2. Point and interval estimation Introduction Properties of estimators Finite sample size Asymptotic...

2. Point and interval estimation

Introduction Properties of estimators

Finite sample size Asymptotic properties

Construction methods Method of moments Maximum likelihood estimation

Sampling in normal populations

1

Interval estimation Asymptotic intervals Intervals for normal populations

2

2. Point and interval estimation

INFERENCIA ESTADÍSTICA

Introduction

3

sample thefrom

about n informatioobtain To :Problem

sample ddistribute

y identicall andt independen ,...,,

parameter unknown ;population ;

21

nXXX

FX

INFERENCIA ESTADÍSTICA

Point estimation

4

n

n

nn

m

S

X

XXX

median Sample

varianceSample

mean Sample

:Examples

ofestimator ),...,,(ˆ

12

21

STATISTICAL INFERENCE

Properties of estimators

5

Unbiased estimator

is an unbiased estimator of if

(bias of )

The bias of an unbiased estimator is zero:

n̂ nE ˆ

n̂ )ˆ()ˆ( nn Eb

0)ˆ(ˆ nn bunbiased

6

Efficiency

122

2

1 ˆˆˆˆ

ˆ

VVbecausepreferwe

E

E

2

1

ˆ

ˆ



7

Mean squared error

2̂E

22 )ˆ(ˆ)ˆ( nnn bVEECM STATISTICAL INFERENCE


8

Mean squared error

If the estimator is unbiased, then and the best one is chosen in terms of variance.

The global criterion to select between twoestimators is:

is preferred to if )()( nn SMSETMSE

nVMSE ̂

nTnS



Standard error

9

nn Vse ˆ)ˆ(


10

Properties of estimators when n

Consistency

is a consistent estimator for parameter ifn̂ P

n̂


Asymptotic behavior

(Weak consistency)

is strongly consistent for ifn̂

csn̂

11

Asymptotically normal

is an asymptotically normal estimator with

parameters if

n̂

),( nn ba

)1,0(ˆ

Nb

a

n

nn


Asymptotic properties

Construction of estimators:method of moments

12STATISTICAL INFERENCE

X with or and we have a sample

The kth moment is

Method of moments:

(i) Equal population moments to sample moments.

(ii) Solve for the parameters.

p

.,...,1 iidXX n

.kk EX

f

13

Properties:

(i) Consistency

Let be a method of moments estimator of Then

n~ .

Pn

~



14

(ii) Asymptotic normality



).( and ),..., ,)',...,,(

,')'(

where

),,0()~

(

11

2

jjkk

dn

gg(ggXXXY

gYYgE

Nn

Construction of estimators:maximum likelihood

15STATISTICAL INFERENCE

X; i.i.d. sample

The maximum likelihood function is the probabilitydensity function or the probability mass function ofthe sample:

nXX ,...,1

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

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

11

11

nn

nn

xfxfxxL

xpxpxxL

16

is the maximum likelihood estimator of ifn̂



),...,;(max),...,;ˆ( 11 nnn xxLxxL

The maximum likelihood estimator of is the valueof making the observed sample most likely.

17

Properties

(i) Consistency

Let be a maximum likelihood estimator of . Then

(ii) Invariance

If is a maximum likelihood estimator of , then is a maximum likelihood estimator of

n̂

n̂

.ˆ Pn

)ˆ( ng ).(g



18

Properties

(iii) Asymptotic normality

(iv) Asymptotic efficiency

The variance of is minimum.n̂

nn VeswithN

es ˆˆ)1,0(

ˆ

ˆ




19INFERENCIA ESTADÍSTICA

)ˆ(

1ˆ

)(

1

n)informatio

(Fisher ));(());(()(

function) (score );( log

);(

11

nnn

n

ii

n

iin

Ies

Ise

XsVXsVI

XfXs

Sampling in normal populations:Fisher’s lemma

20

Let

Given the i. i. d. sample let

Then:(i)

(ii)

(iii) are independent.

).,(~ 2NX

,,...,1 nXX

),(2

nn NX

21

)(2

2

n

XX ni

12, nSX


.1

)(a

12

12

n

XXSndX

nX ni

i

nin

21

distribution

Let independent.

Then

We define

and it verifies

2

niNZ i ,...,1)1,0(

.21

2 iZ

22n

n

iiZY

.2nVY

nEY



22

If the population is normal, the distribution of the estimators is exactly known for any sample size.



Confidence intervals

23

Let , and the sample

Construct an interval with

such that

FX . ,...,1 iidXX n

),...,(),...,(

1

1

n

n

XXbbXXaa

.1)( baP

is the confidence coefficient.1


24

Exact interval:

Asymptotic interval:

1)( baP

1)(n

baP


Confidence intervals

Confidence intervals:asymptotic intervals

25

an asymptotically normal estimator of

Then

n̂

)1,0(ˆ

ˆ e., i. ),1,0(

ˆ

ˆN

esN

esnn

)ˆ

ˆ(1 b

esaP n


26

Define such that

Then

where



2z

.2

)(2

zZP

),ˆ

ˆ()

ˆ

ˆ(1

22

zes

zPbes

aP nn

).ˆˆˆˆ(122

eszeszP nn

27

Then, the confidence interval for is

eszIC n ˆˆ2

1



28

Remark:

For large samples, we can obtain asymptotic confidence intervals.

For small samples, we can obtain exact confidence intervals if the population is normal.

Interval estimation:Asymptotic intervals


29

i. i. d. sample

(i) Confidence interval for with known 02.

Then

),( 2NX .,...,1 nXX

)1,0(),(/0

20 NNX

n

Xn


Intervals for normal populations

nzXIC 0

21

30

(ii) Confidence intervals for with unknown 2.

2 is unknown: we estimate it.

)1,0(),(/

2

NNXn

Xn



31

Student t distribution

Let

be independent. Then

2

)1,0(

nYNZ



)1,0(Nt

nY

Znn

32

Let

Then

.)1()( 2

12

12

2

2

n

ni SnXX



11

)1(

)1(

/

12

21

2

n

nS

n

n

Sn

n

X

tX

tnn

33

The confidence interval is

thus



)(121

22 ;1;1

n

nS

n tX

tPn

nS

nntXIC 1

2

2;11

34INFERENCIA ESTADÍSTICA

We change from an expression with 2 and N(0,1) to another expression with S2

n-1 and tn-1

nS

nntXIC 1

2

2;11


nzXIC 0

21

35

(iii) Confidence interval for 2 with known 0.

Each satisfies:

and for the whole sample:

iX

21

2

)1,0(

oi

oi

X

NX

22

2)(n

oiX



36

and then



))(

(1 22/;2

22

2/1;

n

oin

XP

))()(

(12

2/1;

22

22/;

2

n

oi

n

oi XXP

37

(iv) Confidence interval for 2 with unknown .

If , then applying Fisher’s Lemma:

The confidence interval is:

),( 2NX



212

2)(

n

i XX

))()(

(12

2/1;1

22

22/;1

2

n

i

n

i XXXXP

2. Point and interval estimation Introduction Properties of estimators Finite sample size Asymptotic...

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