Uses and estimation of Student-t distribution

Question

My question relates to a confusion between how the Student-t distribution is often documented versus how it is used.

In the documentation the Student-t is used (from Wikipedia):

when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown

As per this definition, the degrees of freedom parameters is equal to n-1 where n = number of observations.

My confusion comes from the fact that the Student-t is often used to model not the difference between the true mean of a population and the sample mean, but to estimate the distribution of the population itself. As such, is this a misuse of the Student-t distribution? If not:

• The degrees of freedom cannot be defined as n-1 as there is no notion that as the sample size increases the sample mean converges to the population mean. Rather, it is posited that the population is distributed as a t(v) no matter what the sample size, i.e. as n increases it doesn't converge to a normal. Hence, how is the degrees of freedom chosen/estimated in this case? The only answer is I can come up with is iteratively trying different values and selecting the value with the best model fit.
• Similarly, when standardising and un-standardising the practice of multiplying by the square root of n shouldn't be necessary as the underlying assumption of that is that as n increases the difference in mean will go to zero. However, if you are using the Student-t to estimate the population distribution that is not the purpose. As such, I would assume you would standardise and un-standardise without the square root of n part. Is that correct?

E.g. the below is how the t-stat is estimated, however this method isn't valid if one is not estimating the t-stat, i.e. testing if the difference is zero $$\frac{X −μ}{(σ/√n)} \thicksim t(v)$$

Similarly, I have seen other people use the following to un-standardise: $$t*\frac{\sigma}{√(v/(v-2))} + \mu$$ Where v is the degrees of freedom here

Apologies if this is a silly question, the answer me be very obvious. I ask this as I often see the degrees of freedom for Student-t distributions being chosen just as if out of a hat and different ways of un-standardising samples from the Student-t, which don't seem to make sense given the purpose/context.

Answer 1

when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown

This is simply one application of the t-distribution.

The t-distribution can be defined in terms of its density or its distribution function, (or in other ways - say in terms of its characteristic function); in any case, however defined it has a particular functional form that includes a parameter that defines its shape, which we call the degrees of freedom parameter. In the general case there will also be location and scale parameters.

The t-distribution can - like any other distribution - be used for any situation for which it is deemed a suitable model.

As per this definition, the degrees of freedom parameters is equal to n-1 where n = number of observations.

I would not call that a definition of the t-distribution. It describes a circumstance that results in a t-distribution for a particular statistic (which is why it's an application of the t).

My confusion comes from the fact that the Student-t is often used to model not the difference between the true mean of a population and the sample mean, but to estimate the distribution of the population itself. As such, is this a misuse of the Student-t distribution?

Your confusion stems from conflating an application with a definition of a distribution. If I buy a box of crayons that has a drawing of a cat on the front of the box, that doesn't mean the crayons must be used for drawing cats -- I'm still allowed to draw trees with it!

Hence, how is the degrees of freedom chosen/estimated in this case?

The same ways that parameters are estimated, or even chosen more generally.

So you might choose to optimize some criterion for example (e.g. maximum likelihood), or you might match moments, or any number of other ways in which parameters are arrived at.

The only answer is I can come up with is iteratively trying different values and selecting the value with the best model fit.

Well once you have a criterion (you need something that defines what 'better' is in order to select 'best'), that could work but it would be slow. You're unfamiliar with optimization methods?

Similarly, when standardising and un-standardising the practice of multiplying by the square root of n shouldn't be necessary as the underlying assumption of that is that as n increases the difference in mean will go to zero. However, if you are using the Student-t to estimate the population distribution that is not the purpose. As such, I would assume you would standardise and un-standardise with the square root of n part. Is that correct?

Again, you're conflating the application with the tool.

Answer 2

The block quote is incorrect. the t-distribution is used to make inference (pvalues, CIs) about the mean of a normal population, not estimate the mean directly. the point estimate of the mean is the sample mean.

to answer your questions, the t-distribution is a probability distribution indexed by the parameter called the degrees of freedom and the t-distribution can be used for other things, not just the one sample t-test where the degrees of freedom equals n-1. Have a look at the wikipedia page about the t-distribution.