Physical Sciences and Mathematics Commons^™

Computing Highly Accurate Confidence Limits From Discrete Data Using Importance Sampling, Chris Lloyd

Chris J. Lloyd

For discrete parametric models, approximate confidence limits perform poorly from a strict frequentist perspective. In principle, exact and optimal confidence limits can be computed using the formula of Buehler (1957), Lloyd and Kabaila (2003). So-called profile upper limits (Kabaila \& Lloyd, 2001) are closely related to Buehler limits and have extremely good properties. Both profile and Buehler limits depend on the probability of a certain tail set as a function of the unknown parameters. Unfortunately, this probability surface is not computable for realistic models. In this paper, importance sampling is used to estimate the surface and hence the confidence limits. …

Exact One-Sided Confidence Limits For The Difference Between Two Correlated Proportions, Chris Lloyd, Max V. Moldovan

Chris J. Lloyd

We construct exact and optimal one-sided upper and lower confidence bounds for the difference between two probabilities based on matched binary pairs using well-established optimality theory of Buehler (1957). Starting with five different approximate loer and upper limits, we adjust them to have coverage probability exactly equal to the desired nominal level and then compare the resulting exact limits by their mean size. Exact limits based on the signed root likelihood ratio statistic are preferred and recommended for practical use.

Improved Buehler Limits Based On Refined Designated Statistics , Chris Lloyd, Paul Kabaila

Chris J. Lloyd

The Buehler upper confidence limit is as small as possible, subject to the constraints that (a) its coverage probability never falls below nimonal and (b) it is a non-decreasing function of a designated statistic. The designated statistic may have ties among its possible values.

We prove that breaking such ties by a sufficiently small

modification can never increase the Buehler limit. We also prove that, under commonly satisfied conditions, breaking ties by a sufficiently small modification will result in an improved i.e. smaller Buehler limit. We conclude that designated statistics should not contain ties, apart from ties justified by symmetry …