lec25 - .1 Goodness-of-fit for composite...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 25 25.1 Goodness-of-fit 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....
View Full Document

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.

Page1 / 4

lec25 - .1 Goodness-of-fit for composite...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online