Transcript Document

Chapter 6 introduction to Statistical Inference
In this chapter, we will discuss the following problems:
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Point Estimation
Confidence Intervals for Means
Confidence Intervals for Differences of Means
Tests of Statistical Hypotheses
Additional Comments About Statistical Tests
Chi-Square Tests
6.1 Point Estimation
In this chapter we develop statistical inference (estimation and
testing) based on likelihood methods. We show that these procedures
are asymptotically optimal under certain conditions. Suppose that X1,
…, Xn~ (iid) X, with pdf f (x; ), (or pmf p(x; )), .
6.1.1 The Maximum Likelihood Estimation
The parameter  (can be vectors) is unknown. The basis of our
inferential procedures is the likelihood function given by,
n
L( ; x) = L( ; x1 ,, xn )   f ( xi ;  ),   
i 1
(6.1.1)
where x = (x1, … , xn). L( ; x) is the joint pdf or pmf of random
sample X1, …, Xn. We often write L( ; x) as L( ) due to it is a
function of  . The log of L( ) is usually more convenient to work
with mathematically. Denote the logL( ) by
l( ) = logL( ) =
n
 log f ( x ;  ),
i 1
i
  .
(6.1.2)
Note that there is no loss of information in using l( ) because the
log is a one-to-one function.
Example 6.1.1. Let
 x (1   )1 x ,
x  0, 1
X 1 ,, X n ~ p( x;  )  
0,
elsewhere

Where unknown parameter  : 0    1. The problem is how can
we estimate  based on x1, … , xn, the observation of sample?
iid
According to the principle of maximum likelihood, when the
value of  can make L( ), the joint pmf of random sample, get the
maximum value. Then we call this  s value is the best estimate of
. Because it make the event {X1=x1, … , Xn=xn} occur in biggest
probability.
n
n
 L( )   p ( xi ;  )    (1   )
xi
i 1
1 xi
xi
n   xi


(1   )
,
i 1
n
 n



l ( )  log L( )    xi  log    n   xi  log( 1   );
1
 1



dl ( )
 0, We can get the estimator of  is
Let
d
n
1
ˆ   X i  X .
n i 1
Here X is called the maximum likelihood estimator of , and x is
called the maximum likelihood estimate of .
Let 0 denote the true value of . Theorem 6.1.1 gives a theoretical
Reason for maximizing the likelihood function.
Assumptions 6.1.1. (Regularity Conditions)
(R0): The pdfs are distinct: i.e.,    f (xi; )  f (xi; ).
(R1): The pdfs have common support for all .
(R2): The point 0 is an interior point in .
Theorem 6.1.1. Let 0 be the true parameter. Under assumptions
(R0) and (R1),
lim P 0 [ L( 0 ; X )  L( ; X )]  1,
n 
for all    0 .
(6.1.3)
Theorem 6.1.1 says that asymptotically the likelihood function is
maximized at the true value 0. so in considering estimates of 0, it
seems natural to consider the value of  which maximizes the
likelihood.
Definition 6.1.1. (Maximum Likelihood Estimator). We say that
ˆ  ˆ(X) is a maximum likelihood estimator (mle) of  if
ˆ  Arg max L( ; X );
(6.1.4)
The notation Argmax means that L(; X) achieves its maximum
value at ˆ. The mle of  can denoted by ˆMLE .
dL( )
dl ( )
 0, or
 0.
d
d
(6.1.5)
is called estimating equation or likelihood equation.
Example 6.1.2 (Exponential Distribution). Suppose the X1, … , Xn
iid from X~Exp( ). Its pdf is defined by
 1  x /
 e , x0
f ( x;  )  
 0,
x0
Find the mle of  : ˆMLE .
Example 6.1.3(Laplace Distribution). Let X1, … , Xn be iid with
density
1 | x  |
f ( x;  )  e
,    x  ,      .
2
(6.1.6)
This pdf is referred to either the Laplace or the double exponential
distribution. Find the ˆMLE .
n
l ( )  n log 2   | xi   |.
i 1
n
l ( )   sgn( xi   ),
i 1
(6.1.7)
Here sgn(t)=1, 0, 1 depending on whether t>0, t=0, or t<0. Setting
the equation (6.1.7) to 0, the solution for  is median of sample,
which can denoted by Q2 (the 2nd quartile of the sample). i.e
ˆMLE = Q2.
Example 6.1.4(Uniform Distribution). Let X1, … , Xn be iid with
the uniform (0, ) density, i.e.,
1 /  , 0  x  
f ( x;  )  
 0, elsewhere
Find the ˆMLE .
Theorem 6.1.2 Let X1, … , Xn be iid with the pdf f (x; ),  .
For a specified function g, let  = g() be a parameter of interest.
Then g (ˆMLE ) is the mle of =g ().
Theorem 6.1.3. Assume that X1, … , Xn satisfy the regularity
condition (R0) and (R1), 0 is the true parameter, and further that f
(x; ) is differentiable with respect to  in . Then the likelihood
equation,
L( )
l ( )
 0, or
 0.


P
has a solution ˆn such that ˆn  0 .
(6.1.8)
6.1.2 The method of the Moment
Let X1, … , Xn be iid with the pdf f (x; ),  . The parameter
 (can be vectors  = (1, … , r)) is unknown. Setting
k  M k , k  1, 2,
We can find the estimators of the parameters 1, … , r. this
method is called the method of the Moment..It is denoted by ˆM .
It should be done 1, … , r. this method is called the method of
the Moment..It is noted that this could be done in an equivalent
manner by equating
n
  E ( X ) to X and E[( X   ) k ] to  ( X i  X ) k / n, k  2, 3,
1
and so on until unique solutions for 1, … , r are obtained.
6.2 Confidence Intervals for Means
1. Size n sample with unknown meanμbut
known Var. 2
If X1, …, Xn ~ N(, 2), then U  ( X )* 
X 
~ N (0,1)
/ n
 X 


Setting   P | U | z / 2   P 
z
 / 2 
 / n


 P( X  z
/2
/ n    X  z
/2
Then, confidence interval of  is ( x  z
 / n )  1 

/2
/ n ).
If we do not know the distribution of X, then when n is
large, we Still approximately have (C.L.T.)
U  ( X ) ~ N (0, 1)
*
Please look Example 1 on P270.
2. Size n sample with unknown mean  and Var. 2
n
1
 n 2
 E
S 
E[ ( X i  X ) 2 ]   2
 n  1  n  1 i 1
∴ 2 can be replaced by nS2/(n 1). We have an approximate
Confidence interval for . (when n is large)
( X  z / 2 S / n  1)
As a matter of fact, if X1,…,Xn~N(, 2), (i.i.d), then X and S2 are
independent, and
nS 2
  2 ~  2 ( n  1) , T 

X 
~ t ( n  1)
S / n 1
Let’s look Example 2 and 3 on p270-271.
Note: Suppose X1,…, Xn~ (i.i.d) f(x; ), a r. v. Q(X1,…,Xn; ) is a
pivotal quantity (or pivot) if the distribution of Q is independent of
all parameters. The above method to find the c. i. is called pivotal
inference.
As we known, when X1,…,Xn~ (i.i.d)N(, 2),  is unknown, then
X 
U
, ( if  2 is known) is a pivot. Similarly,
/ n
X 
T
, ( if  2 is unknown) is also a pivot.
S / n 1
3. The Confidence interval of p =P(success)
If Y~ b(n, p), we can use the relative frequency Y/n to estimate p.
What is the accuracy.
Y  np


np(1  p)
Y /n p
~ N (0, 1), approximately
p(1  p) / n
(Y / n)  p
 P[ z / 2 
 z / 2 ]  1    , z / 2 is denoted by z0
p(1  p) / n
Y
2
2 1 p
The inequality  K ( p)  (  p )  z0 p
 0.
n
n
That is
2
z
Y z
Y 
2
(1  ) p  ( 2  ) p     0
n
n n
n
2
0
2
0
z02
Y 
1   0   pi  pi  , i  1, 2 s .t .
n
n
p1  p  p2 .
 Y
 Y 
P  p1    p  p2    
 n 
 n
We have an approximate (100)% c. i. [p1, p2] for p. Replacing
p by Y/n in p(1p)/n, we have (P254 Example 1.)
(Y / n)  p
P[ z0 
 z0 ]  
(Y / n)(1  Y / n) / n
Y

(
Y
/
n
)(
1

Y
/
n
)
 for brevity.
∴ we get a c. i.   z0

n
n


6.3 Confidence Intervals for Differences of Means
1. C. I. for differences of means of double normal Distributions
iid
Suppose X1，
 X n1 ~ N ( 1， )，Y1，
，Yn2
2
1
iid
2
N
(

，

~ 2 2 )，
and the samples are independent . Given the confidence coefficient
  1  ，please find c . i . of 1   2 . Where the observed values
of samples are x1，
，
x n1 , y1，
，
yn2 respectively.
I.  12， 22 are known.
The pivot is U 
X  Y  ( 1   2 )
 / n1   / n2
2
1
2
2
~ N (0，
1),
Setting p{U  z / 2 }  1  
The c. i. of 12 can be found from the inequality as follows.
( x  y  z / 2  / n1   / n2 , x  y  z / 2  / n1   / n2 )
2
1
2
2
2
1
2
2
II .  12   22   2 is unknown .
X  Y  ( 1   2 )
The pivot is T 
~ t ( n1  n2  2),
S w 1 / n1  1 / n2
setting P{ T  t / 2 (n1  n2  2)}  1  
The c. i. of 12 can be found from the inequality as follows.
x  y  t / 2 (n1  n2  2) S w 1 / n1  1 / n2
Where Sw is the mixed samples variance.
You can look example 1. on page 278 of the book.
2. The Confidence Intervals for p1 – p2
If Y1~b(n1, p1), Y2 ~ b(n2, p2) and they are independent, Then,
the pivot
(Y1 / n1  Y2 / n2 )  ( p1  p2 )
~ N (0,1)
p1 (1  p1 ) / n1  p2 (1  p2 ) / n2
approximately, n1, n2 are large.
Again replacing pi by Yi/ni in pi(1  pi)/ni, I =1, 2, we have an
approximate (100)% c.i. for p1 p2 as follows :
Y1 Y2
(Y1 / n1 )(1  Y1 / n1 ) (Y2 / n2 )(1  Y2 / n2 )
  z0

n1 n2
n1
n2
You can look example 2. on page 279 of the book.
6.4 Tests of Statistical Hypothesis
Point estimation and confidence intervals are useful statistical
inference procedures. Another type of inference that is frequently
used concerns tests of hypotheses. Suppose a r. v. X~f(x; ) where
, and =01, 01=. We label hypotheses as
H0: 0 versus H1: 1.
(6.4.1)
The hypothesis H0 is referred to as the null hypothesis while H1
is referred to as the alternative hypothesis. Often the null hypothesis
represents no change or no difference from the past while the
alternative represents change or difference. The alternative is often
referred to as the research worker’s hypothesis.
Example 6.4.1 (Zea Mays Data). In 1878 Charles Darwin recorded
some data on the heights of zea mays plants to determine what effect
cross-fertilized or self-fertilized had on the heights of zea mays.
We will represent the data as (X1, Y1), …, (X15, Y15), where Xi and
Yi are the heights of the cross-fertilized and self-fertilized plants,
respectively, in the ith pot. Let Wi=XiYi.  =E(Wi), our hypotheses
are:
H0: =0 versus H1: >0.
(6.4.2)
Hence, 0={0} represents no difference in the treatments and 1=(0,
) represents a difference in the treatments.
The decision rule to take H0 or H1 is based on a sample X1, …, Xn
from the distribution of X and hence, the decision could be wrong.
Table 6.4.1: 22 decision Table for a Test of Hypothesis
Decision
Reject H0
Accept H0
True State of Nature
H0 is true
H1 is true
Type I Error
Correct Decision
Correct Decision
Type II Error
A test of H0 versus H1 is based on a subset C of D. This set C is
called the critical region and its corresponding decision rule (test)
is:
Reject H0, (Accept H1), if (X1, …, Xn)C
(6.4.3)
c
Retain H0, (Reject H1), if (X1, …, Xn)C .
A Type I error occurs if H0 is rejected when it is true while a Type
II error occurs if H0 accepted when H1 is true.
Def. 6.4.1. We say a critical region C is of size  if
  max P [( X 1 ,, X n )  C ].
 0
(6.4.4)
When 1, we want to maximize
1P[Type II Error]= P[(X1, …, Xn)C].
The probability on the right side of this equation is called the
power of the test at . It is the probability that the test detects the
alternative  when 1 is the true parameter. So minimizing the
probability of Type II error is equivalent to maximizing power.
We define the power function of a critical region to be
C()= P[(X1, …, Xn)C]; 1.
(6.4.5)
Hence, given two critical regions C1 and C2 which are both of size
, C1 is better than C2 if C1()  C2() for 1.
Example 6.4.2 (Test for a Binomial Proportion of Success). Let X be
a Bernoulli r. v. with probability of success p. suppose we want to test
at size ,
H0: p=p0 versus H1: p < p0,
(6.4.6)
where p0 is specified. As an illustration, suppose “success” is dying
from a certain disease and p0 is the probability of dying with some
standard treatment.
Remark 6.4.1 (Nomenclature). If a hypothesis completely specifies
the underlying distribution, such as H0: p=p0, in Example 6.4.2, it is
called a simple hypothesis. Most hypotheses, such as H1: p < p0, are
composite hypotheses, because they are composed of many simple
hypotheses and hence do not completely specify the distribution.
Frequently,  is also called the significance level of the test
associated with the critical region, and so on.
Example 6.4.3 (Large Sample Test for the Mean). Let X be a r, v,
with mean  and finite variance 2. We want to test the hypotheses
H0:  = 0 versus H1:  < 0,
(6.4.6)
where 0 is specified. To illustrate, suppose 0 is the mean level on a
standardized test of students who have been taught a course by a
standard method of teaching. Suppose it is hoped that a new method
which incorporates computers will have a mean level  > 0, where
=E(X) and X is the score of a student taught by the new method.
Example 6.4.4 (Test for  under Normality). Let X have a N(,
2) distribution. Consider the hypotheses
H0:  = 0 versus H1:  < 0,
(6.4.6)
where 0 is specified. Assume that the desired size of the test is ,
for 0<  <1. Suppose X1, …, Xn is a r. s. from X. using the
distribution of t(n 1), it is easy to show that the following rejection
rule has exact level :
X  0
 t , n  1 ,
Reject H0 in favor of H1 if T 
(6.4.7)
S/ n
where t, n1 is the upper  critical point of a t distribution with n 1
degrees of freedom; i.e.,  = P(T > t, n1). This is often called the t
test of H0:  = 0.
6.5 Additional Comments About Statistical Tests
All of the alternative hypotheses considered in section 6.4 were
one-sided hypotheses. For illustration, in exercise 6.42 we tested H0:
 =30,000 against the one-sided alternative H1:  >30,000, where 
is the mean of a normal distribution having standard deviation  =
5000. Perhaps in this situation, though, we think the manufacturer’s
process has changed nut are unsure of the direction. That is, we are
interested in the alter-native H1:  30,000. in this section, we
further explore hypotheses testing and we begin with the
construction of a test for a two sided alternative involving the mean
of a r. v.
Example 1 (Large sample Two-Sided Test for the Mean). Let X be a
r. v. with mean  and finite variance 2. We want to test
H0:  =0 versus H1:   0
(6.5.1)
where 0 is specified. Let X1, …, Xn be a r. s. from X. We will use the
decision rule
Reject H0 in favor of H1 if X  h or X  k ,
(6.5.2)
where h and k are such that   PH 0 [ X  h or X  k ].
Clearly h <k, hence, we have
  PH 0 [ X  h or X  k ]  PH 0 [ X  h]  PH 0 [ X  k ].
An intuitive rule is to divide  equally between the two terms on the
right-side of the above expression; that is, h and k chosen by
PH 0 [ X  h]   / 2 and PH 0 [ X  k ]   / 2.
(6.5.3)
By the CLT and the consistency of S2 to 2, we have under H0 that
D
( X  0 ) /( S / n )  N (0, 1).
This and (6.5.3) leads to the approximate decision rule:
X  0
Reject H0 in favor of H1 if
 z / 2 .
S/ n
(6.5.4)
To approximate the power function of the test, we use the CLT.
Upon substituting  for S, it readily follows that the approximate
power function is
 (  )  P ( X  0  z / 2 / n )  P ( X  0  z / 2 / n )
 n ( 0   )

 n ( 0   )
 (6.5.5)
 
 z / 2   1  
 z / 2 ,







 n ( 0   )

n   n ( 0   )
 '( ) 
 z / 2    
 z / 2   ,
 
  




 
() is strictly decreasing for  <0 and strictly increasing for
 >0. Where (z) is the pdf of a standard normal r. v.
Accept H0 if and only if
0  ( X  t / 2,n1 S / n , X  t / 2,n1 S / n )
(6.5.6)
Example 2. Let X1, …, Xn1 iid from N(1, 2); Y1, …, Yn2 iid from
N(2, 2). At  =0.05, reject H0 : 1=2 and accept the one-sided
alternative H1 : 1>2 if
T
Sw
X Y
 t 0.05,n 2 .
1 / n1  1 / n2
where S w2  [( n1  1) S12  (n2  1) S 22 ] /( n  2), n  n1  n2  2.
Example 3. Say X~b(1, p). Consider testing H0: p=p0 against H1:
p<p0. Let X1, …, Xn be a r. s. from X and let
pˆ  X  Y / n, where Y ~ b( n, p).
To test H0 versus H1, we use either
pˆ  p0
Z1 
 c or Z 2 
p0 (1  p0 ) / n
pˆ  p0
 c.
pˆ (1  pˆ ) / n
If n is large, both Z1 and Z2 have approximate N(0, 1) distributions
provided that H0: p=p0 is true. If  is given, c can be decided.
(Y / n  z / 2
(Y / n)(1  Y / n)
(Y / n)(1  Y / n)
, Y / n  z / 2
)
n
n
is a (1 )100% approximate c. i. for p.
Example 4. Let X1, …, X10 be a r. s. from P( ). A critical region for
testing H0:  =0.1 against H1: >0.1 is given by (p290)
10
Y   X i  3.
i 1
Remark (Observed Significance Level). Not many statisticians like
randomized tests in practice, because the use of them means that two
statisticians could make the same assumptions, observe the same
data, apply the same test, and yet make different decisions. Hence
they usually adjust their significance level so as not to randomize.
As a matter of fact, many statisticians report what are commonly
called observed significance level or p-values
6.6 Chi-Square Tests
In this section we introduce tests of statistical hypotheses called
chi-square tests. A test of this sort was originally proposed by Karl
Pearson in 1900, and it provided one of the earlier methods of
statistical inference.
Let’s now discuss some r. vs. that have approximate chi-square
distributions. Let X1 be b(n, p1). Consider the r. v.
X 1  np1
Y
np1 (1  p1 )
which has, as n, a limiting distribution of N(0, 1), we strongly
suspect that the limiting distribution of Z=Y2 is 2(1).
If Gn(y) represents the cdf of Y, we know that (CLT)
lim Gn ( y )  ( y ),    y   ,
n 
Let Hn(z) be the cdf of Z=Y2. Thus, if z  0, (Hn(z)=0, if z < 0)
H n ( z )  P( Z  z )  P(| Y | z )  Gn ( z )  Gn ( z ).
z
1 w 2 / 2
lim H n ( z )  ( z )  (  z )  2
e
dw .
0
n 
2
2
vw
z
1
1 / 2 1  v / 2
 
v
e dv ,
0 (1 / 2)21 / 2
Therefore, the limiting distribution of Y is 2(1).
Now, let X1~b(n, p1), X2 = nX1 and p2=1p1.Then
X1 np1 =(nX2)  n(1 p2) = (X2 np2)
2
2
 1
(
X

np
)
(
X

np
)
1 
2
1
1
1
1
 

Q1  Y 

np1 (1  p1 )
n
 p1 1  p1 
( X 1  np1 ) 2 ( X 2  np2 ) 2


.
np1
np2
We say, when n is positive integer, that Q1 has a limiting chisquare distribution with 1 degree of freedom. This result can be
generalized as follows.
Let X1, …, Xk1 have a multinomial distribution with the
parameters n and p1, …, pk1. Let Xk = n (X1+…+Xk1) and let pk
=1(p1+…+ pk1). Define Qk1 by
( X i  npi )2

.
npi
i 1
k
Qk 1
It is proved in a more advanced course that, as n, Qk1 has a
limiting distribution that is 2(k1). If we accepted this fact, we can
say Qk1 has an approximate chi-square distribution with k1 degree
of freedom when n is a positive integer. This approximation is good
when n is large enough (n  30) and npi  5.
The r. v. Qk1 may serve as basis of the tests of certain statistical
hypotheses which we now discuss. Let the sample space A=A1…
Ak, and AiAj =, ij. Let P(Ai) = pi, i = 1, …, k, where pk
=1(p1+…+pk1), so that pi is the probability that the outcome of
the random experiment of the set Ai. The random experiment is to
be repeated n independent times and X will represent the number of
times the outcome an element of the set Ai. That is, X1, …, Xk1,
Xk=n(X1+…+Xk1) is the multinomial pdf with parameters n, p1,
…, pk1.
Consider the simple hypothesis (concerning this multinomial pmf)
H0: p1=p10, …, pk1=pk1,0, (pk= pk0 =1p10 … pk1,0), where p10 , …,
pk1,0 are specified numbers. It is desired to test H0 against all
alternatives.
If the hypothesis H0 is true, the r. v.
k
( X i  npi 0 ) 2
Qk 1  
.
npi 0
i 1
has an approximate 2(k 1) distribution. If significance level  is
given, then the critical region is Qk1  2(k 1). That is
P(Qk1  2(k 1))= .
This is frequently called a goodness of fit test. There are some
illustrative examples as follows. P280~284 Example 1.~4. P296-299