Abstract
Testing the equality of means of two normally distributed random variables when their variances are unequal is known in the statistical literature as the “Behrens-Fisher problem”. It is well-known that the posterior distributions of the parameters of interest are the primitive of Bayesian statistical inference. For routine implementation of statistical procedures based on posterior distributions, simple and efficient approaches are required. Since the computation of the exact posterior distribution of the Behrens-Fisher problem is obtained using numerical integration, several approximations are discussed and compared. Tests and Bayesian Highest-Posterior Density (H.P.D) intervals based upon these approximations are discussed. We extend the proposed approximations to test of parallelism in simple linear regression models.
Highlights
Suppose that x1 and x2 are two independent normal random variables with means μ1 and μ2, and variances σ 12 and σ, respectively.Samples of sizes n1 and n2 drawn from the corresponding populations are denoted by xij ( i = 1, 2 and j = 1, 2, ni )
Testing the equality of means of two normally distributed random variables when their variances are unequal is known in the statistical literature as the “Behrens-Fisher problem”
It is well-known that the posterior distributions of the parameters of interest are the primitive of Bayesian statistical inference
Summary
Suppose that x1 and x2 are two independent normal random variables with means μ1 and μ2, and variances σ 12 and σ. The Bayesian approach to this problem, viewed as one of the most fascinating approaches of statistical inference on the means of heterogeneous normal populations, will be the main focus of this paper, and we shall give special attention to the problem of testing parallelism of two linear regression lines when the variances of the error terms are not equal. The resulting posterior distribution of U= μ1 − μ2 , which is the primitive of a valid Bayesian procedure, possesses a form requiring an integration that cannot be performed analytically, and direct and routine implementation of this test is impossible.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
