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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). Defn (Joint density function).

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Brief Review

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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)


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)


Note:


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)

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

f2(x2) = =

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


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)=


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 m(p× 1) and

covariance matrix S(p×p)

(written x ~ Np(m,S)) if:


Example (bivariate Normal)

The random vector is said to have the bivariate

Normal distribution with mean vector

and

covariance matrix


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 =f1(x1 ,x2 ,x3 , ... , xn)

y2 =f2(x1 ,x2 ,x3 , ... , xn)

...

yn =fn(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

= 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:


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 m and covariance matrix S.

i.e. x ~ Nn(m, S)

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(Am,ASA')


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


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


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


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


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.


Properties

  • Expectation

  • Variance

  • Covariance

  • Correlation


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

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

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


  • E[UV] = E[h(x1)g(x2)]

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

    if x1 and x2 are independent


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

    or Var[a'x] = a′Sa


  • Cov[a1x1 + a2x2 + ... + anxn ,

    b1x1 + b2x2 + ... + bnxn]

    or Cov[a'x, b'x] = a′Sb


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|q) = f(x1 ,x2 , ... , xn|q1 , q2 ,... , qp) denote the joint density of the data vector x = (x1 ,x2 ,x3 , ... , xn) of observations where the unknown parameter vector qW (a subset of p-dimensional space).


An Example

The data vector

x = (x1 ,x2 ,x3 , ... , xn) a sample from the normal distribution with mean m and variance s2

The model

Then f(x|m , s2) = f(x1 ,x2 , ... , xn|m , s2), the joint density of x = (x1 ,x2 ,x3 , ... , xn) takes on the form:

where the unknown parameter vector q = (m , s2) W ={(x,y)|-∞ < x < ∞ , 0 ≤ y < ∞}.


Defn (Sufficient Statistics)

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

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

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 q 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).

The data can be generated in two equivalent ways:

  • Generating (x1, x2, x3) directly from f (x1, x2, x3|q) or

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


The Sufficiency Principle

Any decision regarding the parameter qshould 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

  • Find sufficient statistics

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


Defn (Minimal Sufficient Statistics)

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

Then S = (S1(x) ,S2(x) ,S3(x) , ... , Sk(x)) is a set of Minimal Sufficient statistics for the parameter vector qif 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|q) where the unknown parameter vector qW.

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

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

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

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|q) where the unknown parameter vector qW.

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

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

then

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


Defn (The Exponential Family)

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

qW,where


  • - ∞ < ai < bi < ∞ are not dependent on q.

    2) W contains a nondegenerate k-dimensional rectangle.

    3) g(q), ai,bi and pi(q) 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(q) is a continuous function of qfor all i = 1, 2,..., k.

8) R = {[p1(q),p2(q), ...,pK(q)] | qW,} 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|q) where the unkown parameter vector qW. Then for a

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

Lx(q) = f(x|q) with qW

The log Likelihood functionlx(q) is defined by:

lx(q) =lnLx(q) = lnf(x|q) with qW


The Likelihood Principle

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

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


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

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


An Example

The data vector

x = (x1 ,x2 ,x3 , ... , xn) a sample from the normal distribution with mean m and variance s2

The joint distribution of x

Then f(x|m , s2) = f(x1 ,x2 , ... , xn|m , s2), the joint density of x = (x1 ,x2 ,x3 , ... , xn) takes on the form:

where the unknown parameter vector q = (m , s2) W ={(x,y)|-∞ < x < ∞ , 0 ≤ y < ∞}.


The Likelihood function

Assume data vector is known

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

The Likelihood function

Then L(m , s)= f(x|m , s) = f(x1 ,x2 , ... , xn|m , s2),


or


hence

Now consider the following data: (n = 10)


70

m

0

s

50

20


70

m

50

20

0

s


Now consider the following data: (n = 100)


70

m

0

s

50

20


70

m

50

20

0

s


The Sufficiency Principle

Any decision regarding the parameter qshould 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 q should be the same.


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

Lx1(q) Lx2(q)

if and only if

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


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

The Likelihood function


Estimation Theory

Point Estimation


Defn (Estimator)

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

Then an estimatorof the parameter f(q) = f(q1 ,q2 , ... , qk) 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|q) where the unknown parameter vector q W.

Let T(x) be an estimator of the parameter

f(q). Then the Mean Square Error of T(x) is defined to be:


Defn (Uniformly Better)

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

Let T(x) and T*(x) be estimators of the parameter f(q). Then T(x) is said to be uniformly better than T*(x) if:


Defn (Unbiased)

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

Let T(x) be an estimator of the parameter f(q). Then T(x) is said to be an unbiased estimator of the parameter f(q) if:


Theorem (Cramer Rao Lower bound)

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

i) exists for all x and for all .

ii)

iii)

iv)


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

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

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


Defn (Uniformly Minimum Variance Unbiased Estimator)

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

1) E[T*(x)] = f(q) for all qW.

2) Var[T*(x)] ≤ Var[T(x)] for all qW whenever E[T(x)] = f(q).


Theorem (Rao-Blackwell)

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

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

Let T(x) be any unbiased estimator of f(q).

Then T*[S1(x), S2(x), ...,Sk(x)] = E[T(x)|S1(x), S2(x), ...,Sk(x)] is an unbiased estimator of f(q) such that:

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

for all qW.


Theorem (Lehmann-Scheffe')

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

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

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

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


Defn (Consistency)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|q) where the unknown parameter vector qW. Let Tn(x) be anestimator off(q). Then Tn(x) is called a consistentestimator of f(q) if for any e > 0:


Defn (M. S. E. Consistency)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|q) where the unknown parameter vector qW. Let Tn(x) be anestimator off(q). Then Tn(x) is called a M. S. E. consistentestimator of f(q) if for any e > 0:


Methods for Finding Estimators

The Method of Moments

Maximum Likelihood Estimation


Methods for finding estimators

  • Method of Moments

  • Maximum Likelihood Estimation


Method of Moments

Let x1, … , xndenote a sample from the density function

f(x; q1, … , qp) = f(x; q)

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


The kth sample moment is defined to be:

To find the method of moments estimator of q1, … , qpwe set up the equations:


We then solve the equations

for q1, … , qp.

The solutions

are called the method of moments estimators


The Method of Maximum Likelihood

Suppose that the data x1, … , xnhas joint density function

f(x1, … , xn; q1, … , qp)

where q = (q1, … , qp) are unknown parameters assumed to lie in W(a subset of p-dimensional space).

We want to estimate the parametersq1, … , qp


Definition: Maximum Likelihood Estimation

Suppose that the data x1, … , xnhas joint density function

f(x1, … , xn; q1, … , qp)

Then the Likelihood function is defined to be

L(q) = L(q1, … , qp)

= f(x1, … , xn; q1, … , qp)

the Maximum Likelihood estimators of the parameters q1, … , qp are the values that maximize

L(q) = L(q1, … , qp)


the Maximum Likelihood estimators of the parameters q1, … , qp are the values

Such that

Note:

is equivalent to maximizing

the log-likelihood function


Application

The General Linear Model


Consider the random variable Y with

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

= b1f1(U1 ,U2 , ... , Uk) + b2f2(U1 ,U2 , ... , Uk) + ... + bpfp(U1 ,U2 , ... , Uk)

=

and

2. var(Y) = s2

  • where b1, b2 , ... ,bp are unknown parameters

  • and f1 ,f2 , ... , fp are known functions of the nonrandom variables U1 ,U2 , ... , Uk.

  • Assume further that Y is normally distributed.


Thus the density of Y is:

f(Y|b1, b2 , ... ,bp, s2) = f(Y| b, s2)

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 = fj(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|b1, b2 , ... ,bp, s2) = f(y|b, s2)


Thus f(y|b,s2) 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 sizea)

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

Letw be any subset ofW.

Consider testing the the Null Hypothesis

H0:q w

against the alternative hypothesis

H1:q w.


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 a if


Defn (Power)

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

Consider testing the the Null Hypothesis

H0:q w

against the alternative hypothesis

H1:qw.

wherew is any subset ofW. Then the Power of the test forqwis defined to be:


Defn (Uniformly Most Powerful (UMP) test of size a)

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

Consider testing the the Null Hypothesis

H0:q w

against the alternative hypothesis

H1:q w.

wherew is any subset ofW.

Let C denote the critical region for the test . Then the test is called the UMP test of sizeaif:


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

Consider testing the the Null Hypothesis

H0:q w

against the alternative hypothesis

H1:q w.

wherew is any subset ofW.

Let C denote the critical region for the test . Then the test is called the UMP test of sizeaif:


and for any other critical region C* such that:

then


Theorem(Neymann-Pearson Lemma)

Let x = (x1 ,x2 ,x3 , ... , xn) denote the vector of observations having joint density f(x|q)where the unknown parameter vectorq W = (q0, q1).

Consider testing the the Null Hypothesis

H0:q=q0

against the alternative hypothesis

H1:q=q1.

Then the UMP test of sizeahas critical region:

where K is chosen so that


Defn (Likelihood Ratio Test of sizea)

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

Consider testing the the Null Hypothesis

H0:q w

against the alternative hypothesis

H1:q w.

wherew is any subset ofW

Then the Likelihood Ratio (LR) test of size a has critical region:

where K is chosen so that


Theorem (Asymptotic distribution of Likelihood ratio test criterion)

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

Consider testing the the Null Hypothesis

H0:q w

against the alternative hypothesis

H1:q w.

wherew is any subset ofW

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


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