Given this fact, the structural and stepwise models outperform the ARIMA, X-11, and Martingale models in terms of audit efficiency by obtaining lower alpha risks for the negative approach.

Alpha risks are significantly lower in the more structured cases for all the models when the negative approach is used as shown in table 6.

In general, for the positive approach, the ARIMA and Martingale models may expose auditors to unnecessary beta risks and their lower alpha risks may simply be due to larger confidence intervals.

Specifically, alpha risk is the likelihood that an auditor will conclude that an account is materially misstated (in error) when it is not and beta risk is the likelihood that an auditor will conclude an account is not materially misstated when it is indeed materially misstated (in error).

Using the positive approach, alpha risk (Type I error) is concluding that the account is in error ([absolute value of E]>0) when it is not in error ([absolute value of E]=0) as shown in table 1 panel A.

Actual State: Actual State: [absolute [absolute Conclusion value of E]=0 value of E]>0 [absolute value of E]=0 Correct Decision Beta Risk (Type II Error) [absolute value of E]>0 Alpha Risk Correct Decision (Type I Error) Panel B: Negative Approach [H.

The alpha risk at [absolute value of E]=0 is the likelihood that an auditor will conclude that an account is in error when it is not.

Similar to the alpha risk, a large prediction error (MPE), such as that for the X-11 model, can lead to anomalies resulting in very low or high beta risks regardless of the width of the confidence interval.

TABLE 4 Relationships Among Confidence Interval Width, Risk, and Testing Approach (a) Confidence Interval Width Risk Level Alpha Risk ([absolute value of E]=0): Positive Approach Wider Lower (Control Point) Negative Approach Wider Higher Beta Risk ([absolute value of E]=M): Positive Approach Wider Higher Negative Approach Wider Lower (Control Point) (a) Assuming the predicted values (P) and actual values (A) are reasonably close.

As a result, the alpha risk for the positive approach is expected to be the same for all the models because a common rejection area ([alpha] =.

The alpha risk for both of these models is closer to the specified control point ([absolute value of E]=0) risk of [alpha] =.

In term of alpha risk, the structural and stepwise models yield the lowest alpha risk for both time periods for the negative approach.