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Unformatted text preview: Lecture 25 25.1 Goodnessoffit for composite hypotheses. (Textbook, Section 9.2) Suppose that we have a sample of random variables X 1 , . . . , X n that can take a finite number of values B 1 , . . . , B r with unknown probabilities p 1 = ( X = B 1 ) , . . . , p r = ( X = B r ) and suppose that we want to test the hypothesis that this distribution comes from a parameteric family { θ : θ ∈ Θ } . In other words, if we denote p j ( θ ) = θ ( X = B j ) , we want to test: H 1 : p j = p j ( θ ) for all j ≤ r for some θ ∈ Θ H 2 : otherwise. If we wanted to test H 1 for one particular fixed θ we could use the statistic T = r X j =1 ( ν j np j ( θ )) 2 np j ( θ ) , and use a simple χ 2 test from last lecture. The situation now is more complicated because we want to test if p j = p j ( θ ) , j ≤ r at least for some θ ∈ Θ which means that we have many candidates for θ. One way to approach this problem is as follows. (Step 1) Assuming that hypothesis H 1 holds, i.e. = θ for some θ ∈ Θ , we can find an estimate θ * of this unknown θ and then (Step 2) try to test whether indeed the distribution is equal to θ * by using the statistics T = r X j =1 ( ν j np j ( θ * )) 2 np j ( θ * ) in χ 2 test....
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.
 Spring '09
 DmitryPanchenko
 Statistics

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