goodtalk

# goodtalk - Introduction Goods 1967 paper Example Good...

This preview shows pages 1–9. Sign up to view the full content.

Introduction Good’s 1967 paper Example Good smoothing Good Smoothing Jim Albert Bowling Green State University December 8, 2009

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

View Full Document
Introduction Good’s 1967 paper Example Good smoothing Outline Introduction Good’s 1967 paper Example Good smoothing
Introduction Good’s 1967 paper Example Good smoothing Introduction General problem in categorical data analysis is how to handle small counts. Wald conﬁdence interval for a proportion ˆ p - 1 . 96 r ˆ p (1 - ˆ p ) n , ˆ p + 1 . 96 r ˆ p (1 - ˆ p ) n ! does not work well for small n . P ( interval covers p ) is not uniformly 0.95.

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

View Full Document
Introduction Good’s 1967 paper Example Good smoothing Ad-hoc solution Add small counts to data, and apply frequentist methods to the adjusted data. John Tukey suggested “starting” counts by 1/6. Agresti and Coull suggest adding “2 successes and 2 failures” to data, and then apply Wald interval estimate. In contingency tables with zero counts, common to add 1/2 to each cell.
Introduction Good’s 1967 paper Example Good smoothing Why not Bayes? Adding imaginary counts corresponds to prior information. Leads to a Bayesian analysis. I. J. Good was one of the ﬁrst to discuss the choice of imaginary counts in smoothing categorical data. Famous 1967 paper by Good “A Bayesian signiﬁcance test for multinomial distributions” discusses his general approach.

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

View Full Document
Introduction Good’s 1967 paper Example Good smoothing Testing problem Observe y = ( y 1 , ..., y t ) from multinomial distribution with sample size n and probabilities p = ( p 1 , ..., p t ). Test hypothesis H : p 1 = ... = p t = 1 t Usual test procedure is Pearson’s statistic: X 2 = t X j =1 ( y j - n t ) 2 n t which is asymptotically χ 2 ( t - 1).
Introduction Good’s 1967 paper Example Good smoothing Motivation for Bayes Accuracy of chi-square approximation for small counts is questionable. Desirable to develop an “exact” Bayesian test free from asymptotic theory. Use procedure with conﬁdence for all t and n .

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

View Full Document
Introduction Good’s 1967 paper Example Good smoothing Bayes factor Ratio of marginal densities under the hypotheses H and A (not H ). Under
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 01/01/2011 for the course STAT 665 taught by Professor Albert during the Spring '10 term at Bowling Green.

### Page1 / 31

goodtalk - Introduction Goods 1967 paper Example Good...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online