STATISTICAL INFERENCE PART VI HYPOTHESIS TESTING 1 TESTS

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STATISTICAL INFERENCE PART VI HYPOTHESIS TESTING 1

STATISTICAL INFERENCE PART VI HYPOTHESIS TESTING 1

TESTS OF HYPOTHESIS • A hypothesis is a statement about a population parameter. •

TESTS OF HYPOTHESIS • A hypothesis is a statement about a population parameter. • The goal of a hypothesis test is to decide which of two complementary hypothesis is true, based on a sample from a population. 2

TESTS OF HYPOTHESIS • STATISTICAL TEST: The statistical procedure to draw an appropriate conclusion

TESTS OF HYPOTHESIS • STATISTICAL TEST: The statistical procedure to draw an appropriate conclusion from sample data about a population parameter. • HYPOTHESIS: Any statement concerning an unknown population parameter. • Aim of a statistical test: test a hypothesis concerning the values of one or more population parameters. 3

NULL AND ALTERNATIVE HYPOTHESIS • NULL HYPOTHESIS=H 0 – E. g. , a treatment

NULL AND ALTERNATIVE HYPOTHESIS • NULL HYPOTHESIS=H 0 – E. g. , a treatment has no effect or there is no change compared with the previous situation. • ALTERNATIVE HYPOTHESIS=HA – E. g. , a treatment has a significant effect or there is development compared with the previous situation. 4

TESTS OF HYPOTHESIS • Sample Space, A: Set of all possible values of sample

TESTS OF HYPOTHESIS • Sample Space, A: Set of all possible values of sample values x 1, x 2, …, xn. (x 1, x 2, …, xn) A • Parameter Space, : Set of all possible values of the parameters. =Parameter Space of Null Hypothesis Parameter Space of Alternative Hypothesis = 0 1 H 0: 0 H 1: 1 5

TESTS OF HYPOTHESIS • Critical Region, C, is a subset of A which leads

TESTS OF HYPOTHESIS • Critical Region, C, is a subset of A which leads to rejection region of H 0. Reject H 0 if (x 1, x 2, …, xn) C Not Reject H 0 if (x 1, x 2, …, xn) C΄ • A test defines the critical region. • A test is a rule which leads to a decision to fail to reject or reject H 0 on the basis of the sample information. 6

TEST STATISTIC AND REJECTION REGION • TEST STATISTIC: The sample statistic on which we

TEST STATISTIC AND REJECTION REGION • TEST STATISTIC: The sample statistic on which we base our decision to reject or not reject the null hypothesis. • REJECTION REGION: Range of values such that, if the test statistic falls in that range, we will decide to reject the null hypothesis, otherwise, we will not reject the null hypothesis. 7

TESTS OF HYPOTHESIS • If the hypothesis completely specify the distribution, then it is

TESTS OF HYPOTHESIS • If the hypothesis completely specify the distribution, then it is called a simple hypothesis. Otherwise, it is composite hypothesis. • =( 1, 2) H 0: 1=3 f(x; 3, 2) Composite Hypothesis H 1: 1=5 f(x; 5, 2) If 2 is known, simple hypothesis. 8

TESTS OF HYPOTHESIS H 0 is True Reject H 0 Do not reject H

TESTS OF HYPOTHESIS H 0 is True Reject H 0 Do not reject H 0 Type I error P(Type I error) = Correct Decision 1 - H 0 is False Correct Decision 1 - Type II error P(Type II error) = Tests are based on the following principle: Fix , minimize . ( )=Power function of the test for all . = P(Reject H 0 )=P((x 1, x 2, …, xn) C ) 9

TESTS OF HYPOTHESIS Type I error=Rejecting H 0 when H 0 is true 10

TESTS OF HYPOTHESIS Type I error=Rejecting H 0 when H 0 is true 10

PROCEDURE OF STATISTICAL TEST 1. Determining H 0 and HA. 2. Choosing the best

PROCEDURE OF STATISTICAL TEST 1. Determining H 0 and HA. 2. Choosing the best test statistic. 3. Deciding the rejection region (Decision Rule). 4. Conclusion. 11

HOW TO DERIVE AN APPROPRIATE TEST Definition: A test which minimizes the Type II

HOW TO DERIVE AN APPROPRIATE TEST Definition: A test which minimizes the Type II error (β) for fixed Type I error (α) is called a most powerful test or best test of size α. 12

MOST POWERFUL TEST (MPT) H 0: = 0 Simple Hypothesis H 1: = 1

MOST POWERFUL TEST (MPT) H 0: = 0 Simple Hypothesis H 1: = 1 Simple Hypothesis Reject H 0 if (x 1, x 2, …, xn) C The Neyman-Pearson Lemma: Reject H 0 if Proof: Available in text books (e. g. Bain & Engelhardt, 1992, p. g. 408) 13

EXAMPLES • X~N( , 2) where 2 is known. H 0 : = 0

EXAMPLES • X~N( , 2) where 2 is known. H 0 : = 0 H 1 : = 1 where 0 > 1. Find the most powerful test of size . 14

Solution 15

Solution 15

Solution, cont. • What is c? : It is a constant that satisfies since

Solution, cont. • What is c? : It is a constant that satisfies since X~N( , 2). For a pre-specified α, most powerful test says, Reject Ho if 16

Examples • Example 2: See Bain & Engelhardt, 1992, p. g. 410 Find MPT

Examples • Example 2: See Bain & Engelhardt, 1992, p. g. 410 Find MPT of Ho: p=p 0 vs H 1: p=p 1 > p 0 • Example 3: See Bain & Engelhardt, 1992, p. g. 411 Find MPT of Ho: X~Unif(0, 1) vs H 1: X~Exp(1) 17

UNIFORMLY MOST POWERFUL (UMP) TEST • If a test is most powerful against every

UNIFORMLY MOST POWERFUL (UMP) TEST • If a test is most powerful against every possible value in a composite alternative, then it will be a UMP test. • One way of finding UMPT is to find MPT by Neyman. Pearson Lemma for a particular alternative value, and then show that test does not depend on the specific alternative value. • Example: X~N( , 2), we reject Ho if Note that this does not depend on particular value of μ 1, but only on the fact that 0 > 1. So this is a UMPT of H 0: = 0 vs H 1: < 0. 18

UNIFORMLY MOST POWERFUL (UMP) TEST • To find UMPT, we can also use Monotone

UNIFORMLY MOST POWERFUL (UMP) TEST • To find UMPT, we can also use Monotone Likelihood Ratio (MLR). • If L=L( 0)/L( 1) depends on (x 1, x 2, …, xn) only through the statistic y=u(x 1, x 2, …, xn) and L is an increasing function of y for every given 0> 1, then we have a monotone likelihood ratio (MLR) in statistic y. • If L is a decreasing function of y for every given 0> 1, then we have a monotone likelihood ratio (MLR) in statistic −y. 19

UNIFORMLY MOST POWERFUL (UMP) TEST • Theorem: If a joint pdf f(x 1, x

UNIFORMLY MOST POWERFUL (UMP) TEST • Theorem: If a joint pdf f(x 1, x 2, …, xn; ) has MLR in the statistic Y, then a UMP test of size • for H 0: 0 vs H 1: > 0 is to reject H 0 if Y c where P(Y c 0)=. • for H 0: 0 vs H 1: < 0 is to reject H 0 if Y c where P(Y c 0)=. 20

EXAMPLE • X~Exp( ) H 0: 0 H 1: > 0 Find UMPT of

EXAMPLE • X~Exp( ) H 0: 0 H 1: > 0 Find UMPT of size . 21

EXAMPLE • Xi~Poi( ), i=1, 2, …, n Determine whether (X 1, …, Xn)

EXAMPLE • Xi~Poi( ), i=1, 2, …, n Determine whether (X 1, …, Xn) has MLR property. Find a UMP level α test for testing H 0: = 0 versus H 1: < 0. 22

GENERALIZED LIKELIHOOD RATIO TEST (GLRT) • GLRT is the generalization of MPT and provides

GENERALIZED LIKELIHOOD RATIO TEST (GLRT) • GLRT is the generalization of MPT and provides a desirable test in many applications but it is not necessarily a UMP test. 23

GENERALIZED LIKELIHOOD RATIO TEST (GLRT) H 0: 0 H 1: 1 and Let MLE

GENERALIZED LIKELIHOOD RATIO TEST (GLRT) H 0: 0 H 1: 1 and Let MLE of under H 0 24

GENERALIZED LIKELIHOOD RATIO TEST (GLRT) GLRT: Reject H 0 if 0 25

GENERALIZED LIKELIHOOD RATIO TEST (GLRT) GLRT: Reject H 0 if 0 25

EXAMPLE • X~N( , 2) H 0 : = 0 H 1 : 0

EXAMPLE • X~N( , 2) H 0 : = 0 H 1 : 0 Derive GLRT of size . 26

EXAMPLE • Let X 1, …, Xn be independent r. v. s, each with

EXAMPLE • Let X 1, …, Xn be independent r. v. s, each with shifted exponential p. d. f. : where λ is known. Find the LRT to test H 0: = 0 versus H 1: > 0.

ASYMPTOTIC DISTRIBUTION OF − 2 ln • GLRT: Reject H 0 if 0 •

ASYMPTOTIC DISTRIBUTION OF − 2 ln • GLRT: Reject H 0 if 0 • GLRT: Reject H 0 if -2 ln >-2 ln 0=c where k is the number of parameters to be tested. Reject H 0 if -2 ln > 28

TWO SAMPLE TESTS Derive GLRT of size , where X and Y are independent;

TWO SAMPLE TESTS Derive GLRT of size , where X and Y are independent; p 0, p 1 and p 2 are unknown. 29