Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors
Problems in statistical auditing are usually one⁻sided. In fact, the main interest for auditors is to determine the quantiles of the total amount of error, and then to compare these quantiles with a given <i>materiality</i> fixed by the auditor, so that the accounting statement...
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| Format: | Article |
| Language: | English |
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MDPI AG
2018-12-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/20/12/919 |
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| author | María Martel-Escobar Francisco-José Vázquez-Polo Agustín Hernández-Bastida |
| author_facet | María Martel-Escobar Francisco-José Vázquez-Polo Agustín Hernández-Bastida |
| author_sort | María Martel-Escobar |
| collection | DOAJ |
| description | Problems in statistical auditing are usually one⁻sided. In fact, the main interest for auditors is to determine the quantiles of the total amount of error, and then to compare these quantiles with a given <i>materiality</i> fixed by the auditor, so that the accounting statement can be accepted or rejected. Dollar unit sampling (DUS) is a useful procedure to collect sample information, whereby items are chosen with a probability proportional to book amounts and in which the relevant error amount distribution is the distribution of the taints weighted by the book value. The likelihood induced by DUS refers to a 201⁻variate parameter <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">p</mi> </semantics> </math> </inline-formula> but the prior information is in a subparameter <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula> linear function of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">p</mi> </semantics> </math> </inline-formula>, representing the total amount of error. This means that partial prior information must be processed. In this paper, two main proposals are made: (1) to modify the likelihood, to make it compatible with prior information and thus obtain a Bayesian analysis for hypotheses to be tested; (2) to use a maximum entropy prior to incorporate limited auditor information. To achieve these goals, we obtain a modified likelihood function inspired by the induced likelihood described by Zehna (1966) and then adapt the Bayes’ theorem to this likelihood in order to derive a posterior distribution for <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula>. This approach shows that the DUS methodology can be justified as a natural method of processing partial prior information in auditing and that a Bayesian analysis can be performed even when prior information is only available for a subparameter of the model. Finally, some numerical examples are presented. |
| first_indexed | 2024-04-14T06:32:12Z |
| format | Article |
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| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-14T06:32:12Z |
| publishDate | 2018-12-01 |
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| record_format | Article |
| series | Entropy |
| spelling | doaj.art-fdd85dcd0cdf482ca8161eb8047162642022-12-22T02:07:35ZengMDPI AGEntropy1099-43002018-12-01201291910.3390/e20120919e20120919Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy PriorsMaría Martel-Escobar0Francisco-José Vázquez-Polo1Agustín Hernández-Bastida2Department of Quantitative Methods, University of Las Palmas de Gran Canaria, 35001 Las Palmas de Gran Canaria, SpainDepartment of Quantitative Methods, University of Las Palmas de Gran Canaria, 35001 Las Palmas de Gran Canaria, SpainDepartment of Quantitative Methods, University of Granada, 18071 Granada, SpainProblems in statistical auditing are usually one⁻sided. In fact, the main interest for auditors is to determine the quantiles of the total amount of error, and then to compare these quantiles with a given <i>materiality</i> fixed by the auditor, so that the accounting statement can be accepted or rejected. Dollar unit sampling (DUS) is a useful procedure to collect sample information, whereby items are chosen with a probability proportional to book amounts and in which the relevant error amount distribution is the distribution of the taints weighted by the book value. The likelihood induced by DUS refers to a 201⁻variate parameter <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">p</mi> </semantics> </math> </inline-formula> but the prior information is in a subparameter <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula> linear function of <inline-formula> <math display="inline"> <semantics> <mi mathvariant="bold">p</mi> </semantics> </math> </inline-formula>, representing the total amount of error. This means that partial prior information must be processed. In this paper, two main proposals are made: (1) to modify the likelihood, to make it compatible with prior information and thus obtain a Bayesian analysis for hypotheses to be tested; (2) to use a maximum entropy prior to incorporate limited auditor information. To achieve these goals, we obtain a modified likelihood function inspired by the induced likelihood described by Zehna (1966) and then adapt the Bayes’ theorem to this likelihood in order to derive a posterior distribution for <inline-formula> <math display="inline"> <semantics> <mi>θ</mi> </semantics> </math> </inline-formula>. This approach shows that the DUS methodology can be justified as a natural method of processing partial prior information in auditing and that a Bayesian analysis can be performed even when prior information is only available for a subparameter of the model. Finally, some numerical examples are presented.https://www.mdpi.com/1099-4300/20/12/919auditingBayesian inferencedollar unit samplingmodified likelihoodpartial prior information |
| spellingShingle | María Martel-Escobar Francisco-José Vázquez-Polo Agustín Hernández-Bastida Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors Entropy auditing Bayesian inference dollar unit sampling modified likelihood partial prior information |
| title | Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors |
| title_full | Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors |
| title_fullStr | Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors |
| title_full_unstemmed | Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors |
| title_short | Bayesian Inference in Auditing with Partial Prior Information Using Maximum Entropy Priors |
| title_sort | bayesian inference in auditing with partial prior information using maximum entropy priors |
| topic | auditing Bayesian inference dollar unit sampling modified likelihood partial prior information |
| url | https://www.mdpi.com/1099-4300/20/12/919 |
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