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Lecture 12 -- Another way to find the

Best Estimator

1. (Regular) Exponential Family

The density function of a regular exponential family is:

( ) ( ) ( ) [ ∑ ( ) ( )

] ( )

Example. Poisson(θ)

( ) ( )

[ ( ) ]

Example. Normal. ( ) ( ) (both unknown).

)

√

( )

√ [

( ) ]

√ [

( )]

√ [

] [

( )]

√ [

] [

( )

( )]

2. Theorem (Exponential family & sufficient

Statistic). Let be a random sample from the

regular exponential family.

Then

( ) (∑ ( )

∑ ( )

)

is sufficient for ( )

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Example. Poisson(θ)

Let be a random sample from Poisson(θ)

Then

( ) ∑

is sufficient for

Example. Normal. ( ) ( ) (both unknown).

Let be a random sample from ( )

Then

( ) (∑

∑

)

is sufficient for ( )

Exercise.

Apply the general exponential family result to all the standard

families discussed above such as binomial, Poisson, normal,

exponential, gamma.

A Non-Exponential Family Example.

Discrete uniform.

( ) is a positive integer.

Another Non-exponential Example.

iid ( ) ( )

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Universal Cases.

are iid with density .

• The original data are always sufficient for .

(They are trivial statistics, since they do not lead any data

reduction)

• Order statistics ( ( ) ( )) are always sufficient for .

( The dimension of order statistics is , the same as the

dimension of the data. Still this is a nontrivial reduction as !

different values of data corresponds to one value of . )

3. Theorem (Rao-Blackwell)

Let be a random sample from the population

with pdf ( ). Let ( ) be a sufficient statistic for

θ, and ( ) be any unbiased estimator of θ.

Let ( ) [ ( ) ], then

(1) ( ) is an unbiased estimator of

(2) ( ) is a function of T,

(3) ( ) ( ) for every , and

( ) ( ) for some unless

with probability 1 .

Rao-Blackwell theorem tells us that in searching

for an unbiased estimator with the smallest

possible variance (i.e., the best estimator, also

called the uniformly minimum variance unbiased

estimator – UMVUE, which is also referred to as

simply the MVUE), we can restrict our search to

only unbiased functions of the sufficient statistic

T(X).

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Proof: Make use of the following equations:

( ) [ ( )]

( ) [ ( )] [ ( )]

Note: The fact that ( ) is a sufficient statistic for θ

will ensure that ( ) is a function of only the sample

and in particular, is independent of θ.

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4. Transformation of Sufficient Statistics

1. If is sufficient for and ( ) a mathematical

function of some other statistic, then is also sufficient.

2. If is sufficient for , and ( ) with being one-to-

one, then is also sufficient.

Remark: When one statistic is a function of the other statistic

and vise verse, then they carry exactly the same amount of

information.

Examples:

• If ∑ is sufficient, so is .

• If (∑ ∑ )

are sufficient, so is ( ).

• If ∑ is sufficient, so is (∑

∑ )

is sufficient,

and so is ∑ ∑ )

.

Examples of non-sufficiency.

Ex. iid Poisson( ). – is not sufficient.

Ex. iid pmf ( ). ( ) is not

sufficient.

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5. Minimal Sufficient Statistics

It is seen that different sufficient statistics are possible. Which

one is the "best"? Naturally, the one with the maximum

reduction.

• For ( ), is a better sufficient statistic for than

( )

Definition:

is a minimal sufficient statistic if, given any other sufficient

statistic , there is a function ( ) such that ( ).

Equivalently, is minimal sufficient if, given any other

sufficient statistic whenever and are two data values

such that ( ) ( ), then ( ) ( ).

Partition Interpretation for Minimal Sufficient Statistics:

• Any sufficient statistic introduces a partition on the sample

space.

• The partition of a minimal sufficient statistic is the coarsest.

• Minimal sufficient statistic has the smallest dimension

among possible sufficient statistics. Often the dimension is

equal to the number of free parameters (exceptions do

exist).

Theorem (How to check minimal sufficiency).

A statistic T is minimal sufficient if the following property

holds: For any two sample points x and y ( ) ( )

does not depend on (i.e. ( ) ( ) is a constant

function of ) if and only if ( ) ( )

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6. Exponential Families & Minimal Sufficient

Statistic:

For a random sample from the regular exponential family with

probability density ( ) ( ) ( ) [∑ ( ) ( ) ],

where is k dimensional, the statistic

( ) (∑ ( )

∑ ( )

)

is minimal sufficient for .

Example. Poisson(θ)

Let be a random sample from Poisson(θ)

Then

( ) ∑

is minimal sufficient for

Example. Normal. ( ) ( ) (both unknown).

Let be a random sample from ( )

Then

( ) (∑

∑

)

Is minimal sufficient for ( )

Remarks:

• Minimal sufficient statistic is not unique. Any two are in one-

to-one correspondence, so are equivalent.

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7. Complete Statistics

Let a parametric family ( ) be given. Let be a

statistic. Induced family of distributions ( ) .

A statistic is complete for the family ( ) or

equivalently, the induced family ( ) is called

complete, if ( ( )) for all implies that ( )

with probability 1.

Example. Poisson(θ)

Let be a random sample from Poisson(θ)

Then

( ) ∑

is minimal sufficient for Now we show that T is also

complete.

We know that ( ) ∑ ( )

Consider any function ( ). We have

[ ( )] ∑ ( )

Because setting [ ( )] requires all the

coefficient

( )

to be zero, which implies ( )

Example. Let be iid from ( ). Show

∑ is a complete statistic. (*Please read our text

book for more examples – but the following result on the

regular exponential family is the most important.)

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8. Exponential Families & Complete Statistics

Theorem. Let be iid observations from the regular

exponential family, with the pdf

( ) ( ) ( ) [∑ ( ) ( ) ],

and ( ) Then

( ) (∑ ( )

∑ ( )

)

is complete if the parameter space ( ( ) ( ))

contains an open set in .

(This is only a sufficient condition, not a necessary condition)

Example. ∼ ( )

Example. Poisson(θ);

( ) ( )

[ ( ) ]

Example. Normal. ( ) ( ) (both unknown).

)

√

( )

√ [

( ) ]

√ [

( )]

√ [

] [

( )]

Example. ∼ ( ) , is not complete.

Example. ∼ ( ) is complete.

Example. {Bin(2, ), p = 1/2, p = 1/4} is not complete.

Example. The family {Bin(2,p), 0 < p < 1} is complete.

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Properties of the Complete Statistics

(i) If is complete and ( ), then is also complete.

(ii) If a statistic T is complete and sufficient, then any minimal

sufficient statistic is complete.

(iii) Trivial (constant) statistics are complete for any family.

9. Theorem (Lehmann-Scheffe). (Complete

Sufficient Statistic and the Best Estimator)

If T is complete and sufficient, then ( ) is

the Best Estimator (also called UMVUE or MVUE)

of its expectation.

Example. Poisson(θ)

Let be a random sample from Poisson(θ)

Then

( ) ∑

is complete sufficient for Since

( )

∑

is an unbiased estimator of θ – by the Lehmann-Scheffe

theorem we know that U is a best estimator

(UMVUE/MVUE) for θ.

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Example. Let . . .

2

1, , ~ ( , )i i d

nX X N , be a random sample from

the normal population where is assumed known. Please

derive

(a) The maximum likelihood estimator for 2 .

(b) Is the above MLE for 2 unbiased?

(c) Is the MLE a best estimator (UMVUE) for 2 ?

Solution:

(a) The likelihood function is

∏ ( ) ∏

√ [

( )

]

( ) [ ] [ ∑ ( )

]

The log likelihood function is

( )

∑ ( )

Solving

∑ ( )

We obtain the MLE for

∑ ( )

(b) Since

[ ] ∑ ( )

It is straight-forward to verify that the MLE

∑ ( )

is an unbiased estimator for .

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(c) Now we calculate the Cramer-Lower bound for the variance of an unbiased estimator for

[ ( )] [

√ [

( )

]]

( )

[ ( )]

( )

[ ( )]

( )

[ [ ( )]

] [

( )

]

[

]

Thus the Cramer-Rao lower bound is:

Therefore we claim that the MLE is an efficient

estimator for . Since the regularity conditions for the

Cramer-Rao lower bound Theorem to be true holds

here, we declare the MLE is also a best estimator

(UMVUE) for .

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(c) Alternatively, we can derive that the MLE is the best estimator directly (rather than using the efficient estimator is also a best estimator argument) as follows: The population pdf is:

( )

√ ( )

√

( )

So it is a regular exponential family, where the red part

is ( ) and the green part is ( ).

Thus ( ) ∑ ( ) is a complete & sufficient

statistic (CSS) for .

It is easy to see that the MLE for is a function of the

complete & sufficient statistic as follows:

∑ ( )

( )

In addition, we know that the MLE is an unbiased

estimator for as we have done in part (b):

( ) ∑ ( )

∑ ( )

∑ ( )

Since the MLE is an unbiased estimator and a

function of the CSS, therefore we claim that is the

best estimator (UMVUE) for by the Lehmann-Scheffe

Theorem.

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10. Theorem (Basu)

A complete sufficient statistic T for the parameter θ is

independent of any ancillary statistic – that is, a

statistic whose distribution does not depend on θ

Example. Consider a random sample of size n from a normal

distribution ( ) ( ).

Consider the MLEs 2

2

ˆ

( )ˆ i

X

X X

n

It is easy to verify that is a complete sufficient statistic for

for fixed values of . Also:

( )

which does not depend on . It follows from the Basu Theorem

that the two MLEs are independent to each other.

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Homework 6:

Problems in our textbook:

5.1, 5.3, 5.5, 5.6, 5.16, 5.21, 5.22, 5.29, 5.30

6.3, 6.6, 6.17, 6.22, 6.30, 7.49, 7.50, 7.52 (a&b), 7.59, 7.60

Read our textbook:

Chapter 5: Sections 5.1, 5.2, 5.3 & 5.4

Chapter 6: Sections 6.1 & 6.2

Chapter 7: Sections 7.3.3 and 7.5 (Miscellanea): 7.5.1 & 7.5.3