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Application of Two Gamma Distributions Mixture to Financial Auditing


The considered problem can be treated as a particular topic in the field of testing some substantive hypothesis in financial auditing. The main theme of the paper is the well-known problem of testing hypothesis on admissibility of the population total of accounting errors amounts. The set of items with non-zero errors amounts is the domain in the accounting population. The book amounts are treated as values of a random variable which distribution is a mixture of the distributions of correct amount and the distribution of the true amount contaminated by error. The mixing coefficient is equal to the proportion of the items with non-zero errors amounts. The mixture of two gamma distributions is taken into account. The well-known method of moments and likelihood method are proposed to estimate parameters of distribution. It let us construct some statistic to test the outlined hypothesis. Moreover, the well-known likelihood ratio test is considered.

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  • Cassel, C.M., Särndal, C.E. and Wretman, J.H. (1977). Foundation of Inference in Survey Sampling. Wiley, New York.

    MATH  Google Scholar 

  • Chen, J., Chen, S.Y. and Rao, J.N.K. (1998). Empirical likelihood confidence intervals for the mean of a population containing many zero values. The Canadian Journal of Statistics 31, 1, 53–68.

    MathSciNet  Article  MATH  Google Scholar 

  • Cox, D.R. and Snell, E.J. (1979). On sampling and the estimation of rare errors. Biometrika 66, 1, 125–132. Errata: Biometrika 69(2), 491 (1982).

    MathSciNet  Article  MATH  Google Scholar 

  • Cramér, H. (1962). Mathematical Methods of Statistics. Asia Publishing House, Bombay.

    MATH  Google Scholar 

  • Dimitrov, B., Green, D. Jr., Rykov, V. and Stanchev, P. (2003). On statistical hypothesis testing via simulation method. International Journal of Information Theories and Applications 10, 408–414.

    Google Scholar 

  • Davison, A.C. and Hinkley, D.V. (1997). Bootstrap Methods and Their Applications. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  • Dufour, J.M. (2006). Monte Carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics. Journal of Econometrics 133, 443–477.

    MathSciNet  Article  MATH  Google Scholar 

  • Dufour, J.M. and Khalaf, L. (2001). Monte Carlo test methods in econometrics, Oxford, Baltagi, B. (ed.), p. 494–519.

  • Fienberg, S.E., Nether, J. and Leitch, R.A. (1977). Estimating the total overstatement error in accounting populations. Journal of the American Statistical Association 72, 295–302.

    Article  Google Scholar 

  • Frost, P.A. and Tamura, H. (1986). Accuracy of auxiliary information interval estimation in statistical auditing. Journal of Accounting Research 24, 57–75.

    Article  Google Scholar 

  • Ghosh, M. and Meeden, G. (1997). Bayesian Methods for Finite Population Sampling. Chapman & Hall, London.

    Book  MATH  Google Scholar 

  • Guy, D.M. and Carmichael, D.R. (1986). Audit sampling: An introduction to statistical sampling in auditing. Wiley, New York.

    Google Scholar 

  • Hall, P. (1992). The Bootstrap and Edgewrth Expansion. Springer-Verlag, New York.

    Book  Google Scholar 

  • Kvanli, A.H., Shen, Y.K. and Deng, L.Y. (1998). Construction of confidence intervals for the mean of a population containing many zero values. Journal of Business and Economic Statistics 16, 362–368.

    Google Scholar 

  • MacKinnon, J. (2007). Bootstrap hypothesis testing. Queen’s Economics Department, Working Paper no 1127.

  • Marazzi, A. and Tillé, Y. (2016). Using past experience to optimize audit sampling design. Review of Quantive Finance Accounting 1-28,

  • McLachlan, G. and Peel, D. (2000). Finite Mixture Models. Wiley, New York.

    Book  MATH  Google Scholar 

  • Meng, X.L. (1977). The EM algorithm and medical studies: A historical link. Statistical Research Methods in Medical Research 6, 3–23.

    Article  Google Scholar 

  • Silvey, S.D. (1959). The Lagrangian multiplier test. The Annals of Mathematical Statistics 30, 2, 389–407.

    MathSciNet  Article  MATH  Google Scholar 

  • Särndal, C.-E., Swensson, B. and Wretman, J. (1989). Statistical models and analysis in auditing. Panel on nonstandard mixtures of distributions. Statistical Science 4, 1, 2–33.

    Article  Google Scholar 

  • Tamura, H. (1988). Estimation of rare errors using judgement. Biometrika 75, 1–9.

    MathSciNet  Article  MATH  Google Scholar 

  • Wywiał, J.L. (2016). Contributions to Testing Statistical Hypotheses in Auditing. PWN, Warsaw.

    Google Scholar 

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The project is supported by the grant of the National Science Centre, Poland, DEC-2012/07/B/HS4/03073.

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Correspondence to Janusz L. Wywiał.

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Wywiał, J.L. Application of Two Gamma Distributions Mixture to Financial Auditing. Sankhya B 80, 1–18 (2018).

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Keywords and phrases

  • Statistical auditing
  • Accounting error
  • Mixture of probability distribution
  • Method of moments
  • Likelihood ratio test

AMS (2000) subject classification

  • Primary: 62H15
  • Secondary: 91B28