A Useful Pivotal Quantity

Panagiotis (Panos) Toulis

The American Statistician, 2017, vol. 71, issue 3, 272-274

Abstract: Consider n continuous random variables with joint density f that possibly dependson unknown parameters θ. If the negative of the logarithm of f is a positive homogenous function of degree p taking only positive values, then that function is distributed as a Gamma random variable with shape n/p and scale 2, and thus it is a pivotal quantity for θ. This provides a general method to construct pivotal quantities, which are widely applicable in statistical practice, such as hypothesis testing and confidence intervals. Here, we prove the aforementioned result and illustrate through examples.

Date: 2017
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DOI: 10.1080/00031305.2016.1237894

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