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?