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Sequential Audit Sampling with Statistical Guarantees
\maketitle
\begin{abstract}
Financial statement auditing is conducted under a risk-based evidence approach to obtain reasonable assurance. In practice, auditors often perform additional sampling or related procedures when an initial sample does not provide a sufficient basis for a conclusion. Across jurisdictions, current standards and practice manuals acknowledge such extensions, while the statistical design of sequential audit procedures has not been fully explored. This study formulates audit sampling with additional, sequentially collected items as a sequential testing problem for a finite population under sampling without replacement. We define null and alternative hypotheses in terms of a tolerable deviation rate, specify stopping and decision rules, and formulate exact sequential boundary conditions in terms of finite-population error probabilities. For practical implementation, we calibrate those boundaries by Monte Carlo simulation at least-favorable deviation rates. The exact design yields ex ante control of decision error probabilities, and the simulation-based implementation approximates that design while allowing the computation of expected stopping times. The framework is most naturally suited to attribute auditing and deviation-rate auditing, especially tests of controls, and it can be extended to one-sided, two-stage, and truncated designs.\footnote{This paper is an English translation of a Japanese paper presented at SIG-FIN (the Special Interest Group on Financial Informatics of the Japanese Society for Artificial Intelligence) as \citet{Kato2025zaimushohyo}, with its content further refined.}
\end{abstract}
\section{Introduction}
Financial statement auditing plays a central role in protecting investors, lenders, and other stakeholders by providing independent assurance on the reliability of financial reporting. In modern large-scale organizations, however, the detailed records underlying account balances and transaction classes are often too numerous to inspect exhaustively. Accordingly, the auditor's objective is not absolute assurance but a high level of \emph{reasonable assurance} that is achievable under limited audit resources.
Under risk-based auditing, audit effort is concentrated on areas that are more likely to contain material misstatement. One of the most important operational tools in this process is audit sampling: the auditor examines a subset of items from a population and uses the resulting evidence to form a conclusion about the population as a whole. This basic architecture is shared internationally. International Standard on Auditing (ISA) 530 and national adoptions such as ISA (UK) 530 and Australian Auditing Standards (ASA) 530 provide the general framework for audit sampling, while Public Company Accounting Oversight Board Auditing Standard (PCAOB AS) 2315 and the GAO/CIGIE \emph{Financial Audit Manual} provide closely related guidance in the United States \citep{IAASB2024Handbook,FRC2025ISAUK530,AUASB2021ASA530,PCAOBAS2315,GAOFAM2025}.\footnote{GAO = Government Accountability Office and CIGIE = Council of the Inspectors General on Integrity and Efficiency.} Similar ISA-based formulations appear in other jurisdictions as well, including New Zealand and Malaysia \citep{XRB2021ISANZ530,MIA2018ISA530}. Japanese audit-sampling practice can also be situated within this broader international landscape. For Japanese practice-oriented accounts and standards, see \citet{BAC2002,BAC2020,ASC2002022,ASC5302022,ASBJ530Study2022,Minami2022practical}.
Importantly, the international institutional setting already contains a sequential element. Under ISA-type standards, if audit sampling has not provided a reasonable basis for a conclusion, the auditor may tailor further procedures; for tests of controls, this may include extending the sample size, testing an alternative control, or modifying related substantive procedures \citep{FRC2025ISAUK530,AUASB2021ASA530}. In the United States, the GAO/CIGIE \emph{Financial Audit Manual} explicitly defines \emph{sequential sampling} and emphasizes that expanding a statistical sample should generally be planned in advance rather than improvised after observing the initial results \citep{GAOFAM2025}. Recent institutional reviews and current standard-setting discussions also show that sampling remains prevalent in practice and that principles-based standards still leave meaningful variation in implementation, together with continuing challenges of consistent application \citep{FRC2023AuditSampling,IAASBISA500Series}.
Behavioral and process-oriented auditing research also emphasizes that auditing is inherently sequential. Normative and descriptive work treats opinion formation, evidence search, and evidence evaluation as iterative activities in which preliminary plans are revised as evidence arrives \citep{Kinney1975DecisionTheory,Felix1982researchin,Cushing1986comparisonof}. For example, \citet{Ashton1988sequentialbelief} studies sequential belief revision and order effects in audit evidence, while \citet{Knechel1990sequentialauditor} explicitly models the auditor as choosing whether to stop or gather additional evidence. These studies explain why sequentiality is natural in auditing, but they do not furnish finite-population stopping boundaries or ex ante error guarantees.
This study contributes to audit-sampling methodology from a statistical perspective, while remaining practically implementable in the institutional environment described above. By refining existing procedures, we formalize sequential audit sampling as a sequential testing problem for a finite population under sampling without replacement. The exact design problem is written in terms of finite-population boundary-crossing probabilities under the hypergeometric model, and Monte Carlo simulation is then used as a practical calibration device. The main ingredients are deliberately close to the standard audit-sampling setup: a finite population, a tolerable deviation rate, and repeated inspection of additional items until a sufficiently reliable conclusion is obtained. The proposed method is general and can also be implemented with procedures other than the basic one discussed in this study.
In summary, this study has the following three goals:
\begin{itemize}
\item to formulate repeated or extended audit sampling as sequential hypothesis testing;
\item to provide ex ante guarantees on the probabilities of incorrect decisions, that is, accepting a hypothesis that does not hold; and
\item to quantify the expected stopping time, which corresponds to the expected number of sampled items.
\end{itemize}
In Section~\ref{sec:setup}, we formulate the problem setting. In Section~\ref{sec:formulationsequential}, we formulate the audit-sampling problem as a sequential testing problem. In Section~\ref{sec:sequentialauditingalgorithm}, we then present the sequential auditing algorithm. In Section~\ref{sec:extensions}, we introduce several practical extensions of the proposed method. In Sections~\ref{sec:numexample} and~\ref{sec:empirical-study}, we present simulation results based on synthetic data and empirical studies using publicly available real-world datasets, respectively. In Appendix~\ref{appdx:relatedwork}, we provide a more detailed discussion of the related work.
\section{Problem Setup}
\label{sec:setup}
Suppose that an auditor examines a company with a population of $n$ audit items, indexed by $1,2,\dots,n$. The entire set of items constitutes a finite population, and $n$ denotes the population size.
The auditor evaluates whether each item contains a problem. We refer to such a problem as a \emph{deviation}. For each item $i=1,\dots,n$, let $X_i\in\{0,1\}$ indicate whether the item contains a deviation, where $X_i=1$ means that the item contains a deviation and $X_i=0$ means that it does not. What qualifies as a deviation depends on the audit objective and the applicable auditing framework. Examples include control failures, fraud, and errors.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\linewidth]{Figure/fig_setup-eps-converted-to.pdf}
\caption{A finite population with population deviation rate $p_0$.}
\label{fig:setup}
\end{figure}
\subsection{Deviation Rate}
We summarize the condition of the population by the number of deviating items,
\[
m \coloneqq \sum^n_{i=1} \mathbbm{1}\left(X_i=1\right),
\]
and especially by the proportion
\[
p_0 \coloneqq \frac{m}{n}.
\]
We call $p_0$ the \emph{population deviation rate}.
\subsection{Tolerable Deviation Rate and Audit Sampling}
Let $r\in(0,1)$ denote a pre-specified benchmark. If the population deviation rate $p_0$ exceeds this benchmark, the auditor regards the population as problematic. We call $r$ the \emph{tolerable deviation rate}.
Although the indicators $X_i$ are unknown before inspection, they are not intrinsically random once an item is examined, because inspection reveals whether the item deviates. Therefore, if the auditor inspects all $n$ items, then $p_0$ is observed without error, and the comparison with $r$ is exact.
When $n$ is large, however, full inspection is often too costly. The practical alternative is to inspect only a subset of the population and use the observed sample to infer the condition of the full population. In auditing, such sampling-based evidence gathering is the standard mode of operation.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\linewidth]{Figure/fig_inference-eps-converted-to.pdf}
\caption{Overview of audit sampling}
\label{fig:inference}
\end{figure*}
In practice, the sample may be collected only once, or additional items may be sampled when the initial results do not provide a sufficient basis for a conclusion. This broad practice is recognized internationally, but it is often described institutionally rather than as a statistically explicit sequential testing problem.
\begin{remark}
The present formulation is most naturally aligned with attribute- and deviation-rate auditing, especially tests of controls and other binary compliance-type tasks \citep{PCAOBAS2315,FRC2025ISAUK530,AUASB2021ASA530}. Other audit-sampling methods, such as monetary-unit sampling for monetary misstatement, require different modeling choices and are outside the direct scope of the binary formulation in this study.
\end{remark}
\subsection{Objective}
Our objective is to turn audit sampling with sequentially added items into a statistically disciplined procedure. We focus on three aspects:
\begin{itemize}[leftmargin=0.5cm]
\item a formulation as sequential hypothesis testing,
\item explicit control of the probabilities of incorrect decisions, and
\item calculation of the expected stopping time.
\end{itemize}
\paragraph{Notation.}
We write ${\mathbb{P}}_p(\cdot)$ for the probability law induced by a finite population with deviation rate $p$ under sampling without replacement, and $[t]=\{1,2,\dots,t\}$. Whenever we write $p$, we implicitly assume that $np$ is an integer; otherwise, one may replace $p$ by the nearest feasible grid point in $\{0,1/n,\dots,1\}$.
\section{Formulation as Sequential Testing}
\label{sec:formulationsequential}
We now formulate audit sampling with additional items as a sequential hypothesis testing problem. We consider two hypotheses:
\begin{align*}
H &: p_0 \le r, \\
K &: p_0 > r.
\end{align*}
Hypothesis $H$ means that the population deviation rate does not exceed the tolerable level, while hypothesis $K$ means that it does.
In practice, it is often useful to introduce an indifference region. Let $\theta_H,\theta_K\in(0,1/2)$. We then consider the separated hypotheses
\begin{align*}
H &: p_0 \le r-\theta_H, \\
K &: p_0 > r+\theta_K.
\end{align*}
When $p_0\in(r-\theta_H,\,r+\theta_K]$, either decision may be regarded as acceptable for the purposes of the testing formulation.
\subsection{Sequential Sampling Procedure}
At each stage $t$, the auditor inspects one previously uninspected item drawn without replacement from the finite population. After relabeling items according to the random inspection order, let $X_1,X_2,\dots$ denote the sequentially observed deviation indicators, and define the sample mean
\[
\widehat p_t \coloneqq \frac{1}{t}\sum_{s=1}^t X_s.
\]
A sequential audit procedure consists of the following steps:
\begin{description}[leftmargin=0.6cm]
\item[Step 1.] A finite population of size $n$ and a tolerable deviation rate $r$ are given.
\item[Step 2.] For each $t=1,2,\dots$, inspect one additional item without replacement.
\begin{description}[leftmargin=0.6cm]
\item[(a)] If a pre-specified stopping condition is satisfied, stop.
\item[(b)] Use the observations up to time $t$ to decide whether to accept $H$, accept $K$, or continue.
\item[(c)] If the procedure continues, inspect another item and return to Step 2.
\end{description}
\end{description}
The condition in Step 2(a) is the \emph{stopping rule}; the mapping in Step 2(b) is the \emph{decision rule}.
\subsection{Control of the Error Probability}
The central design problem is to control the probability of making an incorrect decision. In our setting, two errors matter:
\begin{itemize}[leftmargin=0.5cm]
\item accepting $K$ when $H$ is true, and
\item accepting $H$ when $K$ is true.
\end{itemize}
Let $\alpha,\beta\in(0,1/2)$ be user-specified tolerances. Our goal is to construct a procedure such that
\begin{align}
\sup_{p\in[0,r-\theta_H]} {\mathbb{P}}_p\left(\delta^{\text{SA}}=K\right)
&\le \alpha, \label{eq:type1}\\
\sup_{p\in[r+\theta_K,\,1]} {\mathbb{P}}_p\bigl(\delta^{\text{SA}}=H\bigr) &\le \beta. \label{eq:type2}
\end{align}
For calibration, it is helpful to describe \emph{exact-time} error events. Let
\begin{align*}
E_t^K &\coloneqq \Big\{\widehat p_t > \overline\kappa_r(t), \underline\kappa_r(s) \le \widehat p_s \le \overline\kappa_r(s) \text{ for all } s<t\Big\}, \\
E_t^H &\coloneqq \Big\{\widehat p_t < \underline\kappa_r(t),\ \underline\kappa_r(s) \le \widehat p_s \le \overline\kappa_r(s)\ \text{for all } s<t\Big\}.
\end{align*}
These events are disjoint across $t$, so the overall error probabilities can be written as sums of exact-time boundary-crossing probabilities. This observation underlies the recursive boundary construction below.
\begin{figure*}[ht]
\centering
\includegraphics[width=0.9\linewidth]{Figure/fig_sequentialauditing-eps-converted-to.pdf}
\caption{Illustration of the sequential audit sampling algorithm.}
\label{fig:sequential-auditing}
\end{figure*}
\section{Sequential Auditing Algorithm}
\label{sec:sequentialauditingalgorithm}
Audit sampling in our setting has three features:
\begin{description}[leftmargin=0.6cm]
\item[Sampling from a hypergeometric model.] Under sampling without replacement from a finite population with a fixed number of deviations, the number of observed deviations follows a hypergeometric law.
\item[Finite population.] The audit population consists of finitely many items.
\item[No error under full inspection.] If all items are inspected, then the population deviation rate is known exactly, and the resulting decision is error-free outside the indifference region.
\end{description}
Motivated by these features, we propose a sequential auditing algorithm with explicit error control. Figure~\ref{fig:sequential-auditing} illustrates the procedure of the proposed algorithm.
\subsection{Sequential Auditing Procedure}
We define the sequential auditing procedure as a pair of a stopping rule and a decision rule,
\[
\pi^{\text{SA}} \coloneqq \bigl(\tau^{\text{SA}},\delta^{\text{SA}}\bigr).
\]
Here ``SA'' stands for \emph{sequential auditing}.
\paragraph{Parameters.}
Fix target error levels $0<\alpha,\beta<1/2$. Depending on $\alpha$, $\beta$, the tolerable deviation rate $r$, and the population size $n$, we determine a sequence of lower and upper boundaries
\[
\bigl\{\bigl(\underline\kappa_r(t),\overline\kappa_r(t)\bigr)\bigr\}_{t=1}^n.
\]
\paragraph{Stopping rule.}
We use the stopping time
\begin{align}
\tau^{\text{SA}} \coloneqq \inf\Big\{ t\in[n-1] : \widehat p_t < \underline\kappa_r(t) \text{ or } \widehat p_t > \overline\kappa_r(t) \Big\} \wedge n. \label{eq:stoppingrule}
\end{align}
Thus the auditor stops as soon as the sample mean leaves the continuation region. If the procedure does not stop early, it stops at $t=n$ after full inspection.
\paragraph{Decision rule.}
Upon stopping, the auditor decides as follows:
\begin{itemize}[leftmargin=0.5cm]
\item if $\widehat p_{\tau^{\text{SA}}} < \underline\kappa_r(\tau^{\text{SA}})$, accept $H$;
\item if $\widehat p_{\tau^{\text{SA}}} > \overline\kappa_r(\tau^{\text{SA}})$, accept $K$;
\item if $\tau^{\text{SA}}=n$, inspect all items and set $\delta^{\text{SA}}=H$ when $p_0\le r$, and $\delta^{\text{SA}}=K$ otherwise.
\end{itemize}
This terminal rule guarantees that, if early stopping never occurs, the procedure remains exact under full inspection.
\subsection{Boundary Calibration}
The performance of the proposed procedure depends on the boundaries $\underline\kappa_r(t)$ and $\overline\kappa_r(t)$. We choose them so that the overall error guarantees in (\ref{eq:type1})--(\ref{eq:type2}) are satisfied while the continuation region is made as small as possible.
Let the least-favorable design points be
\[
p_H^* \coloneqq r-\theta_H, \qquad p_K^* \coloneqq r+\theta_K.
\]
We calibrate the upper boundary using $p_H^*$ and the lower boundary using $p_K^*$. Under monotone rejection regions, these are the least-favorable points outside the indifference region. Because the parameter space is finite, one may also validate the resulting design over the entire feasible grid $\{0,1/n,\dots,1\}$ after calibration if one prefers not to rely solely on the least-favorable-point argument.
Fix $t\in[n]$ and suppose that the boundaries up to time $t-1$ have already been chosen. For a candidate upper boundary value $u$, define the exact-time upper-side error probability
\[
A_t(u) \coloneqq {\mathbb{P}}_{p_H^*}\Bigl( \widehat p_t > u,\ \underline\kappa_r(s) \le \widehat p_s \le \overline\kappa_r(s)\ \text{for all } s<t \Bigr).
\]
Likewise, for a candidate lower boundary value $\ell$, define
\[
B_t(\ell) \coloneqq {\mathbb{P}}_{p_K^*}\Bigl( \widehat p_t < \ell,\ \underline\kappa_r(s) \le \widehat p_s \le \overline\kappa_r(s)\ \text{for all } s<t \Bigr).
\]
Because exact-time error events are disjoint across $t$, cumulative control of the sums $\sum^t_{s=1} A_s\left(\overline\kappa_r(s)\right)$ and $\sum^t_{s=1} B_s\left(\underline\kappa_r(s)\right)$ yields the desired overall control.
Three practical difficulties must be handled.
\begin{itemize}[leftmargin=0.5cm]
\item The exact-time error probability at stage $t$ depends on the boundaries chosen at earlier stages.
\item Although the relevant hypergeometric probabilities are, in principle, available, evaluating them for every feasible $p$ and every candidate threshold can become computationally burdensome.
\item The boundaries that satisfy the error constraints need not be unique.
\end{itemize}
We address these difficulties by (i) choosing the boundaries recursively from $t=1$ onward, (ii) calibrating only at the least-favorable rates $p_H^*$ and $p_K^*$, (iii) approximating the required probabilities by Monte Carlo simulation, and (iv) among feasible boundaries, choosing the pair that stops audit sampling as early as possible.
Let
\[
{\mathcal{K}}_t \coloneqq \Bigl\{0,\frac{1}{t},\frac{2}{t},\dots,1\Bigr\}
\]
be the discrete support of $\widehat p_t$. We then select the boundaries recursively according to
\begin{align}
\overline\kappa_r(t)
&\in \operatorname*{arg\,min}_{u\in{\mathcal{K}}_t}
\Bigl\{ u : \sum_{s=1}^{t-1} A_s\bigl(\overline\kappa_r(s)\bigr) + A_t(u) \le \alpha \Bigr\}, \label{eq:upper-calibration}\\
\underline\kappa_r(t)
&\in \operatorname*{arg\,max}_{\ell\in{\mathcal{K}}_t}
\Bigl\{ \ell : \sum_{s=1}^{t-1} B_s\bigl(\underline\kappa_r(s)\bigr) + B_t(\ell) \le \beta \Bigr\}. \label{eq:lower-calibration}
\end{align}
The upper boundary is chosen as the \emph{smallest} feasible threshold, because smaller upper thresholds lead to earlier acceptance of $K$. The lower boundary is chosen as the \emph{largest} feasible threshold, because larger lower thresholds lead to earlier acceptance of $H$.
The calibration problem in (\ref{eq:upper-calibration})--(\ref{eq:lower-calibration}) is exact whenever the probabilities $A_t(u)$ and $B_t(\ell)$ are evaluated exactly from the hypergeometric law. The Monte Carlo method in the next subsection should therefore be understood as a numerical approximation to this exact boundary-design problem: the theoretical guarantees attach to the exact probabilities, and the simulation-based implementation approximates that design increasingly well as the number of replications grows.
\subsection{Monte Carlo Computation of the Boundaries}
In general, the exact probabilities $A_t(u)$ and $B_t(\ell)$ can be computed from the hypergeometric law, but repeated calculation over all times and all candidate thresholds is cumbersome. We therefore use Monte Carlo simulation.
Let $M$ be the number of Monte Carlo replications. For each $m\in[M]$, generate a full random inspection order from a finite population of size $n$ with deviation rate $p_H^*$, and denote the resulting sequence by
\[
\{\widetilde X_i^{H,(m)}\}_{i=1}^n.
\]
Define the corresponding running sample mean by
\[
\widetilde p_t^{H,(m)} \coloneqq \frac{1}{t}\sum_{s=1}^t \widetilde X_s^{H,(m)}.
\]
Similarly, generate sequences
\[
\{\widetilde X_i^{K,(m)}\}_{i=1}^n,
\]
from a finite population with deviation rate $p_K^*$, and define
\[
\widetilde p_t^{K,(m)} \coloneqq \frac{1}{t}\sum_{s=1}^t \widetilde X_s^{K,(m)}.
\]
For a candidate upper boundary $u$, estimate $A_t(u)$ by
\begin{align*}
\widehat A_t(u)
&\coloneqq \frac{1}{M}\sum_{m=1}^M
\1\Bigl\{ \widetilde p_t^{H,(m)} > u,\ \underline\kappa_r(s) \le \widetilde p_s^{H,(m)} \le \overline\kappa_r(s)\ \text{for all } s<t \Bigr\}.
\end{align*}
Likewise, estimate $B_t(\ell)$ by
\begin{align*}
\widehat B_t(\ell)
&\coloneqq \frac{1}{M}\sum_{m=1}^M
\1\Bigl\{ \widetilde p_t^{K,(m)} < \ell,\ \underline\kappa_r(s) \le \widetilde p_s^{K,(m)} \le \overline\kappa_r(s)\ \text{for all } s<t \Bigr\}.
\end{align*}
Replacing $A_t$ and $B_t$ in (\ref{eq:upper-calibration})--(\ref{eq:lower-calibration}) by $\widehat A_t$ and $\widehat B_t$ yields an implementable boundary-construction algorithm.
The recursion can be described explicitly for the first few stages.
\paragraph{The case $t=1$.}
At the first stage, $\widehat p_1\in\{0,1\}$. For the upper boundary, if $\overline\kappa_r(1)=0$, then the procedure accepts $K$ whenever $X_1=1$, so the exact-time upper-side error equals $p_H^*$. If $\overline\kappa_r(1)=1$, then upper-side stopping at $t=1$ is impossible because $\widehat p_1>1$ cannot occur. Therefore, the earliest feasible upper boundary is
\[
\overline\kappa_r(1)=
\begin{cases}
0, & \text{if } p_H^*\le \alpha,\\
1, & \text{otherwise.}
\end{cases}
\]
For the lower boundary, if $\underline\kappa_r(1)=1$, then the procedure accepts $H$ whenever $X_1=0$, so the exact-time lower-side error equals $1-p_K^*$. If $\underline\kappa_r(1)=0$, then lower-side stopping at $t=1$ is impossible because $\widehat p_1<0$ cannot occur. Therefore, the earliest feasible lower boundary is
\[
\underline\kappa_r(1)=
\begin{cases}
1, & \text{if } 1-p_K^*\le \beta,\\
0, & \text{otherwise.}
\end{cases}
\]
\paragraph{The case $t=2$.}
At the second stage, $\widehat p_2\in\{0,1/2,1\}$. Given the stage-1 boundaries, the exact-time error probabilities become
\begin{align*}
A_2(u)
&= {\mathbb{P}}_{p_H^*}\Bigl( \widehat p_2 > u,\ \underline\kappa_r(1) \le \widehat p_1 \le \overline\kappa_r(1) \Bigr),\\
B_2(\ell)
&= {\mathbb{P}}_{p_K^*}\Bigl( \widehat p_2 < \ell,\ \underline\kappa_r(1) \le \widehat p_1 \le \overline\kappa_r(1) \Bigr).
\end{align*}
We then try the upper candidates $u\in\{0,1/2,1\}$ in increasing order and choose the first one for which
\[
A_1\left(\overline\kappa_r(1)\right) + A_2(u)\le \alpha.
\]
Likewise, we try the lower candidates $\ell\in\{1,1/2,0\}$ in decreasing order and choose the first one for which
\[
B_1\bigl(\underline\kappa_r(1)\bigr)+B_2(\ell)\le \beta.
\]
Thus, at each stage, we select the earliest-stopping boundaries among all feasible candidates.
\paragraph{The general case $t=t'$.}
Continue the same recursion for $t'=3,4,\dots,n$:
\begin{align*}
A_{t'}(u)
&= {\mathbb{P}}_{p_H^*}\Bigl( \widehat p_{t'} > u,\ \underline\kappa_r(s) \le \widehat p_s \le \overline\kappa_r(s)\ \text{for all } s<t' \Bigr),\\
B_{t'}(\ell)
&= {\mathbb{P}}_{p_K^*}\Bigl( \widehat p_{t'} < \ell,\ \underline\kappa_r(s) \le \widehat p_s \le \overline\kappa_r(s)\ \text{for all } s<t' \Bigr).
\end{align*}
At each stage, the upper boundary is chosen from the candidate set in increasing order, and the lower boundary is chosen from the candidate set in decreasing order, subject to the cumulative error constraints.
\begin{algorithm}[tb]
\caption{Sequential auditing procedure $\pi^{\text{SA}}$}
\label{alg:sa}
\begin{algorithmic}[1]
\Require Population size $n$; tolerable deviation rate $r$; error levels $\alpha,\beta$; indifference parameters $\theta_H,\theta_K$; Monte Carlo size $M$; observed sequence $\{X_t\}_{t=1}^n$.
\State Set $p_H^* \gets r-\theta_H$ and $p_K^* \gets r+\theta_K$.
\State Generate $M$ synthetic sequences $\{\widetilde X_i^{H,(m)}\}_{i=1}^n$ from a finite population with deviation rate $p_H^*$.
\State Generate $M$ synthetic sequences $\{\widetilde X_i^{K,(m)}\}_{i=1}^n$ from a finite population with deviation rate $p_K^*$.
\For{$t=1,2,\dots,n$}
\State Compute $\underline\kappa_r(t)$ and $\overline\kappa_r(t)$ recursively from the Monte Carlo approximations $\widehat B_t$ and $\widehat A_t$.
\EndFor
\For{$t=1,2,\dots,n$}
\State Compute $\widehat p_t = t^{-1}\sum_{s=1}^t X_s$.
\If{$t<n$ and $\widehat p_t < \underline\kappa_r(t)$}
\State Stop and accept $H$.
\ElsIf{$t<n$ and $\widehat p_t > \overline\kappa_r(t)$}
\State Stop and accept $K$.
\ElsIf{$t=n$}
\State Inspect all items and accept $H$ if $p_0\le r$, otherwise accept $K$.
\EndIf
\EndFor
\end{algorithmic}
\end{algorithm}
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\linewidth]{Figure/fig_montecarlo-eps-converted-to.pdf}
\caption{Monte Carlo simulation and thresholds. The left panel shows the least-favorable finite population when hypothesis $H$ is true, the one that yields the largest upper-side error probability, and the right panel shows the least-favorable finite population when hypothesis $K$ is true, the one that yields the largest lower-side error probability.}
\label{fig:montecalro}
\end{figure*}
\section{Extensions}
\label{sec:extensions}
The sequential auditing algorithm in Section~\ref{sec:sequentialauditingalgorithm} has the following features:
\begin{itemize}[leftmargin=0.5cm]
\item it accepts either $H$ or $K$;
\item it starts evaluating the sample from the first observed item; and
\item if no early decision is made, sampling can continue up to $n$ inspected items.
\end{itemize}
If these features do not match the intended audit workflow, the same basic idea can be adapted. We discuss four extensions.
\subsection{One-Sided Test for Concluding That the Population Is Acceptable}
In some audit settings, one only needs to decide whether the deviation rate is sufficiently low. This is particularly natural when the key audit risk is concluding that a control is effective when it is not \citep{PCAOBAS2315,FRC2025ISAUK530}.
Consider the one-sided hypotheses
\begin{align*}
H_0 &: p_0 \ge r, \\
H_1 &: p_0 < r.
\end{align*}
Here the null hypothesis states that the population is problematic, and the auditor seeks evidence strong enough to reject that null.
To control the probability of wrongly concluding that the population is acceptable when in fact $p_0\ge r$, we impose a type-I error bound $\alpha$. The stopping rule becomes
\begin{align*}
\tau^{\text{SA}} \coloneqq \inf\Big\{ t\ge 1 : \widehat p_t < \underline\kappa_r(t) \Big\} \wedge n.
\end{align*}
If $\tau^{\text{SA}}<n$, reject $H_0$. If $\tau^{\text{SA}}=n$, inspect all items and reject $H_0$ if $p_0<r$, otherwise do not reject $H_0$.
The lower boundary is calibrated so that, under $H_0$, the probability of stopping incorrectly is at most $\alpha$. Among the feasible lower boundaries, the most useful choice is the largest one, because it yields the earliest favorable conclusion subject to the error constraint.
\subsection{One-Sided Test with Power Requirement and a Minimum Sample Size}
The previous one-sided construction controls the probability of a false favorable conclusion, but by itself it does not regulate the frequency of conservative decisions. To address that issue, introduce an indifference parameter $\theta\in(0,1/2)$ and consider
\begin{align*}
H_0 &: p_0 \ge r, \\
H_1 &: p_0 \le r-\theta.
\end{align*}
We then require both a type-I error bound $\alpha$ and power of at least $1-\beta$ at the least-favorable alternative $p=r-\theta$.
Let $\underline t$ denote a minimum sample size before a favorable conclusion is allowed. The stopping rule becomes
\begin{align*}
\tau^{\text{SA}} \coloneqq \inf\Big\{ t\ge \underline t : \widehat p_t < \underline\kappa_r(t) \Big\} \wedge n.
\end{align*}
If $\tau^{\text{SA}}<n$, reject $H_0$. If $\tau^{\text{SA}}=n$, inspect all items and reject $H_0$ if $p_0<r$, otherwise do not reject $H_0$.
The boundary $\underline\kappa_r(t)$ is calibrated exactly as in the previous subsection, and $\underline t$ is chosen as the smallest integer such that the power under $p=r-\theta$ is at least $1-\beta$ according to the Monte Carlo simulation.
\subsection{Two-Stage Testing}
In practice, auditors often inspect an initial batch of items and add more items only if the first-stage results are inconclusive. This pattern is also consistent with international guidance that allows further procedures when the original sample does not provide a sufficient basis for a conclusion \citep{FRC2025ISAUK530,AUASB2021ASA530,GAOFAM2025}.
The proposed framework naturally accommodates such a design. For example, if the auditor wishes to inspect an initial batch of $t_0$ items before allowing any stopping decision, one simply imposes the restriction that the procedure cannot stop before time $t_0$, and calibrates the sequential boundaries subject to that restriction.
\subsection{Truncated Sequential Testing}
The basic procedure can continue up to the full population size $n$. In some applications, however, the auditor may wish to terminate the sequential phase at a fixed truncation time $T<n$. Such truncated sequential tests have been studied in the sequential testing literature \citep{Xiong1995aclass}. Once $T$ is fixed, the same Monte Carlo approach can be used to calibrate the sequential boundaries up to time $T$ together with an explicit terminal decision rule at $T$.
\section{Numerical Example}
\label{sec:numexample}
We illustrate the sequential auditing procedure using synthetic data. Consider a finite population of size $n=100$ and let the tolerable deviation rate be $r=0.2$. Set the indifference-region widths to $\theta_H=\theta_K=0.05$ and the target error levels to $\alpha=\beta=0.05$. We calibrate the boundaries using $M=10{,}000$ Monte Carlo replications.
Figure~\ref{fig:montecalro} displays sample paths from the two least-favorable finite populations used for calibration. In the left panel, the finite population has deviation rate $p=r-\theta_H=0.15$, so $H$ is true; in the right panel, the finite population has deviation rate $p=r+\theta_K=0.25$, so $K$ is true. For illustration, the figure shows $50$ sample paths in each panel together with the calibrated boundaries. These finite populations are used only for calibration and represent the worst cases outside the indifference region in the sense that they maximize the corresponding probabilities of incorrect decisions. For readability, only $50$ paths are plotted, even though the calibration itself uses $10{,}000$ Monte Carlo replications. The calibrated upper and lower boundaries roughly trace the upper and lower envelopes of these least-favorable paths, and only about $5\%$ of the least-favorable paths cross the wrong side.
Next, using the calibrated boundaries, we compute for each feasible $p\in\left\{0,1/n,\dots,1\right\}$ the probability of an incorrect decision and the expected stopping time by a fresh Monte Carlo experiment with $M=10{,}000$ replications. Figure~\ref{fig:error-rate} reports the resulting error curve. Outside the indifference region, the error probability is controlled at approximately the prescribed level. Figure~\ref{fig:expected-stop} reports the expected stopping time. As expected, the procedure takes the longest near the decision boundary and stops more quickly when the deviation rate is well below or well above the tolerable level.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{Figure/fig_error_rate-eps-converted-to.pdf}
\caption{Error probabilities}
\label{fig:error-rate}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{Figure/fig_exp_stop-eps-converted-to.pdf}
\caption{Expected stopping time}
\label{fig:expected-stop}
\end{figure}
\section{Empirical Study}
\label{sec:empirical-study}
This section complements the numerical example by applying the proposed procedure to three \emph{observed finite populations}. The purpose is not to claim that these public datasets are actual records from completed audit engagements. Rather, they provide heterogeneous binary populations on which the finite-population sequential rule can be replayed under random inspection orders. This lets us evaluate whether the main qualitative implications of the method, namely early stopping far from the decision boundary and later stopping near it, also appear in empirical settings.
\subsection{Design}
We use two public data sources. The first is the UCI \texttt{Audit Data}, which contains firm-level suspicious/non-suspicious records collected from the Auditor Office of India \citep{Hooda2018ucimachine}. The second is the public \texttt{FraudDetection} dataset released with \citet{Bao2020detectingaccounting}. From these sources, we construct three binary finite populations:
\begin{enumerate}[leftmargin=0.5cm]
\item \textbf{Audit Risk (UCI).} We define $X_i$ by the binary \texttt{Risk} label in the processed UCI file, giving a finite population of size $n=776$ with $m=305$ deviations, so that $p_0=0.3930$.
\item \textbf{FraudDetection 2014.} We restrict the public-firm-year data to fiscal year 2014 and define $X_i$ by the binary \texttt{misstate} label, giving $n=5{,}627$ and $m=4$, so that $p_0=0.00071$.
\item \textbf{FraudDetection 2000.} We restrict the same data to fiscal year 2000, again using \texttt{misstate} as the deviation indicator, giving $n=6{,}752$ and $m=86$, so that $p_0=0.01274$.
\end{enumerate}
For each population, we calibrate the sequential boundaries with the same Monte Carlo boundary-design routine used in Section~\ref{sec:numexample}. Operationally, the implementation works with the cumulative deviation count $S_t=\sum^t_{s=1} X_s$, which is equivalent to working with the sample-average process $\widehat p_t = S_t/t$. After calibration, we replay the sequential audit over $1{,}000$ random permutations of the observed population and record the stopping time $\tau^{\text{SA}}$ and the final decision. Because the population is fixed and sampling is without replacement, replaying over random orderings is the empirical analogue of sequential simple random sampling without replacement. If no boundary is crossed before $t=n$, we apply the full-population decision at $t=n$.
We set $\alpha=\beta=0.05$ throughout. For the Audit Risk population, we use $(r,\theta_H,\theta_K)=(0.30,0.05,0.05)$ and $M=3{,}000$ Monte Carlo replications for calibration. Since $p_0=0.3930>r+\theta_K=0.35$, this population lies in region $K$. For the two FraudDetection populations, we use a common configuration $(r,\theta_H,\theta_K)=(0.01,0.002,0.002)$ and $M=1{,}500$ calibration replications so that a very low-misstatement year and a higher-misstatement year can be compared under the same tolerable deviation rate. Under this choice, 2014 lies in region $H$ because $p_0=0.00071\leq 0.008$, whereas 2000 lies in region $K$ because $p_0=0.01274>0.012$.
\begin{table}[t]
\centering
\small
\caption{Observed finite populations and sequential-auditing configurations}
\label{tab:empirical-design}
\begin{tabular}{lccccc}
\hline
Population & $n$ & $m$ & $p_0$ & $(r,\theta_H,\theta_K)$ & Region \\
\hline
Audit Risk & 776 & 305 & 0.3930 & $(0.30,0.05,0.05)$ & $K$ \\
Fraud 2014 & 5,627 & 4 & 0.00071 & $(0.01,0.002,0.002)$ & $H$ \\
Fraud 2000 & 6,752 & 86 & 0.01274 & $(0.01,0.002,0.002)$ & $K$ \\
\hline
\end{tabular}
\end{table}
\subsection{Results}
Table~\ref{tab:empirical-replay} reports the replay results. Figures~\ref{fig:emp-auditrisk}--\ref{fig:emp-fraud2000} visualize, for each observed population, one realized cumulative-deviation path under a random inspection order together with the calibrated sequential boundaries, as well as the distribution of stopping times over the $1{,}000$ random-order replay runs.
The Audit Risk population is a high-deviation case far inside region $K$. In this setting, the sequential audit decides toward $K$ in $97.8\%$ of the random orderings and stops after only $34.2$ inspected items on average, corresponding to about $4.4\%$ of the population. The median stopping time is $28$, and the central $10$--$90\%$ range is $11$ to $64$ items. Figure~\ref{fig:emp-auditrisk}(a) shows that a representative cumulative-deviation path rises above the upper boundary quickly, while Figure~\ref{fig:emp-auditrisk}(b) shows a strongly right-skewed stopping-time distribution concentrated at small values.
\begin{figure*}[t]
\centering
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=\linewidth]{Figure/auditrisk_boundary-eps-converted-to.pdf}
\par\vspace{0.3em}
\footnotesize (a) One realized cumulative-deviation path and the calibrated sequential boundaries.
\end{minipage}\hfill
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=\linewidth]{Figure/auditrisk_stoptime-eps-converted-to.pdf}
\par\vspace{0.3em}
\footnotesize (b) Stopping-time distribution over $1{,}000$ random inspection orders.
\end{minipage}
\caption{Empirical replay for the Audit Risk population.}
\label{fig:emp-auditrisk}
\end{figure*}
FraudDetection 2014 provides the opposite extreme: its realized deviation rate is far inside region $H$. Here the sequential audit accepts $H$ in all replayed orderings. The mean stopping time is $428.7$ items and the median is $391$, which corresponds to inspecting about $7.6\%$ of the full population on average. Thus, even in a large population, the planned sequential rule often reaches a stable conclusion well before full inspection. Figure~\ref{fig:emp-fraud2014}(a) shows a representative path crossing the lower boundary after relatively few inspections, and Figure~\ref{fig:emp-fraud2014}(b) shows that the replay stopping times are tightly concentrated around a few moderate stopping points rather than spread over the full population size.
\begin{figure*}[t]
\centering
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=\linewidth]{Figure/jar2014_boundary-eps-converted-to.pdf}
\par\vspace{0.3em}
\footnotesize (a) One realized cumulative-deviation path and the calibrated sequential boundaries.
\end{minipage}\hfill
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=\linewidth]{Figure/jar2014_stoptime-eps-converted-to.pdf}
\par\vspace{0.3em}
\footnotesize (b) Stopping-time distribution over $1{,}000$ random inspection orders.
\end{minipage}
\caption{Empirical replay for the FraudDetection 2014 population.}
\label{fig:emp-fraud2014}
\end{figure*}
FraudDetection 2000 is the most challenging of the three examples because its realized deviation rate is only slightly above the $K$-region cutoff. In this near-boundary case, the procedure still decides toward $K$ in $95.4\%$ of the replayed orderings, but the stopping time becomes much larger and more dispersed: the mean is $912.6$, the median is $721$, and the central $10$--$90\%$ range runs from about $179$ to $1{,}889$ items. The average inspected share rises to $13.5\%$ of the population. Figure~\ref{fig:emp-fraud2000}(a) shows a representative cumulative-deviation path that stays relatively close to the upper boundary for a long period before separating from it, and Figure~\ref{fig:emp-fraud2000}(b) shows the resulting broad right-skewed stopping-time distribution.
\begin{figure*}[t]
\centering
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=\linewidth]{Figure/jar2000_boundary-eps-converted-to.pdf}
\par\vspace{0.3em}
\footnotesize (a) One realized cumulative-deviation path and the calibrated sequential boundaries.
\end{minipage}\hfill
\begin{minipage}[t]{0.48\textwidth}
\centering
\includegraphics[width=\linewidth]{Figure/jar2000_stoptime-eps-converted-to.pdf}
\par\vspace{0.3em}
\footnotesize (b) Stopping-time distribution over $1{,}000$ random inspection orders.
\end{minipage}
\caption{Empirical replay for the FraudDetection 2000 population.}
\label{fig:emp-fraud2000}
\end{figure*}
\begin{table}[t]
\centering
\small
\caption{Replay results over $1{,}000$ random orderings of each observed population}
\label{tab:empirical-replay}
\begin{tabular}{lcccc}
\hline
Population & Mean $\tau^{\text{SA}}$ & Median & Incorrect (\%) & Inspected (\%) \\
\hline
Audit Risk & 34.2 & 28 & 2.2 & 4.4 \\
Fraud 2014 & 428.7 & 391 & 0.0 & 7.6 \\
Fraud 2000 & 912.6 & 721 & 4.6 & 13.5 \\
\hline
\end{tabular}
\end{table}
\subsection{Discussion}
Three points are worth emphasizing. First, the empirical results match the main theoretical intuition of the paper: stopping is fastest when the population deviation rate is far from the decision boundary, and it becomes slower when the realized population is close to the tolerable-deviation threshold. The contrast between Audit Risk and FraudDetection 2000 is especially informative here, and the difference is visible both in the boundary-crossing paths and in the widening of the stopping-time histograms in Figures~\ref{fig:emp-auditrisk} and~\ref{fig:emp-fraud2000}.
Second, the population size alone does not determine the stopping time. Even though FraudDetection 2014 contains more than $5{,}000$ items, its mean inspected share is only $7.6\%$ because the realized deviation rate is far below the tolerable-deviation threshold. By contrast, FraudDetection 2000 requires a substantially larger inspected share because it is near the boundary between acceptable and unacceptable populations. This contrast is also clear in Figures~\ref{fig:emp-fraud2014} and~\ref{fig:emp-fraud2000}: the former has a tight histogram around moderate stopping points, whereas the latter has a much wider right tail.
Third, the observed frequencies of incorrect decisions under random orderings are small and consistent with the design targets. In the two region-$K$ examples, the shares of decisions toward $H$ are $2.2\%$ and $4.6\%$, respectively, and in the region-$H$ example the share of decisions toward $K$ is $0\%$. These replay frequencies should not be confused with the uniform ex ante guarantees derived from the least-favorable calibration problem, but they provide an encouraging population-specific check that the procedure behaves as intended.
Overall, these experiments show that the proposed sequential audit can be implemented on observed heterogeneous finite populations without changing the core methodology. They should be interpreted as empirical finite-population illustrations rather than field evidence from a completed audit engagement, but they nevertheless demonstrate that the method's main practical behavior is not confined to synthetic numerical examples.
\section{Conclusion}
This study formulates audit sampling with sequentially added items as a sequential testing problem for a finite population under sampling without replacement. The resulting planned sequential auditing procedure is designed to control error probabilities while reducing the expected sample size through early stopping. The method calibrates decision boundaries by Monte Carlo simulation at the least-favorable deviation rates and is therefore straightforward to implement at the population sizes commonly encountered in practice. We expect that our approach can refine existing audit-sampling procedures, including sample evaluation, planned sample extension, and further testing when early evidence is inconclusive, by organizing them as explicit sequential procedures with transparent statistical guarantees.
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