4.4 Order Statistics

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Definition: 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 $X_1,\dots ,X_n$

Theorem 4.4.1: Joint pdf of order statistics

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with
  • pdf $f(x)$
  • Support $(a,b)$
• $Y_1<\cdots <Y_n$ : $n$ order statistics of $X_1,\dots ,X_n$

Then the joint pdf of $Y_1,\dots ,Y_n$ is

$$g(y_1,\dots ,y_n)=\{\begin{array}{ll}n!f(y_1)\dots f(y_n)& a<y_1<\cdots <y_n<b\\ 0& \text{elsewhere}\end{array}$$

Marginal pdf of order statistics

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with pdf $f(x),a<x<b$
• $Y_1,\dots ,Y_n$ : Order statistics of $X_1,\dots ,X_n$

Then marginal pdf of $Y_1,\dots ,Y_n$ is given by:

• $Y_1$

$$g_1\left(y_1\right)=\{\begin{array}{ll}n{\left[1-F\left(y_1\right)\right]}^{n-1}f\left(y_1\right),& a<y_1<b\\ 0,& \text{elsewhere}\end{array}$$

• $Y_k$

$$g_k\left(y_k\right)=\{\begin{array}{ll}\frac{n!}{(n-k)!(k-1)!}{\left[1-F\left(y_k\right)\right]}^{n-k}{\left[F\left(y_k\right)\right]}^{k-1}f\left(y_k\right)& a<y_k<b\\ 0& \text{elsewhere}\end{array}$$

• $Y_n$

$$g_n\left(y_n\right)=\{\begin{array}{ll}n{\left[F\left(y_n\right)\right]}^{n-1}f\left(y_n\right),& a<y_n<b\\ 0,& \text{elsewhere}\end{array}$$

The joint marginal pdf of $Y_i$ and $Y_j$ is given by:

$$g_{ij}\left(y_i,y_j\right)=\{\begin{array}{ll}\frac{n!}{(i-1)!(j-i-1)!(n-j)!}{\left[F\left(y_i\right)\right]}^{i-1}{\left[F\left(y_j\right)-F\left(y_i\right)\right]}^{j-i-1}{\left[1-F\left(y_j\right)\right]}^{n-j}f\left(y_i\right)f\left(y_j\right)& a<y_i<y_j<b\\ 0& \text{ elsewhere }\end{array}$$

CDF of order statistics

Theorem

Let

• $X_1,\dots ,X_n$ : Random sample with pdf $f(x)$
• $Y_1,\dots ,Y_n$ : Order statistics of $X_1,\dots ,X_n$

Then cdf of $Y_1,\dots ,Y_n$ is given by

• $Y_1$[^2]

F_{Y_{1}}(x) = 1-[1-F_{X}(x)]$$ - $Y_{k}$[^1]

F_{Y_{k}}(x) = \sum_{j=k}^n \binom n j [F_{X}(x)]^j[1-F_{X}(x)]^{n-j}

- $Y_{n}$[^2]

F_{Y_{n}}(x) = [F_{X}(x)]$$

Exercise

Example 1

Misal $Y_1,Y_2,Y_3$ statistik terurut dengan $n=3$.

Menggunakan Marginal pdf and of order statistics, dapat diperoleh pdf marginal sebagai berikut:

1. pdf marginal untuk $Y_1$ :

$$g_1(y_1)=\{\begin{array}{ll}3[1-F(y_1){]}^{2}f(y_1),& a<y_1<b\\ 0,& \text{lainnya}\end{array}$$

2. pdf marginal untuk $Y_2$ :

Untuk $a<y_2<b$:

$$\begin{array}{rl}{g}_2(y_2)& =\frac{3!}{(2-1)!(3-2)!}[F(y_2){]}^{2-1}[1-F(y_2){]}^{3-2}f(y_2)\\ & =\frac{6}{1!\cdot 1!}F(y_2)[1-F(y_2)]f(y_2)\\ & =6F(y_2)[1-F(y_2)]f(y_2)\end{array}$$

dan $g_2(y_2)=0$ untuk lainnya.

3. pdf marginal untuk $Y_3$ :

$$g_3(y_3)=\{\begin{array}{ll}3[F(y_3){]}^{2}f(y_3),& a<y_3<b\\ 0,& \text{lainnya}\end{array}$$

Exercise 4.56 of Hogg & Craig 5th ed.

Let $Y_1<Y_2<Y_3<Y_4$ be the order statistics of a random sample of size 4 from the distribution having p.d.f. $f(x)=e^{-x}$, zero elsewhere. Find $Pr(3\le Y_4)$.

We will use CDF of order statistics to solve this problem.