Download Power, Neyman-Pearson, and Most Powerful Tests

MATH 4820/5320: Intro Math Stat
Power, Neyman-Pearson,
and Most Powerful Tests
Stephan R Sain
Department of Mathematical Sciences
University of Colorado at Denver
[email protected]
math.cudenver.edu/∼ssain
S PRING 2006
L ECTURE 13 (lec13.pdf)
04/05/06
Outline
• Neyman-Pearson Lemma
– Most powerful tests
• Uniformly most powerful tests
• Announcements:
– Homework 9 due Wednesday.
– Exam II on Wednesday.
– Final Exam
∗ Monday, May 8, 4-6:00 PM, SS 126B.
1
Power
• Recall the definitions of α and β :
–
α = P [Type I Error] =
P [Reject H0 when H0 is true]
–
β = P [Type II Error] =
P [Fail to reject H0 when H0 is false]
• Also, recall the definition of power, i.e. the
probability of correctly rejecting the null hypothesis.
– Power = 1 - Prob[Type II Error] = 1 - β
• Let H0 : θ = θ0 and H1 : θ ∈ θ1 6= θ0 .
– Simple (as opposed to composite) hypothesis.
• Ideally, a test would be able to detect any θ ∈ θ1 ,
i.e. power would be one for all θ ∈ θ1 .
• Clearly, this is not possible.
– sampling variation
• Hence, we want to find the “most powerful test”.
2
Neyman-Pearson Lemma
• Consider testing H0 : θ = θ0 versus
H1 : θ = θ1 based on the information in a random
sample X1 , . . . , Xn from a pdf f (x; θ).
• Let
L(θ) = f (x1 ; θ) · · · f (xn ; θ)
denote the likelihood of the sample for a value of
the parameter θ .
• Then, for a given α, the test statistic that
maximizes the power at θ1 has a critical region
determined by
L(θ0 )
<k
L(θ1 )
where k is chosen so that the test has the desired
value of α.
• Such a test is the most powerful α-level test for H0
versus H1 .
3
Proof
• Let C define the critical region, i.e. C is the set of
all values x1 , . . . , xn such that
L(θ0 )/L(θ1 ) < k .
– If C is the only critical region of size α, then we
are done.
• If another critical region (or equivalently another
test) of size α exists (denoted by A), we need to
show that
Z
Z
C
L(θ1 ) −
A
L(θ1 ) ≥ 0
where
Z
R
Z
L(θ) =
Z
···
R
L(θ; x1 , . . . , xn )dx1 · · · dxn .
4
Proof
• Note that C = (C ∩ A) ∪ (C ∩ A0 ) and
A = (A ∩ C) ∪ (A ∩ C 0 ).
• Then
Z
C
Z
L(θ1 ) −
=
−
=
ZA
L(θ1 )
Z
ZC∩A
ZA∩C
C∩A0
L(θ1 ) +
Z
L(θ1 ) −
C∩A0
A∩C 0
L(θ1 )
L(θ1 )
Z
L(θ1 ) −
A∩C 0
L(θ1 )
• Note that L(θ1 ) ≥ (1/k)L(θ0 ) at each point of
C and L(θ1 ) ≤ (1/k)L(θ0 ) at each point of C 0 .
• Hence,
Z
Z
0
ZC∩A
A∩C 0
L(θ1 ) ≥
L(θ1 ) ≤
5
1
L(θ0 )
k C∩A0
Z
1
L(θ0 )
0
k A∩C
Proof
• These inequalities imply
Z
Z
C∩A0
L(θ1 ) −
A∩C 0
1
k
L(θ1 ) ≥
·Z
¸
Z
C∩A0
L(θ0 ) −
A∩C 0
L(θ0 )
• From earlier,
Z
C
Z
L(θ1 ) −
A
1
k
L(θ1 ) ≥
·Z
¸
Z
C∩A0
L(θ0 ) −
A∩C 0
L(θ0 )
• Similarly,
Z
C∩A0
Z
L(θ0 ) −
ZA∩C
=
C
0
L(θ0 ) −
= α−α
= 0.
6
L(θ0 )
Z
A
L(θ0 )
Proof
• Combining the last two steps gives the desired
result that
Z
C
Z
L(θ1 ) −
A
7
L(θ1 ) ≥ 0.
Example
Suppose Y is a single observations from the
probability density function given by f (y; θ)
for 0
= θy θ−1
< y < 1. Find the most powerful test with
significance level α = 0.05 to test
• H0 : θ = 2 versus H1 : θ = 1
• H0 : θ = 2 versus H1 : θ = 4
8
Another Example
Let X1 , . . . , Xn denote a random sample from a
N (µ, σ 2 ) distribution with σ 2 known. Find the most
powerful test with significance level α for testing
H0 : µ = 0 versus H1 : µ = v where v > 0. How
does the test change if v < 0?
9
Uniformly Most Powerful Tests
• What it the alternative is composite?
– Example:
H0 : θ = θ0 versus H1 : θ > θ0
• No general theorem is available!
• If a test is most powerful for all θ defined by the
alternative, then it is said to be uniformly most
powerful.
– Examine the critical region. In many cases it will
only be a function of θ0 and hence achieves
maximal power for any value of θ under the
alternative.
10
One More Example
Let Y have a binomial(n,p) distribution.
• Find the most powerful test of level α for testing
H0 : p = p0 against H1 : p = p1 where
p1 > p0 .
• If n = 100 determine the critical region for
α = 0.05.
• Is the test uniformly most powerful for testing
H1 : p > p0 .
11
A Last Example
Let X1 , . . . , Xn denote a random sample of size n
from a Poisson distribution with mean λ.
• Find the most powerful test with significance level
α for testing H0 : λ = 2 against H1 : λ = 5.
• If n = 4 determine the critical region for
α = 0.05. How would you determine the critical
region for large n?
• Is the test uniformly most powerful for testing
H1 : λ > 2.
12