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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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