Confidence Distributions, Confidence Nets, and Likelihoods

• Activity Number: 139
• Type: Invited
• Date/Time: Monday, August 9, 2004 : 2:00 PM to 3:50 PM
• Sponsor: General Methodology
• Author(s): Tore Schweder*+
• Companies: University of Oslo
• Keywords: confidence distribution ; confidence net ; prediction ; dimension reduction ; bootstrap ; bias

Abstract

A confidence distribution for a scalar parameter spans nets of confidence intervals, confidence nets, by its quantiles. A particular confidence net is obtained by indexing the likelihood ratio by way of the chi-square distribution, or a simulated null distribution. Confidence distributions for vector parameters are obtained from joint pivots when they exist. The contours of a likelihood ratio for a vector parameter, or of another objective function, make up a confidence net when indexed by a fixed null distribution. Likelihood functions represent cumulative confidence, not confidence density, and integrated and scaled likelihoods are generally not confidence densities. A confidence net for a derived parameter from a profiled or integrated likelihood typically requires simulation to obtain an approximate null distribution, and perhaps to remove bias. A confidence net for an unobserved variable might be found from a predictive likelihood. A simultaneous set of confidence nets is a box-shaped confidence net for the vector parameter or variable. Abundance estimation of bowhead whales off Alaska will illustrate the methodology.