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∙ other than type i and type ii errors there are

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Unformatted text preview: ∙ Other than Type I and Type II errors, there are two other possibilities: we fail to reject H when we should because H is true or we reject H against H 1 when we should because H 1 is true. ∙ Some authors use the language “reject H ” and “accept H .” The latter is less desirable than “fail to reject H ” for reasons we will see. 15 4 . Size and Power ∙ Because we want to make type I errors relatively rare, we need to formalize the notion of what we mean by “rare.” ∙ Suppose first that the null hypothesis is simple: H : . Then the size of a test, usually denoted , is the probability of a Type I error: P Reject H | H is true P Reject H | ∙ I will also use significance level to refer to the size of a test. 16 ∙ If the null hypothesis is composite, the probability of a Type I error generally depends on the value of ∈ Θ . In that case, we define the size as the maximum of the probability of Type I error: max ∈ Θ P Reject H | (where, in our applications, the max will always exist). ∙ By definite, for any ∈ Θ , P Reject H | ≤ . ∙ The probability of a Type II error is a function over Θ 1 : P Fail to Reject H | , ∈ Θ 1 17 ∙ It is very useful to define the power function as P Reject H | , ∈ Θ P Reject H where the second definition is just shorthand notation. ∙ Notice that is the maximum of over ∈ Θ . ∙ For ∈ Θ 1 , 1 − P Type II error| 18 ∙ Ideally, we could have 0, all ∈ Θ 1, all ∈ Θ 1 , which effectively means we would never make a mistake. But this is impossible in practice. ∙ If we want to make the size of a testing procedure “small,” the cost is that power against alternatives is reduced. 19 5 . Test Statistics and Rejection Rules DEFINITION : Given a set of observations X 1 , X 2 ,..., X n , a test statistic T h X 1 ,..., X n is a (measurable) real-valued function. ∙ A test statistic, like an estimator, must be computable for any outcome of data. It cannot depend on unknown parameters. It is viewed as a random variable because it is a function of the random vectors X 1 , X 2 ,..., X n . ∙ For a given sample of data, it is sometimes useful to let t h x 1 ,..., x n denote the particular outcome of T (that is, a number). 20 ∙ Given specification of the null and alternative hypotheses, we will use test statistics and rejection rules to obtain tests with a given size. We can then study power properties to choose among tests....
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