Optimal Bounds Used in Dollar-Unit Sampling: A Comparison of Reliability and Efficiency

Abstract

Auditors typically employ one-sided confidence bounds to estimate the total error in an audit population. This estimate provides an auditor with a given level of assurance that the total error does not exceed the upper confidence bound. This paper summarizes the results of an extensive simulation study using both real and simulated data comparing 14 bounds. No one method was found to be superior in terms of reliability and efficiency. A 95% upper bound is reliable if, when used repeatedly, the bound exceeds the true audit error 95% of the time. Efficiency measures the size of the bound; the smaller the bound is, the more efficient it is said to be. The multinomial-Dirichlet method [Tsui, K. W., Matsamura, E. M., Tsui, K. L. (1985). Multinomial-Dirichlet bounds for dollar-unit sampling in auditing. Acc. Rev. 60(1):76–96] demonstrated the best reliability for a variety of populations. The Bayesian normal bound [Menzefricke, U., Smieliauskas, W. (1984). A simulation study of the performance of parametric dollar unit sampling statistical procedures. J. Acc. Res. 22(2):588–604] and the Cox and Snell bound [Cox, D. R., Snell, E. J. (1979). On sampling and the estimation of rare errors. Biometrika 66(1):125–132] are reliable and more efficient than the multinomial-Dirichlet bound for particular populations. The Augmented Variance Estimator bound [Rohrbach, K. J. (1993). Variance augmentation to achieve nominal coverage probability in sampling from audit populations. Auditing J. Practice Theory 12(2):79–97] is reliable and efficient for populations with error rates of less than 10%. The extended multinominal-Dirichlet bound [Matsumura, E., Tsui, K., Wong, W.K. (1990). An extended multinomial-Dirichlet model for error bounds for dollar-unit sampling. Contemporary Acc. Res. 6:485–500] is reliable and efficient for most of the real populations studied.