Ii the distribution of a pivotal statistic is known

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II. The distribution of a pivotal statistic is known and free of any unknown parameters. It is a common technique to use pivotal statistics to construct confidence intervals and other statistical inference procedures in statistics. Jimin Ding, Math WUSTL Math 494 Spring 2018 25 / 44
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Example: CI for μ Without normality Recall CLT: Let X 1 , X 2 , · · · , X n iid with μ and σ 2 < . Then ¯ X - μ σ/ n D -→ N (0 , 1) , n → ∞ . The result still hold if σ is replaced by a consistent estimator ˆ σ (to be proved in Chapter 5), such as sample standard deviation ˆ σ = S . I Case 1: large sample (Example 4.2.2) Jimin Ding, Math WUSTL Math 494 Spring 2018 26 / 44
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Example: CI for μ Without normality I Case 2: large sample of Bernoulli rv (Example 4.2.3) Jimin Ding, Math WUSTL Math 494 Spring 2018 27 / 44
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Example: CI for μ Without normality I Case 3: small sample ( Poisson Model ) Jimin Ding, Math WUSTL Math 494 Spring 2018 28 / 44
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Example: CI for μ 1 - μ 2 I Case 1: large sample Jimin Ding, Math WUSTL Math 494 Spring 2018 29 / 44
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Example: CI for μ 1 - μ 2 I Case 2: under normality (Example 4.2.4) Jimin Ding, Math WUSTL Math 494 Spring 2018 30 / 44
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Example: CI for p 1 - p 2 in Bernoulli/Binomial Jimin Ding, Math WUSTL Math 494 Spring 2018 31 / 44
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Hypothesis Testing Jimin Ding, Math WUSTL Math 494 Spring 2018 32 / 44
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Motivation I Estimation was concerned with taking the data and making guesses about what the parameter could be. I What if we first guess what the parameter is, and decide whether the data support that guess? I Our initial guess is called the null hypothesis, denoted by H 0 . I The alternative hypothesis , denoted by H 1 , is the opposite of the null hypothesis. I Hypothesis testing : a process to make a decision based data between two opposing hypotheses. I Statistical hypotheses are certain explicit statements about population parameters . Jimin Ding, Math WUSTL Math 494 Spring 2018 33 / 44
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Hypothesis Testing Hypothesis testing is an inference tool to use sample data to examine whether the statistical hypotheses are true. It is usually done in the following 5 steps. Jimin Ding, Math WUSTL Math 494 Spring 2018 34 / 44
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Hypothesis Testing Hypothesis testing is an inference tool to use sample data to examine whether the statistical hypotheses are true. It is usually done in the following 5 steps. I. State null, H 0 , and alternative, H 1 , hypotheses. I H 0 : neutral / no difference / no effect I H 1 : a contradictory claim to H 0 . It depends on the questions we want to answer. Jimin Ding, Math WUSTL Math 494 Spring 2018 34 / 44
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Hypothesis Testing Hypothesis testing is an inference tool to use sample data to examine whether the statistical hypotheses are true. It is usually done in the following 5 steps. I. State null, H 0 , and alternative, H 1 , hypotheses. I H 0 : neutral / no difference / no effect I H 1 : a contradictory claim to H 0 . It depends on the questions we want to answer.
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