Sequential Audit Sampling with Statistical Guarantees

Masahiro Kato and Kei Nakagawa. 2026. Preprint (arXiv:2604.06116). 日本語版

In an audit of financial statements, the first sample often fails to support a conclusion, and the auditor draws more items. We treat the whole procedure, extension included, as a sequential test under sampling without replacement from a finite population, and we bound the probability of a wrong decision in advance. This is an English rewrite and extension of a talk given at the Special Interest Group on Financial Informatics of the Japanese Society for Artificial Intelligence.

The problem

Auditing standards allow the auditor to add procedures when the first sample does not yield sufficient evidence. ISA 530 and its national versions say so explicitly, as do PCAOB AS 2315 in the United States and the financial audit manuals of the GAO and CIGIE. The GAO manual goes further, defining sequential sampling and stating that any extension of the sample should be planned in advance rather than decided once the results are in. How to design such a procedure statistically has received far less attention. Keep adding items without a design and the chance of reaching the wrong conclusion cannot be held down in advance.

The proposed method

We state the null and alternative hypotheses in terms of the tolerable deviation rate, then fix the stopping and decision rules that say when to stop adding items and which hypothesis to take. The population is finite and the sampling is without replacement, so the number of deviations in the sample follows a hypergeometric distribution. Writing the boundary-crossing probability under that distribution gives the conditions an exact sequential boundary must satisfy. For actual use, we calibrate the boundary by Monte Carlo computation at the deviation rate where the decision is hardest.

Main results and conditions

The exact design bounds the probability of a wrong decision in advance. The Monte Carlo implementation only approximates that design, and in return it yields the expected stopping time, meaning the number of items the auditor should expect to examine. We also give extensions to one-sided and two-stage tests, and to designs that stop partway. The scope is attribute sampling and deviation-rate auditing, above all tests of controls, where what is counted is whether a deviation occurred. Extending this to monetary-unit sampling would require a different measurement model, so we leave it out.

Relation to earlier work

Sequential testing itself goes back to Wald (1945, Sequential Tests of Statistical Hypotheses). The closest earlier work in our setting of sampling without replacement from a finite population is Lai, who studied sequential tests under the hypergeometric distribution in detail and gave a simple test whose continuation region is a triangle (1979, Sequential Tests for Hypergeometric Distributions and Finite Populations). On the audit sampling side, the survey by Elder and coauthors points out that sequential sampling carries practical weight, mainly in tests of controls (2013, Audit Sampling Research: A Synthesis and Implications for Future Research). Horgan applied a list-sequential sampling scheme to financial auditing (2003, A list-sequential sampling scheme with applications in financial auditing). This work inherits both the statistical and the auditing line, and gives a boundary-calibration method that explicitly bounds the error probability at the tolerable deviation rate used in auditing, in a form that can be implemented while keeping the population finite.

Where it applies

The setting we have in mind is inspection that counts occurrences and judges a deviation rate, as in tests of controls. How far the sample may be extended can be settled beforehand, and both the error probability when sampling stops early and the number of items likely to be needed can be estimated, so they can be shown as the basis for the audit plan.

Paper and materials

BibTeX

@misc{sequential-audit-sampling,
  author       = {Masahiro Kato and Kei Nakagawa},
  title        = {Sequential Audit Sampling with Statistical Guarantees},
  year         = {2026},
  eprint       = {2604.06116},
  archivePrefix = {arXiv},
  url          = {https://arxiv.org/abs/2604.06116},
}