Order Statistics

Definition

Let $X_1,\dots ,X_n$ : Random Sample with pdf $f(x)$ and support $(a,b)$

If

• $Y_1=\min \{X_1,\dots ,X_n\}$
• $Y_k$ : $k$-th smallest value of $X_1,\dots ,X_n$
• $Y_n=\max \{X_1,\dots ,X_n\}$

Then we say $Y_1,\dots ,Y_n$ is order statistic of $X_1,\dots ,X_n$

About order statistics

Imagine you roll several dice at once. After rolling, you sort the numbers you get.

Now, you wonder:

• What’s the probability of getting the smallest number?
• What’s the probability of getting the second smallest number?
• And so on…

These sorted values are called order statistics.

In a sample of size $n$, the $k$-th order statistic $Y_k$ is the $k$-th smallest value from your sample.

TIP

We are asking questions about:

• “What’s the probability that the smallest value is $\le$ 3?”
• “What’s the expected value of the second-largest number?”