Chapter7. Parameter estimation

§1. Point estimation

Point estimation 点估计 Set the distribution function of population X

is F(x), which contains the unknown parameters ,we call any function of a sample point estimator.

1( ,..., )nX X

1. Introduction

Interval estimation 区间估计

2.1 The Method of Moments 矩估计法 If population distribution contains k unknown paramete

rs, method of moments estimator are found by equating the first k sample moments to the corresponding k population moments, and solving the resulting system of simultaneous equations 联立方程组 .

2.Point estimation

Example 1 suppose are iid ,We

have according to the method of moments, we yield the moments estimator of is .

1 2( , ,..., )nX X X ( )P

1 1, ,X

X

Example 2 Suppose are iid ,

We have ,hence we must solve the equation

Solving for yields the moments estimator of is

1 2( , ,..., )nX X X 2( , )N 2 2 2

1 2 1 21

1, , ,n

ii

X Xn

2 2 2

1

,1 ,

n

ii

X

Xn

2, 2,

2 2 2

1

,1 1 ( ) ,

n

ii

Xn S X Xn n

Example 3 suppose are iid, and the

common pdf is

， is a known positive constant ， we have

Let ,we yield the moment estimator of is .

1 2( , ,..., )nX X X1 1(1 )1 , ,( ; )0,

c x x cf x

其它,

10 c1

1 1(1 )

( ; )

1

,1

c

EX xf x dx

xc x dx

c

X

1 cX

2.2 Maximum likelihood estimate 极大似然估计 Consider a random sample from a

distribution having pdf or probability distribution , are unknown parameters, The joint pdf of is ,this may be regard as a function of ,when so regarded, it is called the likelihood function 似然函数 L of the random sample, and we write

1 2( , ,..., )nX X X

( ; )x 1( ,..., )m

1 2( , ,..., )nX X X

1

( ; )n

ii

x

11

( ,..., ; ) ( ; )n

n ii

L x x x

Suppose that we can find a function of ,say ,when is replaced by ,the likelihood function L is a maximum, then the statistic will be called a maximum likelihood estimator of ,and will be denoted by the symbol .

1 2( , ,..., )nx x x1 2( , ,..., )nu x x x

1 2( , ,..., )nu x x x

1 2( , ,..., )nu X X X

1 2( , ,..., )nu X X X

The function can be maximized by setting the first order partial derivatives of ,with respect to ,equal to zero, that is to say

and solving the resulting equation for ,which is the maximum likelihood estimator of .

L

ln L

1

ln 0,

ln 0.m

L

L

Example 4 Let denote a random sample

from a distribution that is ,we shall find , the maximum likelihood estimator of .

1 2( , ,..., )nX X X( )P

Example 5 –P131,Example 7.6 Let denote a random sample

from a distribution that is are unknown parameters, we shall find , the maximum likelihood estimators of .

1 2( , ,..., )nX X X

2( , ),N

2,

2ˆ ˆ, 2,

Example 6 Let denote a random

sample from a distribution that is , Find the maximum likelihood estimators of .

1 2( , ,..., )nX X X

[ , ]U a b,a b

3.1 Unbiased estimator 无偏估计 If an estimator satisfies

that ,the estimator is called unbiased.

1( ,..., )nX X

( )E

3 The Particular Properties of Estimators

Methods of evaluating estimators

Example 7 Let denote a random sample

from , ,Prove that and are the unbiased estimator of respectively.

1 2( , ,..., )nX X X

X 2,EX DX X 2 2

1

1 ( )1

n

ii

S X Xn

2,

3.2 Efficiency 有效性 Let and are both the unbiased estimator of ,i

f , We say is more efficient than .If the number of s

ample is fixed, and the variance of is less than or equal to the variance of every other unbiased estimator of ,we say is the efficient estimator of .

1

2

1 2D D

1

2

n

Example 8 Consider a random sample from a

distribution having pdf

,Prove that and are both the unbiased estimators of ,and when , is more efficient than .

1 2( , ,..., )nX X X

1 , 0,( )

0 ,

xe xp x

others,1min( , , )nZ X X X

1n XnZ

nZ

3.3 Consistency 一致性 If for any ,we have , We call is the uniform estimator of

.

0

lim (| | ) 1nnP

n