(Received June 22, 1992; revised April 7, 1993)
Abstract. The locally best invariant test for the hypothesis of independence in bivariate distributions with exponentially distributed marginals is derived. The model consists of a family of bivariate exponential distributions with probability density function
ftheta(x1 , x2 ; lambda1 , lambda2) = lambda1lambda2 exp[- (lambda1x1+lambda2x2)] g(lambda1x1 , lambda2x2 ; theta)with unknown scale parameter lambdaj (j=1,2) and association parameter theta which includes the independence situation. The locally best invariant (LBI) test is derived and the asymptotic null and nonnull distributions are also derived under some regularity conditions. The results are applied to the Gumbel (1960, J. Amer. Statist. Assoc., 55, 698-707), Frank (1979, Aequationes Math., 19, 194-226), and Cook and Johnson (1981, J. Roy. Statist. Soc. Ser. B, 43, 210-218) families.
Key words and phrases: Bivariate exponential distribution, locally best invariant test, test of independence, parametric families of Gumbel (type I and II), Frank, and Cook and Johnson.
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