Confidence Interval on the Difference in Means, Variances Unknown and not Assumed Equal

How can I prove that $$T=\dfrac{\overline{X}_1-\overline{X}_2-(\mu_1-\mu_2)}{\sqrt{\dfrac{S_1^2}{n_1}+\dfrac{S_2^2}{n_2}}}$$ has a $t$ distribution?

Assumptions:

1. $X_{11},\ldots,X_{1n_{1}}$ is a random sample from population 1.
2. $X_{21},\ldots,X_{2n_{2}}$ is a random sample from population 2.
3. The two populations represented by $X1$ and $X2$ are independent.
4. $\sigma_{1}$, $\sigma_{2}$ unknown, and $\sigma_1 \neq \sigma_2$.
5. Both populations are normal.

What I know:

• $Z=\dfrac{\overline{X}_1-\overline{X}_2-(\mu_1-\mu_2)}{\sqrt{\dfrac{\sigma_1^2}{n_1}+\dfrac{\sigma_2^2}{n_2}}} \sim N(0,1)$.
• $U=\dfrac{(n_1-1)S_1^2}{\sigma_1^2}+ \dfrac{(n_2-1)S_2^2}{\sigma_2^2} \sim \chi^{2}_{n_1+n_2-2}$
• If $Z \sim N(0,1)$, $U \sim \chi_{n_1+n_2-2}^{2}$ and $Z$ and $U$ are independent, then

$$\dfrac{Z}{\sqrt{\dfrac{U}{n_1+n_2-2}}} \sim t_{n_1+n_2-2}$$

My attempt: I have formed the quotient

$$\dfrac{\dfrac{\overline{X_{1}}-\overline{X_{2}}-(\mu_{1}-\mu_{2})}{\sqrt{\dfrac{\sigma_{1}^{2}}{n_{1}}+\dfrac{\sigma_{2}^{2}}{n_{2}}}}}{\sqrt{\left[\dfrac{(n_{1}-1)S_{1}^{2}}{\sigma_{1}^{2}}+ \dfrac{(n_{2}-1)S_{1}^{2}}{\sigma_{2}^{2}}\right]\dfrac{1}{n_{1}+n_{2}-2}}} \sim t_{n_1+n_2-2}$$

The problem with which I found is that unknown variances can not be eliminated as in the case $\sigma_{1}^{2}=\sigma_{2}^{2}=\sigma^{2}$

• Where you wrote $X_{11},\ldots,X_{1n}$ and $X_{21},\ldots,X_{2n},$ did you intend $X_{11},\ldots,X_{1n_1}$ and $X_{21},\ldots,X_{2n_2} \text{ ?} \qquad$ – Michael Hardy May 25 '18 at 19:13
• @MichaelHardy I already corrected $n_1$ and $n_2$ – Jacob S. May 25 '18 at 19:45

I don't think the proposed theorem is true.

Google the term "Behrens–Fisher problem". The Wikipedia article on this is woefully incomplete, making it look as if one proposed way to model the Behrens–Fisher problem is the Behrens–Fisher problem. But use Google Scholar and Google Books.

• A consideration of the difficulty encountered when the unknown variances of the two normal distributions are not equal is assigned to one of the exercises. – Jacob S. May 25 '18 at 19:50
• 5.4.21 Discuss the problem of finding a confidence interval for the difference $\mu_{1}-\mu_{2}$ between the two means of two normal distributions if the variances $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$ are known but not necessarily equal. – Jacob S. May 25 '18 at 19:56
• 5.4.22. Discuss Exercise 5.4.21 when it is assumed that the variances are unknown and unequal. This is a very difficult problem, and the discussion should point out exactly where the difficulty lies. – Jacob S. May 25 '18 at 19:58
• From the book "Introduction to Mathematical Statistics", Robert V. Hogg – Jacob S. May 25 '18 at 20:02
• Dear Michael, I'd like to discuss with you about the Behrens-Fisher problem. I sent you a Linkedin invitation with my private email address. Looking forward to hearing from you. Best regards. – Fabrice Pautot May 13 '20 at 22:43