On the robustness of the correlation coefficient in sampling from a mixture of two bivariate normals

Abstract

In this paper, the distribution of the correlation coefficient is obtained when a sample of size N is taken from a mixture of two bivariate normal distributions with common covariance matrix ∑ mean vectors and and mixing proportions λ and (1-λ) respectively. Actual values of size for one and two sided tests are obtained for some combinations of the parameters, and some power computation are also given. These calculations reveal that one sided tests are non-robust but the two sided tests are fairly robust. This provides a counter example to the assertion made by Pitman (1937), later confirmed by Duncan and Layard (1973). According to this assertion, when ρ=0 does not imply independence, then the r test should be sensitive to non-normality. In a special case it is shown that the test is not only robust but even the power improves slightly.

Keywords:

• Robustness
• Correlation Coefficient
• Contamination
• Mixture of two bivariate normals
• normality