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Unformatted text preview: Lecture 18 Testing hypotheses. (Textbook, Chapter 8) 18.1 Testing simple hypotheses. Let us consider an i.i.d. sample X 1 , . . . , X n with distribution on some space X , i.e. X ’s take values in X . Suppose that we don’t know but we know that it can only be one of possible k distributions, ∈ { 1 , . . . , k } . Based on the data X , . . . , X n we have to decide among k simple hypotheses: H 1 : = 1 H 2 : = 2 . . . H k : = k We call these hypotheses simple because each hypothesis asks a simple question about whether is equal to some particular specified distribution. To decide among these hypotheses means that given the data vector, X = ( X 1 , . . . , X n ) ∈ X n we have to decide which hypothesis to pick or, in other words, we need to find a decision rule which is a function δ : X n → { H 1 , ··· , H k } . Let us note that sometimes this function δ can be random because sometimes several hypotheses may look equally likely and it will make sense to pick among them ran domly. This idea of a randomized decision rule will be explained more clearly as we go on, but for now we can think of δ as a simple function of the data. 67 LECTURE 18. TESTING HYPOTHESES. 68 Suppose that the i th hypothesis is true, i.e. = i . Then the probability that decision rule δ will make an error is ( δ 6 = H i  H i ) = i ( δ 6 = H i ) , which we will call error of type i or type i error....
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 Spring '09
 DmitryPanchenko
 Statistics

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