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profile likelihood 的理解

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profile likelihood 的理解

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Chapter 3

The Proﬁle Likelihood

3.1 The Proﬁle Likelihood

3.1.1 The method of proﬁling

Let us suppose that the unknown parameters ✓ can be parti t i on ed as ✓

,

), where

are the p-dimensional parameters of interest (eg. mean) and  are the q-dimensional

nuisance parameters (eg. v ariance). We will need to estimate both and ,butour

interest lies only in the parameter .Toachievethisoneoftenproﬁlesoutthenuisance

parameters. To motivate the proﬁle likelihood, we ﬁrst describe a method to estimate the

parameters ( , )intwostagesandconsidersomeexamples.

Let us suppse that {X

} ar e iid random variables, with density f(x; , ) where o ur

objective is to estimate and .Inthiscasethelog-likelihoodis

( , )=

i=1

log f(X

; , ).

To estimate and  one can use (



)=argmax

,

( , ). However, this can be

diﬃcult to directly maximise. Instead let us consider a di↵erent method, which may,

sometimes, be easier to eval u ate. Sup pose, for now, is known, then we rewrite the

likelihood as L

( , )=L

()(toshowthat is ﬁxed but  varies). To estimate  we

maximise L

()withrespectto,i.e.



=argmax



().

In reality is unknown, hence for each we can evaluate



. Note that for each ,we

have a new curve L

()over. Now to estimate , we evaluate the maximum L

(),

over ,andchoosethe ,whichisthemaximumoverallthesecurves. Inotherwords,

we evaluate

=argmax

(



)=argmax

( ,



Abitoflogicaldeductionshowsthat

and 

are the maximum likelihood estimators

(



)=argmax

,

( , ).

We note that we have proﬁled out nuisance parameter ,andthelikelihoodL

(



( ,



)isintermsoftheparameterofinterest .

The advantage of this procedure is best illustrated through some examples.

Example 3.1.1 (The Weibull distribution) Let us suppose that {X

} are iid random

variables from a Weibull distribution with density f(x; ↵, ✓)=

↵y

↵1

✓

↵

exp((y/✓)

↵

).We

know from Example 2.2.2, that if ↵, were known an explicit expression for the MLE can

be derived, it is

✓

↵

=argmax

✓

↵

(✓)

=argmax

✓

i=1

✓

log ↵ +(↵  1) log Y

 ↵ log ✓ 



✓



↵

◆

=argmax

✓

i=1

✓

 ↵ log ✓ 



✓



↵

◆

i=1

↵

)

1/↵

where L

↵

(X; ✓)=

i=1

✓

log ↵ +(↵  1) log Y

 ↵ log ✓ 



✓



↵

◆

. Thus for a given ↵,

the maximum likelihood estimator of ✓ can be derived. The maximum likelihood estimator

of ↵ is

ˆ↵

=argmax

↵

i=1

✓

log ↵ +(↵  1) log Y

 ↵ log(

i=1

↵

)

1/↵





(

i=1

↵

)

1/↵



↵

◆

Therefore, the maximum likelihood estimator of ✓ is (

i=1

ˆ↵

)

1/ˆ↵

. We observe that

evaluating ˆ↵

can be tricky but no worse than maximising the likelihood L

(↵, ✓) over ↵

and ✓.

100

As we mentioned above, we are not interest in the nuisance parameters  and are only

interesting in testing and constructing CIs for .Inthiscase,weareinterestedinthe

limiting distribution of the MLE

. Using Theorem 2.6.2(ii) we have





 

✓





1

◆

where









log f (X

; ,)







log f (X

; ,)

@ @







log f (X

; ,)

@ @







log f (X

; ,)



. (3.1)

To derive an exact expression for the limiting variance of

 ), we use the block

inverse matrix identity.

Remark 3.1.1 (Inverse of a block matrix) Suppose that

is a square matrix. Then

1

(A BD

1

A

1

B(D  CA

1

D

1

CB(A  BD

1

(D  CA

1

. (3.2)

Using (3.2) we have

 )

!N(0, (I

 I

,

1



,

)

1

). (3.3)

Thus if is a scalar we can use the above to construct conﬁdence intervals for .

Example 3.1.2 (Block diagonal information matrix ) If

I( , )=

0 I

,

then using (3.3) we have

 )

!N(0,I

1

101

3.1.2 The score and the log-likelihood ratio for the proﬁle like-

lihood

To ease notation, let us suppose that

and 

are the true parameters in the distribution.

We now consider the log-likelihood ratio

⇢

max

,

( , )  max



(

,)



, (3.4)

where

is the true parameter. However, to derive the limiti n g distribution in this case

for this statistic is a little more compli cated than the log-l i kelihood ratio test that does

not involve nuisance parameters. This is because di r ect l y applying Taylor expansion does

not work since this is usually expanded about the true parameters. We observe that

⇢

max

,

( , )  max



(

,)



⇢

max

,

( , ) L

(

,

)



| {z }



p+q

2

max



(

,) max



(

,

)

| {z }



It seems reasonable that the di↵erence m ay be a 

but it is really not clear by. Below,

we show that by using a few Taylor expansions why this is true.

In the theorem below we will derive the distribution of the score and the nested log-

likelihood.

Theorem 3.1.1 Suppose Assumption 2.6.1 holds. Suppose that (

,

) are the true

parameters. Then we have

( , )



⇡

( , )

,



( ,)

@

,

1



(3.5)

( , )



!N(0, (I

 I



1



)) (3.6)

where I is deﬁned as in (3.1) and

⇢

(



) L

(



)



! 

, (3.7)

where p denotes the dimension of . This result is often called Wilks Theorem.

102

PROOF. We ﬁrst prove (3.5) which is the b a si s of the proofs of (3.6). To avoid, not a t i on a l

diﬃculties we will assume that

( ,)



and

( ,)

@

=

are univariate random

variables.

Our objective is to ﬁnd an expression for

( ,)



in terms of

( ,)

@

=

and

( ,)

=

which will allow us t o obtain its vari an ce and asymptotic distribution .

Making a Taylor expansi on of

( ,)



about

( ,)



gives

( , )



⇡

( , )





 

)

( , )

@@



Notice that we have used ⇡ instead of = because we replace the second derivative

with its true pa r am e te rs . If the sample size is large enough then

( ,)

@@



⇡



( ,)

@@





;eg.intheiidcasewehave

( , )

@@



i=1

log f(X

; , )

@@



⇡ E

✓

log f(X

; , )

@@



◆

= I

,

Therefore

( , )



⇡

( , )



 n(



 



. (3.8)

Next we make a decomposition of (





). We recall that since L

(



)=argmax



(

,)

then

( , )

@



(if the maximum is not on the boundary). Therefore making a Taylor expansion of

(

,)

@



about

(

,)

@



gives

(

,)

@



| {z }

⇡

(

,)

@



(

,)

@



(



 

Replacing

(

,)

@



with I



gives

(

,)

@



 nI



(



 

) ⇡ 0,

103

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