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I read on Wolfram - Student's T-distribution that the Student's t-distribution "is defined as the distribution of the random variable $t$ which is (very loosely) the "best" that we can do not knowing $\sigma$.

I am wondering what this formally means and what criterion is used to gauge what is "best"? More directly, I guess I am asking how the student's t distribution is derived and with what criterion is used to derive it? Any insight would be greatly helpful thanks!

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The statement is too vague. If the samples are normally distributed and you are trying to test a hypothesis about the mean the t statistic provides the best test statistic when the variance is unknown. The link you have on Wolfram shows you what the density looks like along with some of its properties. It has a finite variance when df is 3 or greater. The t test is the likelihood ratio test for the mean of a normal distribution when the mean and variance are unknown. That makes it the most powerful test when testing that the mean is $\mu_0$ versus the simple alternative that the mean is $\mu_1$ different from $\mu_0$ by the Neyman-Pearson lemma. Often $\mu_0$ is taken to be 0.

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  • $\begingroup$ I see, so the notion of being best comes from the likelihood ratio test and NP lemma. Thanks! $\endgroup$ – user123276 Jun 28 '17 at 22:27

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