# Mathematical Statistics Lesson of the Day – Complete Statistics

November 27, 2014 1 Comment

The set-up for today’s post mirrors my earlier Statistics Lesson of the Day on sufficient statistics.

Suppose that you collected data

in order to **estimate** a **parameter** . Let be the **probability density function (PDF)*** for .

Let

be a **statistic** based on .

If

implies that

then is said to be **complete. **To deconstruct this esoteric mathematical statement**, **

- let be a measurable function
- if you want to use to form an unbiased estimator of the zero function,
- and if the only such function is almost surely equal to the zero function,
- then is a complete statistic.

I will discuss the intuition behind this bizarre definition in a later Statistics Lesson of the Day.

**This above definition holds for discrete and continuous random variables.*

I’m glad someone else thinks it is a bizarre definition. I’ve never fully understood the motivation behind it – looking forward to the lesson!