Why is random variate T of student's t-distribution defined the way it is?

Question

I recently learned about t-distribution and I didn't find it very obvious as to why $T$ is defined as $T=\frac{W}{\sqrt{V/r}}$, where W is a standard normal variate and $V$ follows a chi-sqaure distribution. What lead to this definition of T ?

Answer 1

As stated in the link given by drhab, it is a law which is often encountered while making statistical inference.

For example, let's take a sample $(X_{1},X_{2},\dots,X_{n})$ of $n$ observations where the $X_{i}$'s are independent and identically distributed.

We consider the case where $X_{i}\sim\mathcal{N}(\mu,\sigma^{2})$ for all $i=1,\dots,n$. We want to test the following hypotheses (at level $\alpha$):

$$\begin{cases} \mathcal{H}_{0}: &\mu &= &\mu_{0}\\ \mathcal{H}_{1}: &\mu &\neq &\mu_{0} \end{cases}$$

And suppose that we ignore the theoretical variance of the sample.

The sample mean is given by

$$\overline{X}=\frac{\sum_{i=1}^{n}X_{i}}{n}$$

and it is a well-known fact that

$$\overline{X}\sim\mathcal{N}\left(\mu,\frac{\sigma^{2}}{n}\right)$$

The sample variance is given by $s=\frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\overline{X})^{2}$ and we can define $S=(n-1)s/\sigma^{2}$, so that:

$$S=\sum_{i=1}^{n}\left(\frac{X_{i}-\overline{X}}{\sigma}\right)^{2}$$

It is also a well-known fact that $S$ and $\overline{X}$ are independent and that

$$S\sim\chi^{2}_{n-1}$$

Let's define the following statistic:

$$T_{0}=\frac{\sqrt{n}\left(\overline{X}-\mu_{0}\right)}{\sigma\sqrt{S/(n-1)}}$$

One can prove that the so-called $t$-test is equivalent to the maximum likelihood test (which is the most powerful to test the previous hypotheses). This $t$-test rejects $\mathcal{H}_{0}$, at level $\alpha$, when $$|T_{0}|>t^{n-1}_{1-\alpha/2}$$ where $t^{n-1}$ is the law of $T_{0}$ under $\mathcal{H}_{0}$.

When you look at the statistic $T_{0}$, we have $$\begin{align} \frac{\sqrt{n}\left(\overline{X}-\mu_{0}\right)}{\sigma} &\sim \mathcal{N}(0,1)\\ S &\sim \chi_{n-1}^{2} \end{align}$$ And it motivated the definition of the Student's $t$-distribution $$t^{r}\sim \frac{Z}{\sqrt{U/r}}$$ where $Z\sim\mathcal{N}(0,1)$ and $U\sim\chi_{r}^{2}$ are independent.

It is useful to see that $(t^{r})^{2}\stackrel{\mathcal{L}}{=}\mathcal{F}_{1,n-1}$ in order to generalize this test to the multivariate situation.