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The question is easy but it is really hard to find via Google.

Say you have the following hypothesis:

$H0: \mu = 0$

$Ha: \mu \neq 0 $

Now I know that $ \mu $ is called the population mean. But how do you call a variable (or value) that is hypothesized with in general? Because instead of $\mu$ I could also test for a different variable (e.g. $\sigma = 0$ vs. $\sigma \neq 0$).

It's also not the "statistic" (e.g. t-statistic).

So how do you call it?

Melvin Roest
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    Maybe the "population quantity"?! – user103828 Mar 25 '14 at 10:38
  • Really? Are you implying that there is no word for it? Because I could imagine that you could also make a H0 and Ha about p-values or alpha values. Or something not about the population, but which would still be used for hypothesis testing (but now I am going on a limb). – Melvin Roest Mar 25 '14 at 10:48
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    parameter (4 more to go...) – Jimmy R. Mar 25 '14 at 11:19

1 Answers1

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As Stafanos said, it is usually a "population parameter" that you are testing. However, if you are performing a nonparametric hypothesis test, then its more appropriately, and perhaps most generally, called a "population statistic", as this term will apply to any underlying distribution.

So what's the difference between a population statistic and population parameter?

  1. Population statistic: any value that can be can be calculated from the CDF of the distribution (or derivatives thereof). For nonparametric distributions, this is a purely theoretical notion, as you usually cannot write down the distribution in closed form. However, operationally, the above is equivalent to saying that it can be approximated as a function of the empirical distribution function or the underlying data, such that it converged to the true value. Essentially, any consistent function of the data can be viewed as an approximation of an underlying population statistic.

  2. Population parameter: This is a type of population statistic that is applicable to distirbutions that can be completely described by a function characterized by a finite number of parameters.

Most of the time, you are testing a population parameter, but sometimes you care more about population statistics (such as testing equality of variance in non-normal data).