Large sample tests for testing symmetry and independence in some bivariate exponential models

Abstract

Bivariate Exponential Distribution (BVED) were introduced by Freund (1961), Marshall and Olkin (1967) and Block and Basu (1974) as models for the distributions of (X,Y) the failure times of dependent components (C₁,C₂). We study the structure of these models and observe that Freund model leads to a regular exponential family with a four dimensional orthogonal parameter. Marshall-Olkin model involving three parameters leads to a conditional or piece wise exponential family and Block-Basu model which also depends on three parameters is a sub-model of the Freund model and is a curved exponential family. We obtain a large sample tests for symmetry as well as independence of (X,Y) in each of these models by using the Generalized Likelihood Ratio Tests (GLRT) or tests basesd on MLE of the parameters and root n consistent estimators of their variance-covariance matrices.

Keywords:

• Bivariate exponential distributions (BVED)
• Block-Basu model
• freund Model
• Marshall-Olkin model
• MLE
• GLRT
• UMPU test
• Simultaneous failures