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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?

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1 Answer 1

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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.

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