Brief Review Probability and Statistics. Probability distributions Continuous distributions

Brief Review

Probability and Statistics

Probability distributions

Continuous distributions

Defn (density function)

Let x denote a continuous random variable then f(x) is called the density function of x

1) f(x) ≥ 0

2)

3)

( ) 1f x dx

( )

b

a

f x dx P a x b

Defn (Joint density function)

Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables then

f(x) = f(x1 ,x2 ,x3 , ... , xn)

is called the joint density function of x = (x1 ,x2 ,x3 , ... , xn)

if

1) f(x) ≥ 0

2)

3)

1)( xx df

Rxxx PdfR

)(

Note:

nn dxdxdxxxxfdf 2121 ,,)(

xx

n

R

n

R

dxdxdxxxxfdf 2121 ,,)( xx

Defn (Marginal density function)

The marginal density of x1 = (x1 ,x2 ,x3 , ... , xp) (p < n) is defined by:

f1(x1) = =

where x2 = (xp+1 ,xp+2 ,xp+3 , ... , xn)

2)( xx df 221 ),( xxx df

The marginal density of x2 = (xp+1 ,xp+2 ,xp+3 , ... , xn) is defined by:

f2(x2) = =

where x1 = (x1 ,x2 ,x3 , ... , xp)

121 ),( xxx df 1)( xx df

Defn (Conditional density function)

The conditional density of x1 given x2 (defined in previous slide) (p < n) is defined by:

f1|2(x1 |x2) =

conditional density of x2 given x1 is defined by:

f2|1(x2 |x1) =

22

21

22

),()(

x

xx

x

x

f

f

f

f

11

21

11

),()(

x

xx

x

x

f

f

f

f

Marginal densities describe how the subvector xi behaves ignoring xj

Conditional densities describe how the subvector xi behaves when the subvector xj is held fixed

Defn (Independence)

The two sub-vectors (x1 and x2) are called independent if:

f(x) = f(x1, x2) = f1(x1)f2(x2)

= product of marginals

or

the conditional density of xi given xj :

fi|j(xi |xj) = fi(xi) = marginal density of xi

Example (p-variate Normal)

The random vector x (p × 1) is said to have the

p-variate Normal distribution with

mean vector (p × 1) and

covariance matrix (p × p)

(written x ~ Np(,)) if:

)()'(

2

1exp

2

1 12/12/

μxμxxp

f

Example (bivariate Normal) The random vector is said to have the bivariate

Normal distribution with mean vector

and

covariance matrix

2

1

μ

)()'(

2

1exp

2

1 12/12/

μxμxxp

f

2

1

x

xx

2221

2121

2212

1211

)()'(

2

1exp

2

1, 1

2/121 μxμx

xxf

212/12

122211

,exp2

1xxQ

)()'(,1

2212

121121 μxμx

xxQ

2122211

22211221112

21122 )())((2)(

xxxx

21211

21 ,exp12

1, xxQxxf

21, xxQ

2

2

2

22

2

22

1

11

2

1

11

1

2

xxxx

x

y

f(x,y)

x

y

f(x,y)

x

y

f(x,y)

The Bivariate Normal Distribution

x

y y y

x x1

2

1 1

2 2

Contour Plots of the Bivariate Normal Distribution

x

y y y

x x1

2

1 1

2 2

Scatter Plots of data from the Bivariate Normal Distribution

1 21 2 1 2

1 2 1 2 1 2

1 21 2

1 2

Theorem (Transformations)

Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function f(x1 ,x2 ,x3 , ... , xn) = f(x). Let

y1 =1(x1 ,x2 ,x3 , ... , xn)

y2 =2(x1 ,x2 ,x3 , ... , xn)

...

yn =n(x1 ,x2 ,x3 , ... , xn)

define a 1-1 transformation of x into y.

Then the joint density of y is g(y) given by:

g(y) = f(x)|J| where

),...,,,(

),...,,,(

)(

)(

321

321

n

n

yyyy

xxxxJ

y

x

n

n

nn

n

n

y

x

y

x

y

x

y

x

y

x

y

xy

x

y

x

y

x

...

...

...

...

det

21

22

2

2

1

11

2

1

1

= the Jacobian of the transformation

Corollary (Linear Transformations)

Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function f(x1 ,x2 ,x3 , ... , xn) = f(x). Let

y1 = a11x1 + a12x2 + a13x3 , ... + a1nxn

y2 = a21x1 + a22x2 + a23x3 , ... + a2nxn

...

yn = an1x1 + an2x2 + an3x3 , ... + annxn

define a 1-1 transformation of x into y.

Then the joint density of y is g(y) given by:

)det(

1)(

)det(

1)()( 1

AAf

Afg yxy

nnnn

n

n

aaa

aaa

aaa

A

...

...

...

where

21

22221

11211

Corollary (Linear Transformations for Normal Random variables)

Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables having an n-variate Normal distribution with mean vector and covariance matrix .

i.e. x ~ Nn(, ) Let

y1 = a11x1 + a12x2 + a13x3 , ... + a1nxn

y2 = a21x1 + a22x2 + a23x3 , ... + a2nxn ...

yn = an1x1 + an2x2 + an3x3 , ... + annxn define a 1-1 transformation of x into y.

Then y = (y1 ,y2 ,y3 , ... , yn) ~ Nn(A,AA')

Defn (Expectation)

Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function

f(x) = f(x1 ,x2 ,x3 , ... , xn).

Let U = h(x) = h(x1 ,x2 ,x3 , ... , xn)

Then

xxxx dfhhEUE )()()(

Defn (Conditional Expectation)

Let x = (x1 ,x2 ,x3 , ... , xn) = (x1 , x2 ) denote a vector of continuous random variables with joint density function

f(x) = f(x1 ,x2 ,x3 , ... , xn) = f(x1 , x2 ).

Let U = h(x1) = h(x1 ,x2 ,x3 , ... , xp)

Then the conditional expectation of U given x2

1212|11212 )()()( xxxxxxx dfhhEUE

Defn (Variance)

Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function

f(x) = f(x1 ,x2 ,x3 , ... , xn).

Let U = h(x) = h(x1 ,x2 ,x3 , ... , xn)

Then

222 )()( xx hEhEUEUEUVarU

Defn (Conditional Variance)

Let x = (x1 ,x2 ,x3 , ... , xn) = (x1 , x2 ) denote a vector of continuous random variables with joint density function

f(x) = f(x1 ,x2 ,x3 , ... , xn) = f(x1 , x2 ).

Let U = h(x1) = h(x1 ,x2 ,x3 , ... , xp)

Then the conditional variance of U given x2

22

112 )()( xxxx hEhEUVar

Defn (Covariance, Correlation) Let x = (x1 ,x2 ,x3 , ... , xn) denote a vector of continuous random variables with joint density function

f(x) = f(x1 ,x2 ,x3 , ... , xn).

Let U = h(x) = h(x1 ,x2 ,x3 , ... , xn) and

V = g(x) =g(x1 ,x2 ,x3 , ... , xn) Then the covariance of U and V.

)()()()( xxxx gEghEhE

VEVUEUEVUCov ,

ncorrelatio

)()(

, and

VVarUVar

VUCovUV

Properties

• Expectation

• Variance

• Covariance • Correlation

1. E[a1x1 + a2x2 + a3x3 + ... + anxn]

= a1E[x1] + a2E[x2] + a3E[x3] + ... + anE[xn]

or E[a'x] = a'E[x]

2. E[UV] = E[h(x1)g(x2)]

= E[U]E[V] = E[h(x1)]E[g(x2)]

if x1 and x2 are independent

3. Var[a1x1 + a2x2 + a3x3 + ... + anxn]

or Var[a'x] = a′ a

n

jijiji

n

iii xxCovaaxVara ],[2][

1

2

)(...),(),(

...

),(...)(),(

),(...),()(

where

21

2212

1211

nnn

n

n

xVarxxCovxxCov

xxCovxVarxxCov

xxCovxxCovxVar

4. Cov[a1x1 + a2x2 + ... + anxn ,

b1x1 + b2x2 + ... + bnxn]

or Cov[a'x, b'x] = a′ b

n

jijiji

n

iiji xxCovbaxVarba ],[][

1

5.

6.

22xx UEEUE

22 22xx xx UEVarUVarEUVar

Statistical Inference

Making decisions from data

There are two main areas of Statistical Inference

• Estimation – deciding on the value of a parameter– Point estimation– Confidence Interval, Confidence region Estimation

• Hypothesis testing– Deciding if a statement (hypotheisis) about a

parameter is True or False

The general statistical modelMost data fits this situation

Defn (The Classical Statistical Model)

The data vector

x = (x1 ,x2 ,x3 , ... , xn)

The model

Let f(x| ) = f(x1 ,x2 , ... , xn | 1 , 2 ,... , p) denote the joint density of the data vector x = (x1 ,x2 ,x3 , ... , xn) of observations where the unknown parameter vector (a subset of p-dimensional space).

An Example

The data vector

x = (x1 ,x2 ,x3 , ... , xn) a sample from the normal distribution with mean and variance 2

The model

Then f(x| , 2) = f(x1 ,x2 , ... , xn | , 2), the joint density of x = (x1 ,x2 ,x3 , ... , xn) takes on the form:

where the unknown parameter vector ( , 2) ={(x,y)|-∞ < x < ∞ , 0 ≤ y < ∞}.

n

i

iix

nn

n

i

x

eef 1

22

2

2/1

22

2

1

2

1

x

Defn (Sufficient Statistics)

Let x have joint density f(x| ) where the unknown parameter vector .

Then S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is called a set of sufficient statistics for the parameter vector if the conditional distribution of x given S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is not functionally dependent on the parameter vector .

A set of sufficient statistics contains all of the information concerning the unknown parameter vector

A Simple Example illustrating Sufficiency

Suppose that we observe a Success-Failure experiment n = 3 times. Let denote the probability of Success. Suppose that the data that is collected is x1, x2, x3 where xi takes on the value 1 is the ith trial is a Success and 0 if the ith trial is a Failure.

The following table gives possible values of (x1, x2, x3).

(x1, x2, x3) f(x1, x2, x3|) S =xi g(S |) f(x1, x2, x3| S) (0, 0, 0) (1 - )3 0 (1 - )3 1 (1, 0, 0) (1 - )2 1 1/3 (0, 1, 0) (1 - )2 1 1/3 (0, 0, 1) (1 - )2 1

3(1 - )2

1/3 (1, 1, 0) (1 - )2 2 1/3 (1, 0, 1) (1 - )2 2 1/3 (0, 1, 1) (1 - )2 2

3(1 - )2

1/3 (1, 1, 1) 3 3 3 1

The data can be generated in two equivalent ways:

1. Generating (x1, x2, x3) directly from f (x1, x2, x3|) or

2. Generating S from g(S|) then generating (x1, x2, x3) from f (x1, x2, x3|S). Since the second step does involve no additional information will be obtained by knowing (x1, x2, x3) once S is determined

The Sufficiency Principle

Any decision regarding the parameter should be based on a set of Sufficient statistics S1(x), S2(x), ...,Sk(x) and not otherwise on the value of x.

A useful approach in developing a statistical procedure

1. Find sufficient statistics

2. Develop estimators , tests of hypotheses etc. using only these statistics

Defn (Minimal Sufficient Statistics)

Let x have joint density f(x| ) where the unknown parameter vector .

Then S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is a set of Minimal Sufficient statistics for the parameter vector if S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is a set of Sufficient statistics and can be calculated from any other set of Sufficient statistics.

Theorem (The Factorization Criterion)

Let x have joint density f(x| ) where the unknown parameter vector .

Then S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is a set of Sufficient statistics for the parameter vector if

f(x| ) = h(x)g(S, )

= h(x)g(S1(x) ,S2(x) ,S3(x) , ... , Sk(x), ).

This is useful for finding Sufficient statistics

i.e. If you can factor out q-dependence with a set of statistics then these statistics are a set of Sufficient statistics

Defn (Completeness)

Let x have joint density f(x| ) where the unknown parameter vector .

Then S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is a set of Complete Sufficient statistics for the parameter vector if S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is a set of Sufficient statistics and whenever

E[(S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) ] = 0

then

P[(S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) = 0] = 1

Defn (The Exponential Family)

Let x have joint density f(x| )| where the unknown parameter vector . Then f(x| ) is said to be a member of the exponential family of distributions if:

,

0

)()(exp)()(1

Otherwise

bxapSghf iiii

k

ii θxθx

θx

,where

1) - ∞ < ai < bi < ∞ are not dependent on .

2) contains a nondegenerate k-dimensional rectangle.

3) g(), ai ,bi and pi() are not dependent on x.

4) h(x), ai ,bi and Si(x) are not dependent on q.

If in addition.

5) The Si(x) are functionally independent for i = 1, 2,..., k.

6) [Si(x)]/ xj exists and is continuous for all i = 1, 2,..., k j = 1, 2,..., n.

7) pi() is a continuous function of for all i = 1, 2,..., k.

8) R = {[p1(),p2(), ...,pK()] | ,} contains nondegenerate k-dimensional rectangle.

Then

the set of statistics S1(x), S2(x), ...,Sk(x) form a Minimal Complete set of Sufficient statistics.

Defn (The Likelihood function)

Let x have joint density f(x|) where the unkown parameter vector . Then for a

given value of the observation vector x ,the Likelihood function, Lx(), is defined by:

Lx() = f(x|) with

The log Likelihood function lx() is defined by:

lx() =lnLx() = lnf(x|) with

The Likelihood Principle

Any decision regarding the parameter should be based on the likelihood function Lx() and not otherwise on the value of x.

If two data sets result in the same likelihood function the decision regarding should be the same.

Some statisticians find it useful to plot the likelihood function Lx() given the value of x.

It summarizes the information contained in x regarding the parameter vector .

An Example

The data vector

x = (x1 ,x2 ,x3 , ... , xn) a sample from the normal distribution with mean and variance 2

The joint distribution of x

Then f(x| , 2) = f(x1 ,x2 , ... , xn | , 2), the joint density of x = (x1 ,x2 ,x3 , ... , xn) takes on the form:

where the unknown parameter vector ( , 2) ={(x,y)|-∞ < x < ∞ , 0 ≤ y < ∞}.

n

i

iix

nn

n

i

x

eef 1

22

2

2/1

22

2

1

2

1

x

The Likelihood function

Assume data vector is known

x = (x1 ,x2 ,x3 , ... , xn)

The Likelihood function

Then L( , )= f(x| , ) = f(x1 ,x2 , ... , xn | , 2),

22

1 22/ 2

1

1 1

2 2

nii

i

xxn

n ni

e e

2

1

1

2

/ 2

1

2

n

ii

x

n ne

2 2

1

12

2

/ 2

1

2

n

i ii

x x

n ne

or

2 2

1

12

2

/ 2

1,

2

n

i ii

x x

n nL e

2 2

1 1

12

2

/ 2

1

2

n n

i ii i

x x n

n ne

2 2 21

1 22

/ 2

1

2

n s nx nx n

n ne

2 2

2 2 2 21

1

since or 11

n

i ni

ii

x nxs x n s nx

n

1

1

and since then

n

i ni

ii

xx x nx

n

hence

2 2 211 2

2/ 2

1,

2

n s nx nx n

n nL e

221

12

/ 2

1

2

n s n x

n ne

Now consider the following data: (n = 10)

57.1 72.3 75.0 57.8 50.3 48.0 49.6 53.1 58.5 53.7

mean 57.54s 9.2185

2 219 9.2185 10 57.54

25 10

1,

6.2832L e

1

S1

0

5E-17

1E-16

1.5E-16

2E-16

2.5E-16

3E-16

Likelihood n = 10

0

2050

70

1S1

Contour Map of Likelihood n = 100

0 20

50

70

Now consider the following data: (n = 100)

2 2199 11.8571 100 62.02

250 100

1,

6.2832L e

57.1 72.3 75.0 57.8 50.3 48.0 49.6 53.1 58.5 53.7

77.8 43.0 69.8 65.1 71.1 44.4 64.4 52.9 56.4 43.9

49.0 37.6 65.5 50.4 40.7 66.9 51.5 55.8 49.1 59.5

64.5 67.6 79.9 48.0 68.1 68.0 65.8 61.3 75.0 78.0

61.8 69.0 56.2 77.2 57.5 84.0 45.5 64.4 58.7 77.5

81.9 77.1 58.7 71.2 58.1 50.3 53.2 47.6 53.3 76.4

69.8 57.8 65.9 63.0 43.5 70.7 85.2 57.2 78.9 72.9

78.6 53.9 61.9 75.2 62.2 53.2 73.0 38.9 75.4 69.7

68.8 77.0 51.2 65.6 44.7 40.4 72.1 68.1 82.2 64.7

83.1 71.9 65.4 45.0 51.6 48.3 58.5 65.3 65.9 59.6

mean 62.02s 11.8571

1

S1

0

2E-170

4E-170

6E-170

8E-170

1E-169

1.2E-169

1.4E-169

1.6E-169

Likelihood n = 100

0

2050

70

1S1

Contour Map of Likelihood n = 100

0 20

50

70

The Sufficiency Principle

Any decision regarding the parameter should be based on a set of Sufficient statistics S1(x), S2(x), ...,Sk(x) and not otherwise on the value of x.

If two data sets result in the same values for the set of Sufficient statistics the decision regarding should be the same.

Theorem (Birnbaum - Equivalency of the Likelihood Principle and Sufficiency Principle)

Lx1() Lx

2()

if and only if

S1(x1) = S1(x2),..., and Sk(x1) = Sk(x2)

The following table gives possible values of (x1, x2, x3).

(x1, x2, x3) f(x1, x2, x3|) S =xi g(S |) f(x1, x2, x3| S) (0, 0, 0) (1 - )3 0 (1 - )3 1 (1, 0, 0) (1 - )2 1 1/3 (0, 1, 0) (1 - )2 1 1/3 (0, 0, 1) (1 - )2 1

3(1 - )2

1/3 (1, 1, 0) (1 - )2 2 1/3 (1, 0, 1) (1 - )2 2 1/3 (0, 1, 1) (1 - )2 2

3(1 - )2

1/3 (1, 1, 1) 3 3 3 1

The Likelihood function

S = 0

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

S = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8 1

S = 2

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 0.2 0.4 0.6 0.8 1

S = 3

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Estimation Theory

Point Estimation

Defn (Estimator)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector .

Then an estimator of the parameter () = (1 ,2 , ... , k) is any function T(x)=T(x1 ,x2 ,x3 , ... , xn) of the observation vector.

Defn (Mean Square Error)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Let T(x) be an estimator of the parameter (). Then the Mean Square Error of T(x) is defined to be:

2))()((... θxθx TEESM T

xθxθx dfT )|())()(( 2

Defn (Uniformly Better)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Let T(x) and T*(x) be estimators of the parameter (). Then T(x) is said to be uniformly better than T*(x) if:

θθ xx *...... TT ESMESM θwhenever

Defn (Unbiased )

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Let T(x) be an estimator of the parameter (). Then T(x) is said to be an unbiased estimator of the parameter () if:

θxθxxx dfTTE )|()(

Theorem (Cramer Rao Lower bound) Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Suppose that: i) exists for all x and for all . θ

θ

θx

)|(f

ii)

xθ

θxxθx

θd

fdf

)|()|(

iii)

iv)

xθ

θxxxθxx

θd

ftdft

)|()|(

θ

θx allfor

)|(0

2

i

fE

Let M denote the p x p matrix with ijth element.

θ̂

pjif

Emji

ij ,,2,1, )|(ln2

θx

Then V = M-1 is the lower bound for the covariance matrix of unbiased estimators of .

That is, var(c' ) = c'var( )c ≥ c'M-1c = c'Vc where is a vector of unbiased estimators of .

θ̂ θ̂

Defn (Uniformly Minimum Variance Unbiased Estimator)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Then T*(x) is said to be the UMVU (Uniformly minimum variance unbiased) estimator of() if:

1) E[T*(x)] = () for all .2) Var[T*(x)] ≤ Var[T(x)] for all

whenever E[T(x)] = ().

Theorem (Rao-Blackwell)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Let S1(x), S2(x), ...,SK(x) denote a set of sufficient statistics.Let T(x) be any unbiased estimator of (). Then T*[S1(x), S2(x), ...,Sk (x)] = E[T(x)|S1(x), S2(x), ...,Sk (x)] is an unbiased estimator of () such that:

Var[T*(S1(x), S2(x), ...,Sk(x))] ≤ Var[T(x)] for all .

Theorem (Lehmann-Scheffe')

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector .

Let S1(x), S2(x), ...,SK(x) denote a set of complete

sufficient statistics.

Let T*[S1(x), S2(x), ...,Sk (x)] be an unbiased estimator of (). Then:

T*(S1(x), S2(x), ...,Sk(x)) )] is the UMVU estimator of ().

Defn (Consistency)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Let Tn(x) be an estimator of(). Then Tn(x) is called a consistent estimator of () if for any > 0:

θθx allfor 0lim nn

TP

Defn (M. S. E. Consistency)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Let Tn(x) be an estimator of(). Then Tn(x) is called a M. S. E. consistent estimator of () if for any > 0:

0lim...lim 2

θxθ nn

Tn

TEESMn

θ allfor

Methods for Finding Estimators

1. The Method of Moments

2. Maximum Likelihood Estimation

Methods for finding estimators

1. Method of Moments

2. Maximum Likelihood Estimation

Let x1, … , xn denote a sample from the density function

f(x; 1, … , p) = f(x; )

Method of Moments

The kth moment of the distribution being sampled is defined to be:

1 1, , ; , ,k kk p pE x x f x dx

To find the method of moments estimator of 1, … , p we set up the equations:

The kth sample moment is defined to be:

1

1 nk

k ii

m xn

1 1 1, , p m

2 1 2, , p m

1, ,p p pm

for 1, … , p.

We then solve the equations

1 1 1, , p m

2 1 2, , p m

1, ,p p pm

The solutions 1, , p

are called the method of moments estimators

The Method of Maximum Likelihood

Suppose that the data x1, … , xn has joint density function

f(x1, … , xn ; 1, … , p)

where (1, … , p) are unknown parameters assumed to lie in (a subset of p-dimensional space).

We want to estimate the parameters1, … , p

Definition: Maximum Likelihood Estimation

Suppose that the data x1, … , xn has joint density function

f(x1, … , xn ; 1, … , p)

Then the Likelihood function is defined to be

L() = L(1, … , p)

= f(x1, … , xn ; 1, … , p)

the Maximum Likelihood estimators of the parameters 1, … , p are the values that maximize

L() = L(1, … , p)

the Maximum Likelihood estimators of the parameters 1, … , p are the values

1

1 1, ,

ˆ ˆ, , max , ,p

p pL L

1̂ˆ, , p

Such that

Note: 1maximizing , , pL is equivalent to maximizing

1 1, , ln , ,p pl L

the log-likelihood function

Application

The General Linear Model

Consider the random variable Y with

1. E[Y] = g(U1 ,U2 , ... , Uk)

= 11(U1 ,U2 , ... , Uk) + 22(U1 ,U2 , ... , Uk) + ... + pp(U1 ,U2 , ... , Uk)

=

and

2. var(Y) = 2

• where 1, 2 , ... ,p are unknown parameters

• and 1 ,2 , ... , p are known functions of the nonrandom variables U1 ,U2 , ... , Uk.

• Assume further that Y is normally distributed.

k

p

iii UUU ,...,, 2

1

Thus the density of Y is:

f(Y|1, 2 , ... ,p, 2) = f(Y| , 2)

2

2122),...,,(

2

1exp

2

1kUUUgY

s

2

211

22,...,

2

1exp

2

1ki

p

ii UUUY

2

221122...

2

1exp

2

1pp XXXY

kii UUUX ,..., where 21 i = 1,2, … , p

Now suppose that n independent observations of Y,

(y1, y2, ..., yn) are made

corresponding to n sets of values of (U1 ,U2 , ... , Uk) - (u11 ,u12 , ... , u1k),

(u21 ,u22 , ... , u2k),...

(un1 ,un2 , ... , unk).

Let xij = j(ui1 ,ui2 , ... , uik) j =1, 2, ..., p; i =1, 2, ..., n.

Then the joint density of y = (y1, y2, ... yn) is:

f(y1, y2, ..., yn|1, 2 , ... ,p, 2) = f(y|, 2)

n

ikiiiin

uuugy1

22122/2

),...,,(2

1exp

2

1

n

i

p

jkiiijjin

uuuy1

2

12122/2

),...,,(2

1exp

2

1

n

i

p

jijjin

xy1

2

122/2 2

1exp

2

1

XβyXβy

22/2 2

1exp

2

1

n

XβXβXβyyy 2

2

1exp

2

122/2 n

XβyyyXβXβ 2

2

1exp

2

1exp

2

1222/2 n

Xβyyyβy 2

2

1exp,

22

gh

Thus f(y|,2) is a member of the exponential family of distributions

and S = (y'y, X'y) is a Minimal Complete set of Sufficient Statistics.

Hypothesis Testing

Defn (Test of size )

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x| ) where the unknown parameter vector .

Let be any subset of .

Consider testing the the Null Hypothesis

H0:

against the alternative hypothesis

H1: .

Let A denote the acceptance region for the test. (all values x = (x1 ,x2 ,x3 , ... , xn) of such that the decision to accept H0 is made.)

and let C denote the critical region for the test (all values x = (x1 ,x2 ,x3 , ... , xn) of such that the decision to reject H0 is made.).

Then the test is said to be of size if

and allfor )|( θxθxxC

dfCP

0 oneleast at for )|( θxθxxC

dfCP

Defn (Power) Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x| ) where the unknown parameter vector .

Consider testing the the Null Hypothesis

H0:

against the alternative hypothesis

H1: .

where is any subset of . Then the Power of the test for is defined to be:

C

C dfCP xθxxθ )|(

Defn (Uniformly Most Powerful (UMP) test of

size )

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|) where the unknown parameter vector . Consider testing the the Null Hypothesis

H0: against the alternative hypothesis

H1: . where is any subset of .Let C denote the critical region for the test . Then the test is called the UMP test of size if:

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x| ) where the unknown parameter vector . Consider testing the the Null Hypothesis

H0: against the alternative hypothesis

H1: . where is any subset of .Let C denote the critical region for the test . Then the test is called the UMP test of size if:

and allfor )|( θxθxxC

dfCP

0 oneleast at for )|( θxθxxC

dfCP

and for any other critical region C* such that:

and allfor )|(**

θxθxxC

dfCP

0

*

oneleast at for )|(* θxθxxC

dfCP

then

. allfor )|()|(*

θxθxxθxCC

dfdf

Theorem (Neymann-Pearson Lemma)Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x| ) where the unknown parameter vector = (0, 1).

Consider testing the the Null Hypothesis

H0: = 0

against the alternative hypothesis

H1: = 1.

Then the UMP test of size has critical region:

Kf

fC

)|(

)|(

1

0

θx

θxx

where K is chosen so that C

df xθx )|( 0

Defn (Likelihood Ratio Test of size )Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x| ) where the unknown parameter vector .

Consider testing the the Null Hypothesis

H0:

against the alternative hypothesis

H1: .

where is any subset of Then the Likelihood Ratio (LR) test of size a has critical region:

where K is chosen so that

Kf

fC

)|(max

)|(max

θx

θxx

θ

θ

and allfor )|( θxθxxC

dfCP

0 oneleast at for )|( θxθxxC

dfCP

Theorem (Asymptotic distribution of Likelihood ratio test criterion)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x| ) where the unknown parameter vector .

Consider testing the the Null Hypothesis

H0:

against the alternative hypothesis

H1: .

where is any subset of

Then under proper regularity conditions on U = -2ln(x) possesses an asymptotic Chi-square distribution with degrees of freedom equal to the difference between the number of independent parameters in and .

)|(max

)|(maxLet

θx

θxx

θ

θ

f

f