Improving the accuracy of estimates for complex sampling in auditing 1.

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1 Improving the accuracy of estimates for complex sampling in auditing 1. Y. G. Berger 1 P. M. Chiodini 2 M. Zenga 2 1 University of Southampton (UK) 2 University of Milano-Bicocca (Italy) The research leading to these results has received support under the European Commissions 7th Framework Programme (FP7/ ) under grant agreement n312691, InGRID Inclusive Growth Research Infrastructure Diffusion. Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

2 Overview 1 Audit Sampling 2 Definitions 3 Simulation study 4 Conclusions and Future research Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

3 AUDIT SAMPLING In auditing the goal is to verify if the values of the accounts reported by the company are not materially misstated To determine the total error in the amount reported by the company auditors should audit all accounts in the population of accounts. This is not possible as it is too costly and time expensive! In practice auditors verify only a sample of accounts to estimate the error of the total population of accounts (total amount error & error rate) Common statistical methods to select an audit sample are by without replacement or by probability proportional to size (PPS) Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

4 SAMPLING PLANS IN AUDITING For the auditor the only interest is to verify that the error rate is within a pre-assigned value, if the observed rate is greater than the pre-assigned value a census is made. Practically speaking all the sampling methods are reliable for audit sampling, by the way Monetary Unit Sampling (MUS) - a particular case of the PPS - is the most popular! This sampling method directs efforts towards high-valued items which contain the greatest potential of large overstatement Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

5 SAMPLING PLANS IN AUDITING The way the auditors make their choices in terms of sampling strategies is frequently based on personal experience! In general: book values distributions are highly positively skewed with different percentage of errors Two main scenarios can be met: - a population with relatively large number of small accounts combined with high rate of mistakes - a population with relatively large number of accounts combined with a small rate of mistakes Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

6 DEFINITIONS An accounting population consist of N line items with book (or recorded) values, y 1, y 2,, y N and total book amount T y defined by: T y = N y i. i=1 The audited (true) amount of the N line items in the population is denoted by x 1, x 2,..., x N and the total audited amount is: T x = N x i. i=1 Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

7 DEFINITIONS The error in item i, is z i = y i x i, 1 i N. When z i > 0, the i-th item is said to be overstated and when z i < 0, it is understated. When z i = 0, the account is said to be error free. The total error amount is defined as: N T z = z i. For y i 0, t i = z i y i = y i x i y i is called the fractional error or taint. The values (x 1, x 2,..., x N ) are unknown before sampling, whereas (y 1, y 2,..., y N ) are known. It is assumed that the amount of any overstatement does not exceed the stated recorded value. i=1 Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

8 DEFINITIONS The purpose of the audit is to estimate the total error amount T z : n n T z = z i = t i y i i=1 i=1 obtained by the examination of a sample of n items of the account. Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

9 Stringer Bound (Stringer, 1963) Let T 1,..., T n be the independent random variables which represent the taintings. The distribution of these taintings is some unknown mixture of distributions on the interval [0; 1], so that Pr(0 T i 1) = 1. Let denote µ = E(T i ) and let 0 t 1:n t 2:n... t n:n 1 be the ordered statistics of (T 1, T 2,..., T n ). For α (0; 1) and i = 0, 1,..., n 1 let p = p n (i; 1 α) be the unique solution of: with p n (n; 1 α) = 1. i k=0 ( ) n p k (1 p) n k = α k Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

10 Stringer Bound (Stringer, 1963) The Stringer method for obtaining an upper bound for the total overstatement error can be obtained by combining the upper limits for the sample error rates with the taints: ˆT z = T y p n(0, 1 α) + T y n [p n(i, 1 α) p n(i 1, 1 α)]t n i+1:n i=1 then p n(i, 1 α) is the (1 α) upper confidence limit for the binomial parameter when i errors are observed in a sample of size n. Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

11 Stringer Bound (Stringer, 1963) Equivalently for a given α, n and number of errors i, it is possible to find the value p n (i, 1 α) that satisfies: i j=0 ( ) n [p n (i, 1 α)] j [1 p n (i, 1 α)] n j = α. j The Stringer bound is sometimes calculated using the Poisson approximation for obtaining the upper confidence limits p n (i, 1 α). Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

12 Empirical Likelihood estimator EL is a non-parametric likelihood. Hartley and Rao in 1968 first introduced it in the context of survey sampling as scale-load approach. From early 2000 the EL approach has been introduced also in survey sampling literature. EL approach provides non-parametric confidence intervals similar to the parametric likelihood ratio intervals. Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

13 Empirical Likelihood estimator The shape and the orientation of the EL intervals are completely determined by the data. Chen et al. (2003) obtained EL intervals on the population mean for populations containing many zero values thats the case of audit sampling. Parametric likelihood ratio intervals based on parametric mixture distributions perform better than the standard normal theory intervals in terms of coverage, but EL intervals perform better under deviations from the assumed mixture model, by providing non-coverage rate below lower bound closer to the nominal value and also larger lower bound. Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

14 EL approach (Berger, 2016) Let U be a finite populations of N units. θ 0 is the unique solution G(θ) = 0, G(θ) = g i (θ) i U The empirical log-likelihood function is defined l (m) = log( m i ) = log(m i ) (1) i S i S m i is estimated by the value ˆm i which maximize l (m) subject m i 0 m i c = C i S Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

15 EL approach (Berger, 2016) The solution is given by where π i = np i. ˆm i = {(t + η) T c i } 1 = (π i + η T c i ) 1 Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

16 Maximum EL estimator The empirical log-likelihood ratio function is defined by ˆr(θ) = 2{l( ˆm) l( ˆm, θ)} where l( ˆm) = log( ˆm i ) i S l( ˆm, θ) = log( ˆm i (θ)) i S ˆm i (θ) is the value that maximize (1) subject to - m i 0 - m i ci = C with ci = (ci T, 0) T and C = (C T, 0) T i S Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

17 HOW TO EVALUATE AUDITING RISK Previous studies has already demonstrated that different sampling plans such as Systematic Sampling or Probability Proportional to Size plans (such as Unrestricted Random Sampling, Lahiri Sampling,...) are all reliable even for all sample size! We evaluate the efficiency of EL Bound respect the Stringer Bound. The parametr of interest is the ratio of error-per-euro (Taint) Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

18 SIMULATION STUDY We simulated data using a real accounting population of credit invoices of an audited society. 40,0 30,0 Frequency 20,0 10,0 0, Value Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

19 SIMULATION STUDY X LogNormal(7.001, 1.71). N = m = 1000 samples. Error randomly associated to the X i (no fraudulent hypothesis) Error rates : + 5%, + 10% + 20%. Taint simulated values: ; ; ; Sample fractions: * 0.05 * 0.1 Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

20 SIMULATION STUDY To compare different estimators Tightness, MSE and Coverage Probability have been computed Tightness is used to indicate how close the bound value is to the true error rate: (AV (ˆt) t) t 100 If it is small the sampling method is said to be tight and if it is large is said to be conservative Variability of the bound: this is an indicator of the uncertainty of the bound. This is measured by Mean Squared Error (MSE): 1 m m (ˆt i t) 2 where ˆt i is the estimated value for the error rate at the ith replicate. i=1 Coverage probability: for a specific bound it refers to the proportion of replications for which a bound is greater than or equal to the true population error amount. A bound is considered unreliable if its coverage is significantly below the specified nominal coverage, otherwise it is reliable. Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

21 Case Study I: Taint EL Stringer % Error % n/n Tight. MSE COV Tight. MSE COV % 8.745E % 8.012E % 3.275E % 3.121E % 1.311E % 6.532E % 2.003E % 9.041E % 3.096E % 8.521E % 1.281E % 8.301E Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

22 Case Study II: Taint EL Stringer % Error % n/n Tight. MSE COV Tight. MSE COV % 9.451E % 1.713E % 7.921E % 1.312E % 4.251E % 9.101E % 1.741E % 1.700E % 7.096E % 3.512E % 2.283E % 7.192E-04 1 Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

23 Case Study III: Taint EL Stringer % Error % n/n Tight. MSE COV Tight. MSE COV % 2.217E % 1.653E % 4.628E % 7.510E % 1.115E % 1.210E % 4.941E % 2.612E % 2.140E % 1.300E % 8.162E % 6.010E Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

24 Results * EL is generally better than SB for estimating error rate * EL is tighter for the estimation of the real error rate. SB is more conservative! * MSE of SB is greater respect EL. * EL and SB can be considered reliable. Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

25 Conclusions and Future research * We introduced di EL Bound to estimate the Upper Bound for the estimation of the error rate * In general EL Bound perform better respect the Stringer Bound. + Other distributions (Dagum, Gamma,...) for the X values. + Other hypothesys on the generation of the error (on the queues,...) Berger, Chiodini, Zenga Improving the accuracy of estimates / 26

26 References Arens, A.A., Loebbecke, J.K., (1981). Applications of Statistical sampling to Auditing, Prentice-Hall, Inc.. Berger, Y. G., De La Riva Torres, O. (2015). Empirical likelihood confidence intervals for complex sampling designs, Journal of the Royal Statistical Society: Series B (Statistical Methodology),78(2), Chen, J., Chen, S. Y., and Rao, J. N. K. (2003). Empirical likelihood confidence intervals for the mean of a population containing many zero values. The Canadian Journal of Statistics 31, Chiodini P.M., Zenga M. (2014) Efficiency of the sample plans for symmetric and non-symmetric distributions in auditing: a comparison. in Contribution to sampling Statistics, ed Mecatti F., Conti PL., Ranalli MG Horgan, J.M. (2008). Monetary-unit sampling old and new, School of Computing, Dublin City University, Dublin,Ireland. Nandram, H.N., (2009). Applications of Statistical sampling to Auditing, Monetary unit sampling: Improving estimation of the total audit error, Advances in Accounting, incorporating Advances in International Accounting, 25, Owen Art B. (1988). Empirical likelihood ratio confidence intervals for a single functional, Biometrika 75(2), Rao, J. N. K. (2006). Empirical Likelihood Methods for Sample Survey Data:An Overview AUSTRIAN JOURNAL OF STATISTICS, 35(2-3), Stringer, K.W. (1963). Practical aspects of statistical sampling in auditing. Proceedings of the Business and Economics Statistics Section, ASA Berger, Chiodini, Zenga Improving the accuracy of estimates / 26