Parametric Empirical Bayes Tolerance Intervals

Abstract

The theory of a parametric empirical Bayes (PEB) normal distribution tolerance bound is derived. Classical tolerance intervals are based on the assumption of a fixed unknown standard deviation, σ, and mean, μ. Such intervals assume no prior knowledge about the possible value of σ. The empirical Bayes approach assumes that there is prior knowledge of the value of σ, which can be characterized in terms of a translated beta density (with shape parameters α, β and range parameters A. B). The resulting PEB K factors are smaller than those of the classical approach, so fewer samples are required to achieve equal precision. An example illustrates potential cost savings. Extensive tables are not provided, since the PEB K factors depend on (α, β, A, B), as well as the sample size, confidence, and coverage probability. Since the exact computation is intensive, I suggest an approximation based on the noncentral t distribution. The degrees of freedom and noncentrality parameters are obtained by matching the first two moments of the PEB conditional t distribution to those of a noncentral t distribution. The approximate K factors can then be evaluated by a simple computer program.

KEY WORDS:

• Beta distribution
• Posterior distribution
• Prior distribution