# Order statistics

## Order statistics

### Order statistics

#### Defining order statistics

The $$k$$th order statistic is the $$k$$th smallest value in a sample.

$$x_{(1)}$$ is the smallest value in a sample, the minimum.

$$x_{(n)}$$ is the largest value in a sample, the maximum.

#### Probability distributions of order statistics

The probability distribution of order statistics depends on the underlying probability distribution.

#### Probability distribution of sample maximum

If we have:

$$Y=\max \mathbf X$$

The probability distribution is:

$$P(Y\le y)=P(X_1\le y, X_2\le y,...,X_n\le y)$$

If these are iid we have:

$$P(Y\le y)=\prod_i P(X_i\le y)$$

$$F_y(y)=F_X(y)^n$$

The density function is:

$$f_y(y)=nF_X(y)^{n-1}f_x(y)$$

#### Probability distribution of the sample minimum

If we have:

$$Y=\min \mathbf X$$

The probability distribution is:

$$P(Y\le y)=P(X_1\ge y, X_2\ge y,...,X_n\ge y)$$

If these are iid we have:

$$P(Y\le y)=\prod_i P(X_i\ge y)$$

$$F_y(y)=[1-F_X(y)]^n$$

The density function is:

$$f_y(y)=-n[1-F_X(y)]^{n-1}f_x(y)$$