/mo> 2 α 2 5 ] ( n 2 + 2 α 2 4 ) / 2 (12)

where clearly, (12) is the kernel of the joint distribution of two independent t random variables ${t}_{1}$ and ${t}_{2}$ with ${n}_{1}+2{\alpha }_{1}-5$ and ${n}_{2}+2{\alpha }_{2}-5$ degrees of freedom respectively.

Also, we have that

$\begin{array}{c}{\mu }_{i}={\stackrel{¯}{x}}_{i}-\frac{{t}_{i}{S}_{i}\sqrt{{a}_{i}\left({n}_{i}-1\right)}}{\sqrt{{n}_{i}\left({n}_{i}+2{\alpha }_{i}-5\right)}}\\ ={\stackrel{¯}{x}}_{i}-\frac{{t}_{i}\sqrt{{S}_{i}^{2}\left({n}_{i}-1\right)\left(1+\frac{2{\beta }_{i}}{\left({n}_{i}-1\right){S}_{i}^{2}}\right)}}{\sqrt{{n}_{i}\left({n}_{i}+2{\alpha }_{i}-5\right)}}\\ ={\stackrel{¯}{x}}_{i}-\frac{{t}_{i}\sqrt{\left({n}_{i}-1\right){S}_{i}^{2}+2{\beta }_{i}}}{\sqrt{{n}_{i}\left({n}_{i}+2{\alpha }_{i}-5\right)}}\end{array}$

And this implies that the posterior distribution of $\theta$ , the difference of the two means ${\mu }_{1}$ and ${\mu }_{2}$ is given as

$\theta |x~{\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}-\left({T}_{1}\sqrt{\frac{\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}}{{n}_{1}\left({n}_{1}+2{\alpha }_{1}-5\right)}}-{T}_{2}\sqrt{\frac{\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}}{{n}_{2}\left({n}_{2}+2{\alpha }_{2}-5\right)}}\right)$ (13)

and

$E\left(\theta |x\right)={\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}$

where ${T}_{i}$ is a random variable that follows a t distribution with ${n}_{i}+2{\alpha }_{i}-5$ degrees of freedom, where $i=1,2$ . The Bayesian measure of evidence under a Gamma Prior is then given by

${P}_{Ga}^{BF}\left(x\right)=P\left(|{T}_{1}\sqrt{\frac{\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}}{{n}_{1}\left({n}_{1}+2{\alpha }_{1}-5\right)}}-{T}_{2}\sqrt{\frac{\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}}{{n}_{2}\left({n}_{2}+2{\alpha }_{2}-5\right)}}|\ge |{\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}|\right)$ (14)

To establish that the Bayesian measure of evidence of Yin (2012) solves the paradox in Lindley (1957) when a Gamma prior is assigned to the nuisance parameters, we need to show that

$\underset{\left({n}_{1},{n}_{2}\right)\to \left(\infty ,\infty \right)}{\mathrm{lim}}{P}_{Ga}^{BF}\left(x\right)=0$ (15)

Recall that

$\begin{array}{c}{P}^{B}\left(x\right)=P\left(|\theta -E\left(\theta |x\right)|\ge |{\theta }_{0}-E\left(\theta |x\right)||x\right)\\ =2\left[1-P\left(Z<{Z}_{0}\right)\right]\\ =2\left[1-P\left({\chi }_{1}^{2}<{Z}_{0}^{2}\right)\right]\end{array}$

where Z is a standard normal random variable. Now, under the Gamma prior, we have that

$\begin{array}{l}{Z}_{0}^{2}=\frac{{\theta }_{0}-E\left(\theta |x\right)}{\sqrt{Var\left(\theta |x\right)}}\\ {Z}_{0}^{2}=\frac{{n}_{1}\left({n}_{1}+2{\alpha }_{1}-7\right){n}_{2}\left({n}_{2}+2{\alpha }_{2}-7\right){\left[{\theta }_{0}-\left({\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}\right)\right]}^{2}}{{n}_{2}\left({n}_{2}+2{\alpha }_{2}-7\right)\left[\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}\right]+{n}_{1}\left({n}_{1}+2{\alpha }_{1}-7\right)\left[\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}\right]}\end{array}$ (16)

Then, it can easily be shown that

$\underset{\left({n}_{1},{n}_{2}\right)\to \left(\infty ,\infty \right)}{\mathrm{lim}}{Z}_{0}^{2}=\infty$ (17)

which implies that (15) holds. To show that (17) holds, we now need to show that

$\underset{{n}_{1}\to \infty }{\mathrm{lim}}\left[\underset{{n}_{2}\to \infty }{\mathrm{lim}}{Z}_{0}^{2}\right]=\underset{{n}_{2}\to \infty }{\mathrm{lim}}\left[\underset{{n}_{1}\to \infty }{\mathrm{lim}}{Z}_{0}^{2}\right]=\infty$ (18)

Let ${f}_{1}={\mathrm{lim}}_{{n}_{1}\to \infty }\left[{\mathrm{lim}}_{{n}_{2}\to \infty }{Z}_{0}^{2}\right]$ , ${a}_{1}=2{\alpha }_{1}-7$ , ${a}_{2}=2{\alpha }_{2}-7$ and $A={\left[{\theta }_{0}-\left({\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}\right)\right]}^{2}$ then we have that

$\begin{array}{l}\underset{{n}_{2}\to \infty }{\mathrm{lim}}{Z}_{0}^{2}=\underset{{n}_{2}\to \infty }{\mathrm{lim}}\left[\frac{\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)A}{\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)\left[\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}\right]+\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left[\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}\right]}\right]\\ =\underset{{n}_{2}\to \infty }{\mathrm{lim}}\left[\frac{{n}_{2}^{2}\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left(1+\frac{{a}_{2}}{{n}_{2}}\right)A}{{n}_{2}^{2}\left(1+\frac{{a}_{2}}{{n}_{2}}\right)\left[\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}\right]+{n}_{2}\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left[\left(1-\frac{1}{{n}_{2}}\right){S}_{2}^{2}+\frac{2{\beta }_{2}}{{n}_{2}}\right]}\right]\\ =\underset{{n}_{2}\to \infty }{\mathrm{lim}}\left[\frac{\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left(1+\frac{{a}_{2}}{{n}_{2}}\right)A}{\left(1+\frac{{a}_{2}}{{n}_{2}}\right)\left[\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}\right]+{n}_{2}^{-1}\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left[\left(1-\frac{1}{{n}_{2}}\right){S}_{2}^{2}+\frac{2{\beta }_{2}}{{n}_{2}}\right]}\right]\\ =\frac{\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)A}{\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}}\end{array}$

${f}_{1}=\underset{{n}_{1}\to \infty }{\mathrm{lim}}\left[\frac{\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)A}{\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}}\right]=\underset{{n}_{1}\to \infty }{\mathrm{lim}}\left[\frac{\left({n}_{1}+{a}_{1}\right)A}{\left(1-\frac{1}{{n}_{1}}\right){S}_{1}^{2}+\frac{2{\beta }_{1}}{{n}_{1}}}\right]=\infty$ (19)

Secondly, let ${f}_{2}={\mathrm{lim}}_{{n}_{2}\to \infty }\left[{\mathrm{lim}}_{{n}_{1}\to \infty }{Z}_{0}^{2}\right]$ , then we have that

$\begin{array}{l}\underset{{n}_{1}\to \infty }{\mathrm{lim}}{Z}_{0}^{2}=\underset{{n}_{1}\to \infty }{\mathrm{lim}}\left[\frac{\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)A}{\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)\left[\left({n}_{1}-1\right){S}_{1}^{2}+2{\beta }_{1}\right]+\left({n}_{1}^{2}+{n}_{1}{a}_{1}\right)\left[\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}\right]}\right]\\ =\underset{{n}_{1}\to \infty }{\mathrm{lim}}\left[\frac{{n}_{1}^{2}\left(1+\frac{{a}_{1}}{{n}_{1}}\right)\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)A}{\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right){n}_{1}\left[\left(1-\frac{1}{{n}_{1}}\right){S}_{1}^{2}+\frac{2{\beta }_{1}}{{n}_{1}}\right]+{n}_{1}^{2}\left(1+\frac{{a}_{1}}{{n}_{1}}\right)\left[\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}\right]}\right]\end{array}$

$=\underset{{n}_{1}\to \infty }{\mathrm{lim}}\left[\frac{\left(1+\frac{{a}_{1}}{{n}_{1}}\right)\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)A}{{n}_{1}^{-1}\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)\left[\left(1-\frac{1}{{n}_{1}}\right){S}_{1}^{2}+\frac{2{\beta }_{1}}{{n}_{1}}\right]+\left(1+\frac{{a}_{1}}{{n}_{1}}\right)\left[\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}\right]}\right]$

$\underset{{n}_{1}\to \infty }{\mathrm{lim}}{Z}_{0}^{2}=\frac{\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)A}{\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}}$

${f}_{2}=\underset{{n}_{2}\to \infty }{\mathrm{lim}}\left[\frac{\left({n}_{2}^{2}+{n}_{2}{a}_{2}\right)A}{\left({n}_{2}-1\right){S}_{2}^{2}+2{\beta }_{2}}\right]=\underset{{n}_{2}\to \infty }{\mathrm{lim}}\left[\frac{\left({n}_{2}+{a}_{2}\right)A}{\left(1-\frac{1}{{n}_{2}}\right){S}_{2}^{2}+\frac{2{\beta }_{2}}{{n}_{2}}}\right]=\infty$ (20)

Since from (19) and (20) we have that ${f}_{1}={f}_{2}=\infty$ , it has been shown that the Bayesian measure of evidence of Yin (2012) under the Gamma prior solves the paradox in Lindley  .

Consequently, since it can be easily seen from (13) that the posterior distribution of $\theta$ is symmetric about its expected value, $E\left(\theta |x\right)={\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}$ , then Theorem 2 of Yin and Li  applies here. This implies that under the Gamma Prior, the Bayesian measure of evidence of Yin  yields the $1-\alpha$ credible intervals for $\theta ={\mu }_{1}-{\mu }_{2}$ centered at ${\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}$ .

Lemma 1. Let Gamma Priors be assigned to the precisions ${\tau }_{1}$ and ${\tau }_{2}$ . Then, for values of ${\beta }_{1}\ll 1,{\beta }_{2}\ll 1$ and ${\alpha }_{1}={\alpha }_{2}=2$ , the Posterior distribution of $\theta$ , denoted by $\theta |x$ is the same as the Posterior distribution under Jeffreys’ independent prior given by $\pi \left({\mu }_{1},{\mu }_{2},{\sigma }_{1}^{2},{\sigma }_{2}^{2}\right)\propto {\sigma }_{1}^{-2}{\sigma }_{2}^{-2}$

Proof. By considering the values of ${\beta }_{1}$ and ${\beta }_{2}$ that satisfy ${\beta }_{1}\ll 1$ and ${\beta }_{2}\ll 1$ , we can safely assume that

$\frac{{\beta }_{1}}{\left({n}_{1}-1\right){S}_{1}^{2}}\approx 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\beta }_{2}}{\left({n}_{2}-1\right){S}_{2}^{2}}\approx 0$

especially where ${S}_{i}^{2}\gg 1$ , and ${n}_{i}$ is sufficiently large, $\forall i=1,2$ . Then by setting ${\alpha }_{1}={\alpha }_{2}=2$ ,we have from (10) that

$\begin{array}{c}{\pi }^{Ga}\left({\mu }_{1},{\mu }_{2}|x\right)\propto {\left[1+\frac{{n}_{1}{\left({\stackrel{¯}{x}}_{1}-{\mu }_{1}\right)}^{2}}{\left({n}_{1}-1\right){S}_{1}^{2}}\right]}^{-{n}_{1}/2}{\left[1+\frac{{n}_{2}{\left({\stackrel{¯}{x}}_{2}-{\mu }_{2}\right)}^{2}}{\left({n}_{2}-1\right){S}_{2}^{2}}\right]}^{-{n}_{2}/2}\\ \propto {\left[1+\frac{{t}_{1}^{2}}{{n}_{1}-1}\right]}^{-{n}_{1}/2}{\left[1+\frac{{t}_{2}^{2}}{{n}_{2}-1}\right]}^{-{n}_{2}/2}\end{array}$ (21)

which is the kernel of the joint distribution of two independent t random variables with ${t}_{1}$ having ${n}_{1}-1$ degrees of freedom and ${t}_{2}$ having ${n}_{2}-1$ degrees of freedom respectively. The nit can be easily seen that

${\mu }_{i}={\stackrel{¯}{x}}_{i}-\frac{{t}_{i}{S}_{i}}{\sqrt{{n}_{i}}},i=1,2.$

and consequently,

$\theta |x~{\stackrel{¯}{x}}_{1}-{\stackrel{¯}{x}}_{2}-\left(\frac{{S}_{1}{T}_{1}}{\sqrt{{n}_{1}}}-\frac{{S}_{2}{T}_{2}}{\sqrt{{n}_{2}}}\right)$ (22)

where ${T}_{i}$ is a t random variable with ${n}_{i}-1$ degrees of freedom.+

Lemma 1 shows that the posterior distribution of the difference in means under Jeffreys’ independent prior is a special case of the posterior distribution of the difference in means under the Gamma prior.

4. Simulation Results and Discussion

For the purpose of this discussion, we shall refer to the methodology of Yin  as the New Bayesian measure of evidence. The Metropolis-Hastings algorithm was used for the simulation with a thinning length of 12. The values in Table 1 were obtained by fixing the following values: ${\alpha }_{1}=1.5,{\alpha }_{2}=2.0,{\beta }_{1}=0.5,{\beta }_{2}=0.5,{S}_{1}^{2}=11,{S}_{2}^{2}=14$ . These results reveal that for the different sample sizes, whether large or small, equal or unequal, the conclusions of a hypothesis test based on either the Generalized p-value, the Posterior Predictive p-value, the New Bayesian measure of evidence under the objective prior, or the New Bayesian measure of evidence under the Gamma prior are in the same direction. However, the New Bayesian measure of evidence under the Gamma prior gives consistently smaller evidence against the null hypothesis, whether the sample sizes are equal or unequal except for large sample sizes where the new Bayesian measure of evidence gives stronger evidence against the null compared to the Posterior Predictive P-value.

On the other hand, the values in Table 2 were obtained by fixing the following values: ${\alpha }_{1}=2.0,{\alpha }_{2}=2.0,{S}_{1}^{2}=20,{S}_{2}^{2}=25$ . The values in this table reflect the accuracy of the approximation of the new Bayesian measure of evidence under Jeffreys’ independent prior to the new Bayesian measure of evidence under the Gamma prior. Results here show that when the sample sizes are at least 30, the approximation seems to be good and the values of the ${{\beta }^{\prime }}_{i}s$ do not need to be far less than 1. The approximation is good only contingent on the fact that the values of the ${{\beta }^{\prime }}_{i}s$ are less than 1.

Thirdly, the values in Table 3 were obtained by fixing the following values: ${\alpha }_{1}=2.0,{\alpha }_{2}=2.0,{S}_{1}^{2}=3,{S}_{2}^{2}=5$ . The values in this table also reflect for smaller variances, the accuracy of the approximation of the new Bayesian measure of evidence under Jeffreys’ independent prior by the new Bayesian measure of evidence under the Gamma prior. In a similar manner, results here show that the approximation is equally good for smaller variances. In fact, the approximation is good where samples sizes can be at least as large as 10 so long as the values of the ${{\beta }^{\prime }}_{i}s$ are considerably less than 1. Note that the parameter values are fixed to demonstrate the behaviour of the conclusion from the New Bayesian measure of evidence under different circumstances like when the sample variances are small or moderate or large. Also, in Table 2, the values were fixed to see how well the New Bayesian measure of evidence under Jeffreys’ prior can be approximated by the New Bayesian measure of evidence under the Gamma prior.

Table 1. The four different probability values for different values of n1 and n2.

Table 2. The four different probability values for different values of ${\alpha }_{1},{\alpha }_{2},{\beta }_{1}$ and ${\beta }_{2}$ .

Table 3. The four different probability values for different values of ${\alpha }_{1},{\alpha }_{2},{\beta }_{1}$ and ${\beta }_{2}$ .

Finally, Lehmann’s data on measures of driving times from following two different routes and Sahu’s data on scores of surgical and non-surgical treatments both displayed as Table 1 and Table 2 respectively in Yin and Li  were used as real examples to demonstrate the performance of the four measures of evidence (results not shown). All conclusions were in the same direction for all four measures of evidence against the null hypothesis.

5. Conclusions

In this paper, we looked at the Bayesian analysis of the Behrens-Fisher problem using the methodology of Yin  by assigning Gamma Priors to the two unknown variances. We were able to show analytically, that the Bayesian measure of evidence of Yin  solves simultaneously, the Behrens-Fisher problem and Lindley’s paradox when Gamma Priors are assigned to the unknown variances. In fact, we were able to show that the solution obtained by Yin and Li  is a special case of ours, where Gamma Prior is used instead of Jeffreys’ independent prior.

Simulation results further confirm the fact that extending the methodology of Yin  while assigning Gamma Prior to each of the nuisance parameters also solves Lindley’s paradox. This implies that the prowess of the methodology of Yin  does not only lie in the use of noninformative priors. In fact, simulation results reveal that for large sample sizes, the measure of evidence against the null hypothesis is stronger when the nuisance parameters are assigned Gamma Priors with carefully selected parameter values.

Cite this paper
Goltong, N. and Doguwa, S. (2018) Bayesian Analysis of the Behrens-Fisher Problem under a Gamma Prior. Open Journal of Statistics, 8, 902-914. doi: 10.4236/ojs.2018.86060.
References
   Yin, Y. (2012) A New Bayesian Procedure for Testing Point Null Hypothesis. Computational Statistics, 27, 237-249.
https://doi.org/10.1007/s00180-011-0252-6

   Yin, Y. and Li, B. (2014) Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence. Journal of Applied Mathematics, 2014, Article ID: 978691.
https://doi.org/10.1155/2014/978691

   Lindley, D.V. (1957) A Statistical Paradox. Biometrika, 44, 187-192.
https://doi.org/10.1093/biomet/44.1-2.187

   Spanos, A. (2013) Who Should Be Afraid of the Jeffreys-Lindley Paradox? Philosophy of Science, 80, 73-93.
https://doi.org/10.1086/668875

   Robert, C.P. (2014) On the Jeffreys-Lindley’s Paradox. Philosophy of Science, 81, 216-232.
https://doi.org/10.1086/675729

   Scheffe, H. (1944) A Note on the Behrens-Fisher Problem. Annals of Mathematical Statistics, 15, 430-432.
https://doi.org/10.1214/aoms/1177731214

   Fraser, D.A.S. and Streit, F. (1972) On the Behrens-Fisher Problem. Australian Journal of Statistics, 14, 167-171.
https://doi.org/10.1111/j.1467-842X.1972.tb00354.x

   Robinson, G.K. (1976) Properties of Students t and of the Behrens-Fisher Solution to the Two Means Problem. The Annals of Statistics, 4, 963-971.
https://doi.org/10.1214/aos/1176343594

   Tsui, K.-W. and Weerahandi, S. (1989) Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters. Journal of American Statistical Association, 84, 602-607.
https://doi.org/10.2307/2289949

   Zheng, S., Shi, N.-Z. and Ma, W. (2009) Statistical Inference on the Difference or Ratio of Means from Heteroscedastic Normal Populations. Journal of Statistical Planning and Inference, 140, 1236-1242.
https://doi.org/10.1016/j.jspi.2009.11.010

   Ozkip, E., Yazici, B. and Sezer, A. (2014) A Simulation Study on Tests for the Behrens-Fisher Problem. Turkiye Klinikleri Journal of Biostatistics, 6, 59-66.

   Degroot, M.H. (1982) Comment. Journal of the American Statistical Association, 77, 336-339.
https://doi.org/10.1080/01621459.1982.10477811

   Berger, J.O. and Sellke, T. (1987) Testing a Point null Hypothesis: The Irreconcilability of p-Values and Evidence. Journal of the American Statistical Association, 82, 112-122.
https://doi.org/10.2307/2289131

   Casella, G. and Berger, R.L. (1987) Reconciling Bayesian and Frequentist Evidence in One Sided Testing Problem. Journal of the American Statistical Association, 82, 106-111.
https://doi.org/10.1080/01621459.1987.10478396

   Berger, J.O. and Delampady, M. (1987) Testing Precise Hypotheses. Statistical Science, 2, 317-352.
https://doi.org/10.1214/ss/1177013238

   Meng, X.-L. (1994) Posterior Predictive p-Values. The Annals of Statistics, 22, 1142-1160.
https://doi.org/10.1214/aos/1176325622

   Ghosh, M. and Kim, Y.-H. (2001) The Behrens-Fisher Problem Revisited: A Bayes-Frequentist Synthesis. The Canadian Journal of Statistics, 29, 5-17.
https://doi.org/10.2307/3316047

   Yin, Y. (2012) A New Bayesian Procedure for Testing Point Null Hypothesis. Computational Statistics, 27, 237-249.
https://doi.org/10.1007/s00180-011-0252-6

   Yin, Y. and Li, B. (2014) Analysis of the Behrens-Fisher Problem Based on Bayesian Evidence. Journal of Applied Mathematics, 2014, Article ID: 978691.
https://doi.org/10.1155/2014/978691

   Lindley, D.V. (1957) A Statistical Paradox. Biometrika, 44, 187-192.
https://doi.org/10.1093/biomet/44.1-2.187

   Spanos, A. (2013) Who Should Be Afraid of the Jeffreys-Lindley Paradox? Philosophy of Science, 80, 73-93.
https://doi.org/10.1086/668875

   Robert, C.P. (2014) On the Jeffreys-Lindley’s Paradox. Philosophy of Science, 81, 216-232.
https://doi.org/10.1086/675729

   Scheffe, H. (1944) A Note on the Behrens-Fisher Problem. Annals of Mathematical Statistics, 15, 430-432.
https://doi.org/10.1214/aoms/1177731214

   Fraser, D.A.S. and Streit, F. (1972) On the Behrens-Fisher Problem. Australian Journal of Statistics, 14, 167-171.
https://doi.org/10.1111/j.1467-842X.1972.tb00354.x

   Robinson, G.K. (1976) Properties of Students t and of the Behrens-Fisher Solution to the Two Means Problem. The Annals of Statistics, 4, 963-971.
https://doi.org/10.1214/aos/1176343594

   Tsui, K.-W. and Weerahandi, S. (1989) Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters. Journal of American Statistical Association, 84, 602-607.
https://doi.org/10.2307/2289949

   Zheng, S., Shi, N.-Z. and Ma, W. (2009) Statistical Inference on the Difference or Ratio of Means from Heteroscedastic Normal Populations. Journal of Statistical Planning and Inference, 140, 1236-1242.
https://doi.org/10.1016/j.jspi.2009.11.010

   Ozkip, E., Yazici, B. and Sezer, A. (2014) A Simulation Study on Tests for the Behrens-Fisher Problem. Turkiye Klinikleri Journal of Biostatistics, 6, 59-66.

   Degroot, M.H. (1982) Comment. Journal of the American Statistical Association, 77, 336-339.
https://doi.org/10.1080/01621459.1982.10477811

   Berger, J.O. and Sellke, T. (1987) Testing a Point null Hypothesis: The Irreconcilability of p-Values and Evidence. Journal of the American Statistical Association, 82, 112-122.
https://doi.org/10.2307/2289131

   Casella, G. and Berger, R.L. (1987) Reconciling Bayesian and Frequentist Evidence in One Sided Testing Problem. Journal of the American Statistical Association, 82, 106-111.
https://doi.org/10.1080/01621459.1987.10478396

   Berger, J.O. and Delampady, M. (1987) Testing Precise Hypotheses. Statistical Science, 2, 317-352.
https://doi.org/10.1214/ss/1177013238

   Meng, X.-L. (1994) Posterior Predictive p-Values. The Annals of Statistics, 22, 1142-1160.
https://doi.org/10.1214/aos/1176325622

   Ghosh, M. and Kim, Y.-H. (2001) The Behrens-Fisher Problem Revisited: A Bayes-Frequentist Synthesis. The Canadian Journal of Statistics, 29, 5-17.
https://doi.org/10.2307/3316047

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