Incorporating systematic uncertainties into upper limits Elton S. Smith, Jefferson Lab Review of Feldman Cousins unified method for constructing confidence limits Integrating over background level and detection efficiency Summary and conclusions

Central Confidence Interval

Upper limits

Probability density Wish to determine parameter m Measure X, with known constant background b Estimate for m is x Function of x/s and m/s only

Ordering principle Likelihood ratio relative to probability with m=max(0,x)

Likelihood Ratio: R(x) = P(x|m)/P(x/mbest) se=0

Ordering Principle: Picking the 90% interval Feldman Cousins Phys Rev D57 (1998) 3873

Probability density, uncertainty in background Background b, estimated at mb with uncertainty smb Integrate over “true” background b Function of x/s and m/s only

Probability density, scale uncertainty Ŝ is the measured detection efficiency S is the “true” detection efficiency Define se = sS/Ŝ, the relative scale uncertainty

Probability distribution

Ordering principle Likelihood ratio relative to probability with m=max(0,x)

90% CL including scale uncertainties

Example 13

Correct Coverage se 90% CL

Dependence on se Rolke NIMA 551 (2005) 493

Scaling from nominal FC limits Simple procedure for scaling FC limits to include systematics

Summary and Conclusions We have investigated how uncertainties in the estimation of background and detection efficiency affect the 90% confidence intervals in the unified approach of Feldman and Cousins. Assumption: Gaussian statistics Systematic uncertainties included using Bayesian approach Confidence intervals have reasonable limiting behavior depend quadratically on se can be obtained by scaling the nominal FC intervals have correct coverage CLAS-NOTE 2007-019 and CLAS-NOTE 2008-20