Name:_
SAMPLE FINAL EXAM
Directions: Mark your final answer to each question on the Parscore form.
In an effort to improve customer service, Dell monitors the time customers wait on hold
when calling in for support. The following summary data represents t
III. Non-UMP Tests (Continued)
2. Locally Most Powerful Tests
Unbiased tests are designed to give high power over all parameter values in the region of
the alternative hypothesis. An alternative is to restrict the test to a specified region and
seek the m
III. Non-UMP Tests
Suppose that UMP tests do not exist.
1. Unbiased Tests
Definition: A test with power function ( ) is unbiased if ( ' ) ( " ) for every
' co , " o .
In common language, the power is greater in the region that is the complement of the nu
OTHER HYPOTHESIS TESTING ISSUES
I. Size of U-I and I-U Tests
Recall from our previous lecture:
1). Union Intersection Tests
U-I: H 0 :
True only if All are true.
Specify:
R : cfw_ x : T ( x ) R set of
Sometimes the U-I test region is simple. For exam
OTHER HYPOTHESIS TESTING ISSUES
II. Asymptotic Results (Ch 10.3)
1) It can often be very difficult to derive the actual distribution of a LRT statistic so that yu
can get the value of c in ( x ) < c . Sometimes asymptotic results can be used.
Theorem: Giv
EVALUATING HYPOTHESIS TESTS
I. Best Tests of Hypothesis
C is a best critical region of size for testing simple hypotheses
H 0 : = '
H1 : = '
if for every subset A of the sample space for which P [ ( X 1 . X n ) A; H 0 ] = .
a) P [ ( X 1 . X n ) C ; H 0 ]
II. Types of Hypothesis Tests
A. Likelihood Ratio Tests
The statistic test (x ) is computed as:
(x ) =
sup L ( |x )
0
sup L ( |x )
The supremum in the numerator is relative to the domain of the null hypothesis, while the
supremum in the denominator is re
B. Bayesian Tests
-Utilize the posterior distribution ( | x )
-Calculate the P ( 0 | x ) and P ( 0c | x )
-Set up tests to reflect desired probabilities of error
For example:
cfw_x : P( 0c | x) > 1/ 2
i.e.
cfw_P(
0
| x ) > cfw_ P( 0c | x )
Bayes Test P
HYPOTHESIS TESTING
I. Introduction
Def: A hypothesis test is a statement about a population parameter(s)
Two complementary hypotheses:
H 0 : Null Hypothesis (Simple/Composite)
H1 : Alternative Hypothesis (Simple/Composite)
Example:
H 0 : = 0
H1 : > 0
Def:
METHODS OF EVALUATING ESTIMATORS
There are often multiple point estimators for a parameter(s). Several criteria are used to
compare estimators:
1. Bias: The difference between the expected value of the estimator W and the
parameter being estimated.
Exampl
Asymptotic variances of MLEs and of functions of MLEs
a) From Casella and Berger, one approach is to use the Rao-Cramer Lower Bound.
h ' ( )
Var h =
.
I n ( )
() )
2
This looks almost like the Rao-Cramer lower bound, except for the numerator. There, we
II. Corollaries and Extensions to the Neyman Pearson Lemma
A. Recall Neyman Pearson Lemma
Test H0: = o vs H1: = 1 , where X ~ f ( x i ) , i = 0 ,1
(Since two alternatives), using
x R if f ( x | 1 ) > k f ( x | 0 )
x R c if f ( x | 1 ) < k f ( x | 0 )
k 0,