This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 dont 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 : X n { H 1 , , H k } . Let us note that sometimes this function 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 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 will make an error is ( 6 = H i  H i ) = i ( 6 = H i ) , which we will call error of type i or type i error....
View
Full
Document
 Spring '09
 DmitryPanchenko
 Statistics

Click to edit the document details