## Mathematical Statistics Lesson of the Day – Basu’s Theorem

July 21, 2015 1 Comment

Today’s Statistics Lesson of the Day will discuss **Basu’s theorem**, which connects the previously discussed concepts of minimally sufficient statistics, complete statistics and ancillary statistics. As before, I will begin with the following set-up.

Suppose that you collected data

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

Let

be a statistics based on .

**Basu’s theorem** states that, if is a complete and minimal sufficient statistic, then is independent of every ancillary statistic.

Establishing the independence between 2 random variables can be very difficult if their joint distribution is hard to obtain. **This theorem allows the independence between minimally sufficient statistic and every ancillary statistic to be established without their joint distribution **– and this is the great utility of Basu’s theorem.

However, establishing that a statistic is complete can be a difficult task. In a later lesson, I will discuss another theorem that will make this task easier for certain cases.

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