Tolerance limits under Poisson regression based on small-sample asymptotic methodology

ABSTRACT

This article explores the calculation of tolerance limits for the Poisson regression model based on the profile likelihood methodology and small-sample asymptotic corrections to improve the coverage probability performance. The data consist of n counts, where the mean or expected rate depends upon covariates via the log regression function. This article evaluated upper tolerance limits as a function of covariates. The upper tolerance limits are obtained from upper confidence limits of the mean. To compute upper confidence limits the following methodologies were considered: likelihood based asymptotic methods, small-sample asymptotic methods to improve the likelihood based methodology, and the delta method. Two applications are discussed: one application relating to defects in semiconductor wafers due to plasma etching and the other examining the number of surface faults in upper seams of coal mines. All three methodologies are illustrated for the two applications.

KEYWORDS:

• Count data
• Modified signed log-likelihood ratio test statistic
• Poisson regression
• Small-sample asymptotics
• Upper tolerance limit

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 62 - Statistics
• Secondary - General applied mathematics

Acknowledgments

This work has benefitted greatly from the critical comments and suggestions by Dr. Thomas Mathew and Dr. DoHwan Park at the University of Maryland Baltimore County statistics department. I am grateful to them for their input and suggestions. The author also acknowledges the helpful suggestions and improvements provided by the reviewers.