Asymptotic distribution of independent two-sample t-test

Question

Consider two independent random samples of sizes $n_1$ and $n_2$ ($n_1\neq n_2$ may be the case) on independent random variables $x_1$ and $x_2$. That is, we have one iid sample of size $n_1$ from the distribution that $x_1$ follows, another iid sample of size $n_2$ from the distribution that $x_2$ follows, and these two samples were independently taken. Assume that $x_1$ and $x_2$ have finite fourth moments.

Let $\overline{x}_1=n_j^{-1}\sum_{i=1}^{n_j}x_j$ and $s^2_j=(n_j-1)^{-1}\sum_{i=1}^{n_j}(x_{ji}-\overline{x_j})^2$ for $j=1,2$. Define the test statistic $$t=\frac{\overline{x}_1-\overline{x_2}}{s}$$ where $s=\sqrt{\frac{s^2_1}{n_1}+\frac{s^2_2}{n_2}}$.

Under the null hypothesis $\mathbb{H}_0:\mathbb{E}(x_1)=\mathbb{E}(x_2)$, is $t$ asymptotically standard normal as $n_1$ and $n_2 $ diverges to infinity? If not, when is $t$ asymptotically standard normal? Is there another test statistic that has an asymptotic normal distribution under the null?

I note that $t$ is a function of $\{x_{1i}\}_{i=1}^{n_1}\cup\{x_{2i}\}_{i=1}^{n_2}$.

Compare with the article "Student's t-test" on Wikipedia.

Answer 1

Consider $$W \equiv \frac {\sqrt{n_2}}{n_1} \sum_{i=1}^{n_1}X_{1i} - \frac 1{\sqrt{n_2} }\sum_{i=1}^{n_2}X_{2i}$$

Let $v_1, v_2$ denote the true variances of the two distributions. Given the various assumptions and under the null we have

$$E(W) = 0, \text{Var}(W) = \frac{n_2}{n_1}v_1 + v_2$$

Assume in addition that $n_2/n_1 \to c < \infty,\;\;\; n_1, n_2 \to \infty$

Again given the assumptions, as $n_1, n_2 \to \infty$

$$\frac {W}{\sqrt{\frac{n_2}{n_1}v_1+v_2}} \to_p \frac {W}{\sqrt{cv_1+v_2}}\to_d N(0,1)$$

More over, by the assumptions and Slutsky's theorem

$$\frac {W}{\sqrt{\frac{n_2}{n_1}s_1+s_2}} - \frac {W}{\sqrt{cv_1+v_2}} \to_p 0$$