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### Anna Janicka

Lecture VII, 1.04.2019

ESTIMATOR PROPERTIES, PART III

CONFIDENCE INTERVALS – INTRO

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Plan for Today

1. Asymptotic properties of estimators – cont.

asymptotic normality asymptotic efficiency

2. Consistency, asymptotic normality and asymptotic efficiency of MLE estimators

3. Interval estimation – confidence intervals

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Asymptotic normality

2

1

2

n

1

2

2

n

D

ˆ( , ,..., ) ( )

### )

( )

)

lim (n g X1 X2 X g a a

P n

n = Φ

θ

θ

θ σ

1

2

n

n2

### N θ

σ

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Asymptotic normality – properties

2

2

### σ

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Asymptotic normality – what it is not

1 2

θ

n

n

1

2

n

n

2

### n

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Asymptotic normality – example

1

2

n

2

2

### n −  →

D

n σ 2

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Asymptotic normality – how to prove it

n

### and let h:R→R be a function differentiable at point µ such that h’( µ )≠0. Then

µ, σ2 are functions of θ

usually used when estimators are functions of statistics Tn, which can be easily shown co converge on the base of CLT

2

n

D

2

2

n

### −  →

D

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Asymptotic normality – examples cont.

X1

1 12

λ

λ

D

2

) / 1 (

1 1

1

2

2 λ

λ

### n

X D

X 1

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Asymptotic efficiency

2

1

2

2

n

D

1

2

n

2

2

n

### = ⋅

( )

) ( ) (

) ( ) '

( ˆ as.ef

1 2

2

θ θ

σ

θ I g g

=

modification of the definition of efficiency to the limit case, with the asymptotic

variance in place of the normal variance

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Relative asymptotic efficiency

2 1 2

1 2 2 2

1

1

2

### ( X )

Note. A less (asymptotically) efficient estimator may have other properties, which will make it preferable to a more efficient one.

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Relative asymptotic efficiency – examples.

Is the mean better than the median?

2

### c) some distributions do not have a mean...

Theorem: For a sample from a continuous distribution with density f(x), the sample median is an asymptotically normal estimator for the median m

(provided the density is continuous and ≠0 at point m):

X µ

N(0,σ 2)

n →D

md µ

N(0,πσ22 )

n →D

1 )

, d eˆ m (

as.ef X = π2 <

X µ

N(0, λ22 )

n →D

md

### )

(0, 2 )

1

µ N λ

n →D as.ef(meˆd, X ) = 2 > 1

md

### )

(0, 4( ( ))2 )

1 m f

D N

m

n →

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Consistency of ML estimators

Let X1, X2, ..., Xn,... be a sample from a distribution with density fθ (x). If Θ ⊆ R is an open set, and:

all densities fθ have the same support;

the equation has exactly one solution, .

Then is the MLE(θ ) and it is consistent

Note. MLE estimators do not have to be unbiased!

0 ) (

ln θ = θ L

d d

θˆ

### θ ˆ

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Asymptotic normality of ML estimators

Let X1, X2, ..., Xn,... be a sample with density fθ (x), such that Θ ⊆ R is open, and is a consistent

m.l.e. (for example, fulfills the assumptions of the previous theorem), and

exists

Fisher Information may be calculated, 0<I1(θ )<∞

the order of integration with respect to x and derivation with respect to θ may be changed

then is asymptotically normal and

) (

2 ln

2

θ L θ d

d

## ( θˆθ )

D

I1(1θ )

### n −  →

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Asymptotic normality of ML estimators

1

2

( '(( )))

1

2

θ

D gI θ

n

1

2

n

### g

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Asymptotic efficiency of ML estimators

### g( θ )) is asymptotically efficient.

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Asymptotic normality and efficiency of ML estimators – examples

### In the Laplace model: the median is an asymptotically efficient estimator of µ

Examples

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Summary: basic (point) estimator properties

### asymptotic efficiency

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Interval estimation – confidence intervals

### We estimate with given precision

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Confidence interval

1

2

n

θ

1

2

n

1

2

n

1

2

n

### [ gg

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Confidence intervals – use and interpretation

Typically:

### α

is a small number, for example 1-

= 0,95 or 1-

### α

= 0,99

The condition from the definition means: the random interval includes the unknown value g(

### θ

) with given (high) probability.

If we calculate the realization of the

confidence interval (e.g. ) then

we CAN’T say that the unknown parameter is included in the range with probability 1-

### α

anymore!

the parameter is either in the interval or not – the event is not random, it is just something we don’t know.

] , [g g

3 ,

1 =

= g g

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Confidence intervals – construction

### observed in nature)

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Confidence intervals – construction cont.

Convenient method: we look for random

variables which depend on sample data and

parameter values, but whose distributions do not depend on unknown parameters (pivotal method) If U = U(X1, X2, ..., Xn, θ ) is such a function, then we look for confidence intervals [a,b] such that

Usually we look for „symmetric” CI

θ aUb ≥ 1− P

2 2 ,

### α α

θ

θ U < aP U > b

P

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