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I would also be interested to know why it is one or the other.

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up vote 13 down vote accepted

A p-value is a random variable, so it's not a population parameter.

You could certainly argue that it's a statistic:

A statistic (singular) is a single measure of some attribute of a sample (e.g., its arithmetic mean value). It is calculated by applying a function (statistical algorithm) to the values of the items of the sample, which are known together as a set of data.

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Could you argue that it isn't? – Scortchi Apr 25 '14 at 15:43
@Scortchi Someone might argue that it isn't usually thought of in those terms. I'd say that I wouldn't think of it as an estimator*, but that it is a sample statistic. *(at least not one I'd care to use for anything in general; in particular instances perhaps) – Glen_b Apr 25 '14 at 23:04
Actually, on reflection while strictly speaking any sample statistic could be called an estimator of any population quantity (though many are terrible for that purpose), I don't think it can be a reasonable estimator of anything useful. There's no "true p-value" that it converges to for example. – Glen_b Jul 3 at 23:40

A $p$-value is the probability of observing a test static value as or more extreme than the test statistic created from one's data if the null hypothesis is true. You can therefore interpret the $p$-value as a measure of how extreme your test statistic is under H$_{0}$ and the probability distribution attached to H$_{0}$. The $p$-value is therefore a statistic that is a function of one's data, and one's choice of H$_{0}$.

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If the test statistic can be called a statistic, then so must the $p$-value: the test statistic is a function of the data under the assumption that the null hypothesis is true. The $p$-value is simply a probability associated with that test statistic.

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