1. Likelihood ratio tests
3. Hypotheses testing
Introduction
Wald tests
p – values
Likelihood ratio tests 1
STATISTICAL INFERENCE

2. Hypotheses testing: introduction
Goal:
not finding a parameter value, but deciding on
the validity of a statement about the
parameter .
This statement is the null hypothesis and the
problem is to retain or to reject the hypothesis using the
sample information.
Null hypothesis :
Alternative hypothesis : 2
STATISTICAL INFERENCE

3. a Hypotheses testing: introduction Four different outcomes:
TRUE
ACCEPT
Type I
error
Type II H0 H1 a
Type I error : reject H0 | H0 is true
Type II error : accept H0 | H0 is false 3
STATISTICAL INFERENCE

4. Hypotheses testing: introduction
To decide on the null hypothesis, we define the rejection
region:
e. g.,
It is a size  test if
i. e., if 4
STATISTICAL INFERENCE

5. Hypotheses testing: introduction
Simple hypothesis
Composite hypothesis
Two-sided hypothesis
One-sided hypothesis 5
STATISTICAL INFERENCE

6. Hypotheses testing: Wald test
Let and the sample
Consider testing
Assume that is asymptotically normal: 6
STATISTICAL INFERENCE

7. Hypotheses testing: Wald test
The rejection region for the Wald test is:
and the size is asymptotically .
The Wald test provides a size  test for the
null hypothesis 7
STATISTICAL INFERENCE

8. Hypotheses testing: p-value
We want to test if the mean of
is zero.
Let and denote by
the values of a particular sample.
Consider the sample mean as the test statistic: 8
INFERENCIA ESTADÍSTICA

9. Hypotheses testing: p-value
We use a distance to test the null hypothesis: 9
INFERENCIA ESTADÍSTICA

10. Hypotheses testing: p-value
H0 is rejected when is large, i. e.,
when is large.
This means that is in the distribution tail.
The probability of finding a value more extreme than
the observed one is
This probability is the p-value. 10
INFERENCIA ESTADÍSTICA

11. Hypotheses testing: p-value
Remark:
The p-value is the smallest size  for which
H0 is rejected.
The p-value expresses evidence against H0: the smaller
the p-value, the stronger the evidence against H0.
Usually, the p-value is considered small when p < 0.01
and large when p > 0.05. 11
STATISTICAL INFERENCE

12. Hypotheses testing: likelihood ratio test
Given , we want to test a hypothesis
about with a sample
For instance:
Under each hypothesis, we obtain a different likelihood: 12
INFERENCIA ESTADÍSTICA

13. Hypotheses testing: likelihood ratio test
We reject H0 if, and only if,
i. e., 13
STATISTICAL INFERENCE

14. Hypotheses testing: likelihood ratio test
The general case is
where is the parametric space.
We reject H0 14
STATISTICAL INFERENCE

15. Hypotheses testing: likelihood ratio test
Since
the likelihood ratio is 15
STATISTICAL INFERENCE

16. Hypotheses testing: likelihood ratio test
and the rejection region is 16
STATISTICAL INFERENCE

17. Hypotheses testing: likelihood ratio test
The likelihood ratio statistic is 17
STATISTICAL INFERENCE

18. Hypotheses testing: likelihood ratio test
Theorem
Assume that Let
Let λ be the likelihood ratio test statistic. Under
where r-q is the dimension of Θ minus the dimension of
Θ0. The p-value for the test is P{χ2r-q >λ}. 18
STATISTICAL INFERENCE