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If a variable has a normal distribution, we say it is normally distributed. If it has an exponential distribution, we say it is exponentially distributed.

What is the proper way to reference a variable with a Student's $t$-distribution?

For example, how would one complete this sentence: "... and we consider the observations to be <> random variables"?

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    $\begingroup$ $t$-distributed maybe? Hey, the difference in the number of characters between $t$-distributed and exponentially is statistically insignificant! $\endgroup$ – Dilip Sarwate Sep 3 '15 at 13:23
  • $\begingroup$ @DilipSarwate, shouldn't we be comparing t-distributed with exponentially distributed instead? The number of characters differs more then. $\endgroup$ – Richard Hardy Sep 3 '15 at 13:41
  • $\begingroup$ @RichardHardy No, exponentially distributed is a misnomer and a waste of characters. We should all be using $e$-distributed and $n$-distributed and $g$-distributed etc in the interest of brevity (and the 600 character limitation on comments enforced by stackexchange etc)...... $\endgroup$ – Dilip Sarwate Sep 3 '15 at 13:49
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    $\begingroup$ ... and why not say normal random variable and exponential random variable and Rayleigh random variable and Cauchy random variable and Gamma random variable instead of the mouthful of "$X$ is a normally distributed random variable" etc? $\endgroup$ – Dilip Sarwate Sep 3 '15 at 13:53
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    $\begingroup$ It isn't even clear to me why it is $t$-distributed, with the hyphen. It isn't "normally-distributed". Additionally, we can have normal variables. But, there are also no $t$ variables (at least, I don't think I've seen that). $\endgroup$ – pixels Sep 3 '15 at 15:10
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Scortchi's prescriptive answer is very good. From a purely descriptive viewpoint, I have found all of the following on Google scholar, but my conclusion is these forms are relatively unusual, particularly when we compare how often a phrasing with "normally distributed" is used instead of "normal distribution" — in comparison to the relatively small numbers found below, there were tens of thousands of results for both "Student's t distribution" and "Student t distribution".

So actual practice varies widely! I think the most common variant I found overall was "Student-t distributed", and if you were to go with this then your usage would not be unusual. Something I found puzzling was how the location of the hyphen is very inconsistent: while "Student's t-distributed" is more common than "Student's-t distributed" (which does look very unnatural), when the "'s" is omitted I found that "Student-t distributed" was massively more common than "Student t-distributed". It seems that people feel a hyphen is needed somewhere, but do not necessarily agree on where it should go. On the other hand, before "distributed" the "Student t" or "Student's t" is acting as an adverb, and style guides generally declare along the lines that "most adverbial phrases do not need hyphens".

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    $\begingroup$ (+1, grudgingly). If you were to go with "principle component analysis", your usage wouldn't be unusual: 46,900 hits. $\endgroup$ – Scortchi - Reinstate Monica Sep 9 '15 at 16:37
  • $\begingroup$ @Scortchi I thought your answer was stylistically better but my curiosity was piqued as to what forms were "out there". Interestingly pretty much none of these combinations were common enough in the Google Books corpus to show up on Google n-grams. In contrast it is possible to do a decent n-gram of "Student's t distribution" vs "Student t distribution" vs "t-Student distribution" to see trends in their relative prevalence (primarily in textbooks). I'm not convinced that counting papers is a great metric (how many papers does Google index?) but all these forms can be at least be attested. $\endgroup$ – Silverfish Sep 9 '15 at 18:53
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    $\begingroup$ (I'll admit to a bias: personally I'm a descriptivist not a prescriptivist! I try to follow Postel's Law: I make an effort to be conservative with adherence to traditional rules in the way I write, but generally liberal in interpreting English written by others. For me, "English" is what people "out there" actually write, rather than what the grammar books say they should. So my answer should be taken in that vein, I'm not suggesting any of these are great style.) $\endgroup$ – Silverfish Sep 9 '15 at 19:08
  • $\begingroup$ As for "principle component analysis": should "principle" and "principal" eventually appear with equal frequency in such settings, then they will have effectively been rendered synonymous. Sad will be the day, for a useful distinction will be lost. Till then, "principle components" is a minority usage, regarded as mistaken by majority users (and hence best avoided if you wish not to be regarded as mistaken). I don't think "a Student-t distributed random variable" is so bad, but writing "a random variable following a Student's t distribution" isn't so clunky either, and sidesteps the issue. $\endgroup$ – Silverfish Sep 9 '15 at 19:15
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    $\begingroup$ I subscribe to that mode of thinking about language as well, @Silverfish, which is why I marked this one as "correct", though both answers are great! $\endgroup$ – pixels Sep 15 '15 at 12:32
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"Normal" & "exponential" are adjectives, already furnished with the adverbs "normally" & "exponentially". When a noun or noun phrase is pressed into service as an adjective there won't in general be a corresponding adverb to hand. So "Weibull distributed" or "gamma distributed", though you sometimes see them, are already doing some violence to grammar (I'd avoid those). "Student's t distributed" seems even worse; I think because the possessive modifier is supposed to apply to the "t distribution", or if not just because it reminds us that "Student's t" is really a noun phrase. So don't say that.

"Student's t random variable" seems fine—here "t" is being used attributively, as an adjective; & why not? "Has a Student's t distribution" or "follows a Student's t distribution" are other options.

† Even when we've a proper adjective, like "Gaussian", we're still not guaranteed an adverb. "Gaussianly distributed"? No.

‡ I'm uncertain whether "Student's t distribution" is better parsed as "the t distribution, invented by Student", or "the distribution of Student's t statistic".

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    $\begingroup$ Gosset-ian ... makes you cringe. There's no solution. $\endgroup$ – Antoni Parellada Sep 9 '15 at 15:51

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