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

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2 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 : Hypotheses testing: introduction STATISTICAL INFERENCE

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3 Four different outcomes: TRUE ACCEPT Type I error Type II error H0H0 H0H0 H1H1 H1H1 Type I error : reject H 0 | H 0 is true Type II error : accept H 0 | H 0 is false STATISTICAL INFERENCE Hypotheses testing: introduction

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

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

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

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

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Hypotheses testing: p-value 8 INFERENCIA ESTADÍSTICA 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:

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

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Hypotheses testing: p-value 10 INFERENCIA ESTADÍSTICA H 0 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.

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11 Remark: The p-value is the smallest size for which H 0 is rejected. The p-value expresses evidence against H 0 : the smaller the p-value, the stronger the evidence against H 0. Usually, the p-value is considered small when p < 0.01 and large when p > STATISTICAL INFERENCE Hypotheses testing: p-value

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

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13 We reject H 0 if, and only if, i. e., STATISTICAL INFERENCE Hypotheses testing: likelihood ratio test

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14 The general case is where is the parametric space. We reject H 0 STATISTICAL INFERENCE Hypotheses testing: likelihood ratio test

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

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

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

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18 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 Θ 0. The p-value for the test is P{χ 2 r-q >λ}. STATISTICAL INFERENCE Hypotheses testing: likelihood ratio test

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