From Wikipedia

In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters whose probability distribution does not depend on the unknown parameters 1 (also referred to as nuisance parameters). Note that a

pivot quantity need not be a statistic—the function and its value can depend on the parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic.

I don't understand why a pivot quantity may not be a statistic?

How is a statistic defined? Is it just a measurable function of random variables, and does it not depend on the model parameters?

**Contents**hide

#### Best Answer

The reason is that a pivot is a function of data *and (unknown) parameters*, while a statistic is only a function of data. For example, if $Z_1, dotsc, Z_n$ is an iid sample from the distribution $text{Normal}(mu, 1)$, then a pivot will be $(bar{Z} – mu)sqrt{n}$, since this function of data and unknown parameter $mu$ has (under this model) the known distribution $text{Normal}(0,1)$. But this is **not** a statistic, since it does depend as a function on $mu$, which is unknown, and not part of the data.