Department

Statistical SciencesCourse Code

STA261H1Professor

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Statistical Inference

RANDOM SAMPLE

Definition: Random Sample

n

XX ,,

1 is called a random sample from a distribution with pdf

(

)

xf (or pf

(

)

xP) if n

XX ,,

1 are

independent and have identical distribution with pdf

(

)

xf (or pf

(

)

xP). It is often denoted “iid”

(independent-identically-distributed).

Note

Let n

XX ,,

1 be a sample from a distribution with pdf

(

)

xf. The joint pdf of

(

)

n

XXX ,,

1= is

(

)

(

)

(

)

(

)

(

)

nnnn xfxfxfxfxxf 1111 ,== .

SAMPLE MEAN

Definition: Sample Mean and Sample Variance

Let n

XX ,,

1 be a random sample. The sample mean is defined by n

XX

n

X

Xn

n

i

i

n

++

==

=

11 , and the

sample variance is defined by

( )

1

12

−

−

=

=

n

XX

S

n

i

ni

.

Theorem

If n

XX ,,

1 are iid each with a

(

)

2

,

σµ

N distribution, then nn XkXk++

11 has a normal distribution

with mean

(

)

µµµ

nn kkkk ++=++ 11 and variance

(

)

2

22

1

σ

n

kk ++.

More generally, if n

XX ,,

1 are independent and each i

X has a

(

)

2

,ii

N

σµ

, then nn XkXk++

11 has a

normal distribution mean nn

kk

µµ

++

11 and variance 222

1

2

1nn

kk

σσ

++.

USEFUL DISTRIBUTION

Theorem

Suppose n

XX ,,

1 is a random sample from

(

)

2

,

σµ

N distribution. Then

n

X and

( )

=

−

n

i

niXX

1

2 are

independent, and

( )

2

1

2

σ

=

−

n

i

niXX

has a 2

1−n

χ

distribution.

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The t and F Distribution

Two important distributions useful in statistical inference are:

1) If

(

)

1,0~NW and 2

~r

V

χ

are independent, then

r

V

W has a t-distribution.

2) 2

1

~r

U

χ

and 2

2

~r

V

χ

are independent, then

2

1

r

V

r

U

has a F-distribution.

THE CENTRAL LIMIT THEOREM

Let n

XX ,,

1 is a random sample with finite mean

µ

and variance 0

2>

σ

. Let nn XXS++=

1. Then

(

)

( )

(

)

( )

( )

1,0

var

var 2

N

n

X

X

XEX

S

SES

n

n

n

nn

n

nn →

−

=

−

=

−

∞→

σ

µ

.

STATISTICAL MODEL

Consider a random sample n

XX ,,

1 from a distribution with pdf

(

)

xf

θ

. The family

(

)

{

}

Ω∈

θ

θ

|xf (where

(

)

xf

θ

is pdf (or

(

)

xP

θ

a pf),

θ

is an unknown parameter, Ω is a parameter space) is a statistical model.

We know that the distribution under investigation is in the family, but don’t know which one. Based on the

sample values n

xx ,,

1, we find an estimate for

θ

; Once we find

θ

, we know the distribution.

LIKELIHOOD FUNCTION

Definition: The Likelihood Function

Let n

XX ,,

1 be a random sample from a distribution with pdf

(

)

xf

θ

(or pf

(

)

xP

θ

). The likelihood is

defined by R

→

Ω

:L given by

(

)

(

)

(

)

(

)

nnn xfxcfxxcfxxL

θθθ

θ

111 ,,,|== where 0

>

c(or

(

)

(

)

(

)

(

)

nnn xPxcPxxcPxxL

θθθ

θ

111 ,,,|== ).

Definition: The Maximum Likelihood Estimate (MLE)

The function Ω→S:

ˆ

θ

is called the maximum likelihood estimator.

(

)

s

θ

ˆ is called the maximum likelihood estimate of

θ

if for each

Ω

∈

θ

,

(

)

(

)

(

)

nn xxLxxsL,,|,,|

ˆ11

θθ

≥.

The Algorithm

This suggests that in order to obtain the MLE of

θ

, we maximum the likelihood function. Since a version of

the likelihood version with 1

=

c gives the same maximum value, we use this version. In most cases, this is

done by differentiation.

1) Write the likelihood function

( ) ( )

∏

=

=

n

i

inxfxxL

1

1,|

θ

θ

.

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2) Write the log likelihood function defined by

( ) ( )( ) ( )

=

==

n

i

inn xfxxLxxl

1

11 ln,|ln,|

θ

θθ

.

3) Write the score function

( )

(

)

θ

θ

θ

∂

∂

=n

n

xxl

xxS,|

,|1

1

.

4) Write the score equation

(

)

0,|1=

n

xxS

θ

and solve for

θ

.

5) Check that the solution is the global maximum. If it is, then it is the MLE of

θ

.

Theorem

If

(

)

n

xx ,

ˆ1

θ

is the MLE in Ω, and Ω′

→Ω1-1

:

φ

, then the MLE in the new parameterization is

(

)

(

)

(

)

nn xxxx ,

ˆ

,

ˆ11

θθφ

′

=.

Proof:

(

)

(

)

( )

(

)

( )

( )

(

)

( )

( ) ( )

( )

( ) ( ) ( ) ( )

nnnn

nnn

xx

n

xx

n

xx

nn

xxLxxgxxfxxL

xxxxLxxf

xxgxxgxxxxL

n

nn

,,|,,,,,,|

,,|,,

ˆ

,,

,,,,,,|,,

ˆ

1

*

111

111

,,

ˆ

1

,,

ˆ

1

,,

ˆ

11

*

1

11

θθ

θ

θ

θθ

θ

θφθ

′

===≥

==

==

′

′

′

Hence for every Ω

′

∈

′

θ

,

(

)

(

)

(

)

nnn xxLxxxxL,,|,,|,,

ˆ1

*

11

*

θθ

′

≥

′, and so

(

)

n

xx ,,

ˆ1

θ

′ is the MLE

of the new parameterization.

The Algorithm: The Multidimensional Case

In the multidimensional case, the parameter space is

(

)

{

}

1,,,

1>=Ωk

k

θθ

.

1) Write the likelihood function

(

)

(

)

nkxxL,|,, 11

θθ

.

2) Write the log likelihood function defined by

(

)

(

)

(

)

(

)

(

)

nknkxxLxxl,|,,ln,|,, 1111

θθθθ

=.

3) Write the score function

( )( )

(

)

(

)

( )( )

∂

∂

∂

∂

=

k

nk

nk

nkxxl

xxl

xxS

θ

θθ

θ

θθ

θθ

,|,,

,|,,

,|,,

11

1

11

11

.

4) Write the score equation

( )( )

(

)

(

)

( )( )

=

∂

∂

∂

∂

=

0

0

,|,,

,|,,

0,|,,

11

1

11

11

k

nk

nk

nkxxl

xxl

xxS

θ

θθ

θ

θθ

θθ

and solve.

5) Check that the solutions are the global maximum (the matrix of the second partial derivatives evaluated at

(

)

k

θθ

ˆ

,,

ˆ1 must be negative definite, or equivalently, all eigenvalues negative).

STANDARD ERROR AND BIAS

Suppose

θ

ˆ is the MLE;

(

)

(

)

(

)

nn xxxx ,,

ˆ

,,

ˆ11

θφφ

= is the estimate of

(

)

θφ

. How reliable are the estimates?

One measure of accuracy commonly used is MSE (mean squared error).

Definition: Mean Squared Error

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