Abstract
In this paper we have demonstrated the ability of the new Bayesian measure of evidence of Yin (2012, Computational Statistics, 27: 237-249) to solve both the Behrens-Fisher problem and Lindley's paradox. We have provided a general proof that for any prior which yields a linear combination of two independent t random variables as posterior distribution of the di erence of means, the new Bayesian measure of evidence given that prior will solve Lindleys' paradox thereby serving as a general proof for the works of Yin and Li (2014, Journal of Applied Mathematics, 2014(978691)) and Goltong and Doguwa (2018, Open Journal of Statistics, 8: 902-914). Using the Pareto prior as an example, we have shown by the use of simulation results that the new Bayesian measure of evidence solves Lindley's paradox.
Highlights
In this paper we have demonstrated the ability of the new Bayesian measure of evidence of Yin (2012, Computational Statistics, 27: 237-249) to solve both the Behrens-Fisher problem and Lindley’s paradox
We have provided a general proof that for any prior which yields a linear combination of two independent t random variables as posterior distribution of the difference of means, the new Bayesian measure of evidence given that prior will solve Lindleys’ paradox thereby serving as a general proof for the works of Yin and Li (2014, Journal of Applied Mathematics, 2014(978691)) and Goltong and Doguwa (2018, Open Journal of Statistics, 8: 902-914)
We propose to give a general result of the methodology of [5] in solving simultaneously, the Behrens-Fisher problem and Lindley’s paradox when any given prior is assigned to the unknown variances which yields a posterior distribution that is a linear combination of two independent t random variables
Summary
In an attempt to solve the Behrens-Fisher Problem, [8] derived generalized pivotal quantities and generalized p-values for hypothesis testing in the presence of nuisance parameters. [5] developed a Bayesian testing procedure for testing a precise null hypothesis in the one sample case that avoids the dichotomy of the parameter space, thereby solving Lindley’s paradox. This new Bayesian measure of evidence is given by. [7] 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 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]. [19] proved that Lindley’s Paradox could be reversed given that the probability density is persistently unbounded or where sufficient regularity of likelihood or prior is absent
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