Approximate Tolerance Limits and Confidence Limits on Reliability for the Gamma Distribution

No exact method is known for determining tolerance limits or s-confidence limits for reliability for the gamma distribution when both parameters are unknown. Perhaps the simplest approximate method is to determine a tolerance limit assuming the shape parameter known and then replace the shape parameter with its ML estimate to obtain approximate limits. Simulated values of the true probability levels, achieved by this method, indicate that this method is not suitable, contrary to what has been anticipated. A second approach is to consider the corresponding tolerance limits assuming the distribution mean known and the shape parameter unknown, and then replace the distribution mean by the sample mean. This approach gives useful results for many practical cases. Simulated values of the true probability levels achieved are presented for some typical cases and limiting values are provided. This method appears satisfactory for all values of the shape parameter, for the common s-confidence levels, and moderate sample sizes.