2
$\begingroup$

I have a question about the distribution of a pivot. Different sources give slightly different definitions of a pivot. Some of them define it as a random variable $Q(\mathbf{X}, \theta)$ whose distribution is the same for all $\theta$, where $\theta$ is a vector of parameters of $X$'s distribution (see definitions from wikipedia and Casella-Berger below; here $X$ is a r.v. which generate observations). But some other sources define pivot as a random variable whose distribution doesn't depend on unknown parameters of $X$'s distribution (see definition from "probability course" below), this means that theoretically pivot's distribution can depend on known parameters of $X$'s distribution. All examples of pivots for different population distributions, that I have seen in textbooks, didn't depend on all (known and unknown) parameters of $X$'s distribution.
So, my question is – can the distribution of a pivot depend on known parameters of $X$'s distributiion?
I mean cases when $X \sim P_\theta; ~\theta = (\theta_1, \ldots, \theta_k),$ and some of the components of vector $\theta$ are known (known parameters) and other components are unknown (unknown parameters); and we need to build confidence interval for some unknown parameter $\theta_j$ using sample from $P_\theta$ (typically we construct a pivot for this purpose).

Below definitions of a pivot from three different sources are given.

First one is the definition of a pivot from wikipedia:

In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters). 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.
More formally, let $\mathbf{X} = (X_1,X_2,\ldots,X_n)$ be a random sample from a distribution that depends on a parameter (or vector of parameters) $\theta$. Let $g(\mathbf{X},\theta)$ be a random variable whose distribution is the same for all $\theta$. Then $g$ is called a pivotal quantity (or simply a pivot).

And here is the definition of a pivot from Casella-Berger's book "Statistical inference": enter image description here

Finally, here is the definition of a pivot from the website probabilitycourse.com: probcourse

$\endgroup$

1 Answer 1

3
$\begingroup$

The usefulness of a pivot is that you know its distribution, so you can actually do calculation with it, such as in finding a confidence interval. If its distribution depends on known parameters, you still know its distribution, and still can do the calculations ... so it really behaves like a pivot, yes?

So, yes, the distribution of a pivot can depend on known parameters.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.