Bayesian Analysis of the Behrens-Fisher Problem under a Gamma Prior

Abstract

Yin [1] has developed a new Bayesian measure of evidence for testing a point null hypothesis which agrees with the frequentist p-value thereby, solving Lindley's paradox. Yin and Li [2] extended the methodology of Yin [1] to the case of the Behrens-Fisher problem by assigning Jeffreys' independent prior to the nuisance parameters. In this paper, we were able to show both analytically and through the results from simulation studies that the methodology of Yin [1] solves simultaneously, the Behrens-Fisher problem and Lindley's paradox when a Gamma prior is assigned to the nuisance parameters.

On Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence

“…It can easily be seen that the posterior distributions of the difference of means derived by [6] and [7] under the different circumstances satisfy the conditions for Lemma 1 to hold. As such, Lemma 1 then proves mathematically that using either the Bayesian measure of evidence under Jeffreys' independent prior as obtained by [6] or under the Gamma prior as obtained by [7], both of which are extensions of the methodology of [5], will solve Lindley's paradox. Note that [6] did not provide a mathematical proof of the ability of the new Bayesian measure of evidence of [5]…”

“…By the use of simulation results, this approach was shown to solve the two problems and also, it was shown mathematically to yield credible intervals that actually possess 1 − α coverage probability. [7] also extended the methodology of [5] by assigning Gamma priors to the precisions and showed mathematically that the paradox in [1] is avoided while solving the Behrens-Fisher problem. They also showed that results obtained by assigning Gamma priors to the precisions could coincide with results obtained by assigning Jeffrey's independent prior to the variances as is done in [6].…”

“…Using Jeffreys' independent prior, [6] extended the methodology of [5] to solve the Behrens-Fisher Problem so as to avoid Lindleys' Paradox. Also, [7] made an attempt to solve the Paradox in [1] by assigning Gamma priors to the precisions using the methodology of [5]. It is noteworthy that the posterior distribution of the difference of means derived by [6] and that derived by [7] are similar in the sense that both are a linear combination of two t distributed random variables.…”

“…Also, [7] made an attempt to solve the Paradox in [1] by assigning Gamma priors to the precisions using the methodology of [5]. It is noteworthy that the posterior distribution of the difference of means derived by [6] and that derived by [7] are similar in the sense that both are a linear combination of two t distributed random variables. Also these two posterior distributions differ in the sense that, the coefficients of these linear combinations are not the same and also, the distributions of the t random variables differ in their degrees of freedom.…”

Bayesian Estimation of the Parameters of the Odd Generalized Exponentiated - Inverse Exponential Distribution (OGE -IED)

“…A prior distribution could be informative or non-informative, improper or conjugate. For a demonstration on the use of priors, see Nawaz and Aslam [8], Chandra and Rathavr [9], Aslam et al [10], Guure and Bosomprah [5], Yin [11], Yin and Zhao [12], Yin and Li [13], Goltong and Doguwa [14] and Goltong and Doguwa [15].…”