AMS570 Order Statistics - Stony Brook

AMS570

Order Statistics

1. Definition: Order Statistics of a sample.

Let X1, X2, ..., be a random sample from a population with p.d.f. f(x). Then,

2. p.d.f.'s for

W.L.O.G.(W thout Loss of Ge er l ty), let's ssu e s continuous.

(

)

f

f

(

)

=

f

f

Example 1. Let

exp( ), = ,...,

Please (1). Derive the MLE of (2). Derive the p.d.f. of (3). Derive the p.d.f. of

Solutions. (1).

L f

( e ) e

lL l

1

l

Is an unbiased estimator of ? ()

t

t

t ( t)

f

y ey

Let

() y

y e yy

y e yy

() (

)

s ot u se

(2).

(

)

=

f

f

f

e

f u u e u u [ e u]

e

f

e

e

e

e e

2

(3).

(

)=

f

f

e

e

3. Order statistics are useful in deriving the MLE's.

Example 2. Let X be a random variable with pdf.

f

{ f

Derive the MLE of .

other se

Solution. Uniform Distribution important!!

L f

{ f ll other se

MLE : max lnL -> max L e s

...

Now we re-express the domain in terms of the order statistics as follows:

Therefore,

If

[

] the L

Therefore, any [

] is an MLE for .

3

4. The pdf of a general order statistic

Let

sample, and pdf

denote the order statistics of a random

, from a continuous population with cdf

. Then the pdf of

is

Proof: Let Y be a random variable that counts the number of

less than or equal to x. Then we have

(

). Thus:

()

5. The Joint Distribution of Two Order Statistics

Let

denote the order statistics of a

random sample,

, from a continuous

population with cdf

and pdf

. Then the

joint pdf of and ,

is

6. Special functions of order statistics

(1) Median (of the sample):

{

(2) Range (of the sample):

4

7. More examples of order statistics

Example 3. Let X1,X2, X3 be a random sample from a distribution of the continuous type having pdf f(x)=2x, 0 ................
................

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