Order Statistics 10/30

$X_1,X_2,\ldots,X_n$ are n iid continuous random variables with a common density f, a distribution function F.

Define:

$$X_{(1)}=min(X_1,X_2,\ldots,X_n)$$

$$X_{(2)}=min\{\{(X_1,X_2,\ldots,X_n)\} -\{X_{(1)}\}\}$$

$$X_{(n)}=max(X_1,X_2,\ldots,X_n)$$

The ordered values of the iid sample are known as the order statistics.

The question we will try to reply to is:
What is the formula for the density of the kth order statistic?

We start with the extremes that are easy to handle:

First method

$$\begin{eqnarray*}F_{max}(x)=F_{(n)}(x) &=& P(X_1\leq x, X_2\leq x, \ldots, X_n\l... ...& F^n(x)\\ f_{max}(x)=\frac{d}{dx}(F(x))^n=n(F(x))^{n-1}f(x)\\ \end{eqnarray*}$$

$$\begin{eqnarray*}F_{min}(x)=F_{(1)}(x) &=& 1-P(X_{min}>x)\\ &=& 1-P(X_1> x, X_2... ...\\ f_{min}(x) &=&-\frac{d}{dx}(1-F(x))^n=n(1-F(x))^{n-1}f(x)\\ \end{eqnarray*}$$

Second Method

$$\begin{eqnarray*}f_{(n)}(x)dx &=& P(X_{(n)}\in dx)\\ &=& P(\mbox{ one of the }X... ...n dx)P(\mbox{ all others } < x)\\ &=& n f(x) dx (F(x))^{n-1}\\ \end{eqnarray*}$$

$$\begin{eqnarray*}f_{(1)}(x)dx &=&P(X_{(1)}\in dx)\\ &=& P(\mbox{ one of the }X'... ...dx)P(\mbox{ all others } > x)\\ &=& n f(x) dx (1-F(x))^{n-1}\\ \end{eqnarray*}$$

The advantage of this method is that it can be generalized: X_(k) is the k smallest of $X_1,X_2,\ldots,X_n$.

$$\begin{eqnarray*}f_{(k)}(x)dx &=&P(X_{(k)}\in dx)\\ &=& P(\mbox{ one of the }X'... ...)}(x)&=&n f(x)\binom{n-1}{k-1} (F(x))^{(k-1)}(1-F(x))^{(n-k)}\\ \end{eqnarray*}$$