Approximate tolerance limits and prediction limits for the gamma distribution

The problem of deriving one-sided tolerance intervals and prediction intervals is investigated for a two-parameter gamma distribution when both parameters are unknown. The shape parameter κ of the gamma distribution is a nuisance parameter for this problem. If κ is known, an exact tolerance limit or prediction limit can be easily derived. When κ is Unknown, the following startegy is recommended. Since the gamma distribution is well approximated by a lognormal distribution for large values of κ, the tolerance limit based on the normal distribution is recommended for large values of κ̂ (the maximum likelihood estimate of κ). Based on numerical results, this procedure is recommended when κ̂ > 7. If κ̂ ≤ 7, the recommendation is to divide the interval [0, 7] for κ̂ into sub-intervals, and replace κ with cκ̂ in the expression for the exact tolerance limit. A different constant c is to be used depending on the sub-interval two which κ belongs. Tables are provided giving such sub-intervals and constants for sample sizes 8 and 20. The above strategy is also recommended for computing the prediction limit. Numerical results are given to show that the recommended procedure is satisfactory in terms of providing coverages close to the nominal level. An example is given to illustrate the results.