1. Hypothesis testing Some general concepts: Null hypothesisH 0 A statement we “wish” to refute Alternative hypotesisH 1 The whole or part of the complement of H 0 Common case: The statement is about an unknown parameter,   H 0 :    H 1 :    –  (  \  ) where  is a well-defined subset of the parameter space 

2. Simple hypothesis:  (or  –  ) contains only one point (one single value) Composite hypothesis: The opposite of simple hypothesis Critical region (Rejection region) A subset C of the sample space for the random sample X = (X 1, …, X n ) such that we reject H 0 if X  C (and accept (better phrase: do not reject ) H 0 otherwise ). The complement of C, i.e. C will be referred to as the acceptance region C is usually defined in terms of a statistic, T(X), called the test statistic

3. Simple null and alternative hypotheses Errors in hypothesis testing: Type I errorRejecting a true H 0 Type II errorAccepting a false H 0 Significance level  The probability of Type I error Also referred to as the size of the test or the risk level Risk of Type II error  The probability of Type II error Power  The probability of rejecting a false H 0, i.e. the probability of the complement of Type II error = = 1 – 

4. Writing it more “mathematically”: Classical approach: Fix  and then find a test that makes  desirably small A low value of  does not imply a low value of , rather the contrary Most powerful test A test which minimizes  for a fixed value of  is called a most powerful test (or best test) of size 

5. Neyman-Pearson lemma x = (x 1, …, x n ) a random sample from a distribution with p.d.f. f (x;  ) We wish to testH 0 :  =  0 (simple hypothesis) versus H 1 :  =  1 (simple hypothesis) The the most powerful test of size  has a critical region of the form where A is some non-negative constant. Proof: Se the course book Note! Both hypothesis are simple

6. Example:

8. How to find B ? If  1 >  0 then B must satisfy

9. If the sample x comes from a distribution belonging to the one-parameter exponential family:

10. “Pure significance tests” Assume we wish to test H 0 :  =  0 with a test of size  Test statistic T(x) is observed to the value t Case 1: H 1 :  >  0 The P-value is defined as Pr(T(x)  t | H 0 ) Case 2: H 1 :  <  0 The P-value is defined as Pr(T(x)  t | H 0 ) If the P-value is less than  H 0 is rejected

11. Case 3: H 1 :    0 The P-value is defined as the probability that T(x) is as extreme as the observed value, including that it can be extreme in two directions from H 0 In general: Consider we just have a null hypothesis, H 0, that could specify the value of a parameter (like above) a particular distribution independence between two or more variables … Important is that H 0 specifies something under which calculations are feasible Given a test statistic T = t the P-value is defined as Pr (T is as extreme as t | H 0 )

12. Uniformly most powerful tests (UMP) Generalizations of some concepts to composite (null and) alternative hypotheses: H 0 :    H 1 :    –  (  \  ) Power function: Size:

13. A test of size  is said to be uniformly most powerful (UMP) if If H 0 is simple but H 1 is composite and we have found a best test (Neyman- Pearson) for H 0 vs. H 1 ’:  =  1 where  1   – , then if this best test takes the same form for all  1   – , the test is UMP. Univariate cases: H 0 :  =  0 vs. H 1 :  >  0 (or H 1 :  <  0 ) usually UMP test is found H 0 :  =  0 vs. H 1 :    0 usually UMP test is not found

14. Unbiased test: A test is said to be unbiased if  (  )   for all    –  Similar test: A test is said to be similar if  (  ) =  for all    Invariant test: Assume that the hypotheses of a test are unchanged if a transformation of sample data is applied. If the critical region is not changed by this transformation, thes test is said to be invariant. Consistent test: If a test depends on the sample size n such that  (  ) =  n (  ). If lim n   n (  ) = 1 the test is said to be consistent. Efficiency: Two test of the pair of simple hypotheses H 0 and H 1. If n 1 and n 2 are the minimum sample sizes for test 1 and 2 resp. to achieve size  and power  , then the relative efficiency of test1 vs. test 2 is defined as n 2 / n 1

15. (Maximum) Likelihood Ratio Tests Consider again that we wish to test H 0 :    H 1 :    –  (  \  ) The Maximum Likelihood Ratio Test (MLRT) is defined as rejecting H 0 if 0    1 For simple H 0  gives a UMP test MLRT is asymptotically most powerful unbiased MLRT is asymptotically similar MLRT is asymptotically efficient

16. If H 0 is simple, i.e. H 0 :  =  0 the MLRT is simplified to Example

17. Distribution of  Sometimes  has a well-defined distribution: e.g.   A can be shown to be an ordinary t-test when the sample is from the normal distribution with unknown variance and H 0 :  =  0 Often, this is not the case. Asymptotic result: Under H 0 it can be shown that –2ln  is asymptotically  2 -distributed with d degrees of freedom, where d is the difference in estimated parameters (including “nuisance” parameters) between

18. Example Exp (  ) cont.

19. Score tests

20. Wald tests Score and Wald tests are particularly used in Generalized Linear Models

21. Confidence sets and confidence intervals Definition: Let x be a random sample from a distribution with p.d.f. f (x ;  ) where  is an unknown parameter with parameter space , i.e.   . If S X is a subset of , depending on X such that then S X is said to be a confidence set for  with confidence coeffcient (level) 1 –  For a one-dimensional parameter  we rather refer to this set as a confidence interval

22. Pivotal quantities A pivotal quantity is a function g of the unknown parameter  and the observations in the sample, i.e. g = g (x ;  ) whose distribution is known and independent of . Examples:

23. To obtain a confidence set from a pivotal quantity we write a probability statement as (1) For a one-dimensional  and g monotonic, the probability statement can be re- written as where now the limits are random variables, and the resulting observed confidence interval becomes For a k-dimensional  the transformation of (1) to a confidence set is more complicated but feasible.

24. In particular, a point estimator of  is often used to construct the pivotal quantity. Example:

26. Using the asymptotic normality of MLE:s One-dimensional parameter  : k-dimensional parameter  :

27. Construction of confidence intervals from hypothesis tests: Assume a test of H 0 :  =  0 vs. H 1 :    0 with critical region C(  0 ). Then a confidence set for  with confidence coefficient 1 –  is