparison of the Type I and Type II Error Rates of Tests of Multivariate Normality. Journal of Statistical Computation and Simulation, 75
(2):93{107, 2005. doi: 10.1080/0094965042000193233. URL http://www.tandfonline.com/doi/
abs/10.1080/0094965042000193233.
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