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Exact fit of simple finite mixture models


  • Dirk Tasche


How to forecast next year's portfolio-wide credit default rate based on last year's default observations and the current score distribution? A classical approach to this problem consists of fitting a mixture of the conditional score distributions observed last year to the current score distribution. This is a special (simple) case of a finite mixture model where the mixture components are fixed and only the weights of the components are estimated. The optimum weights provide a forecast of next year's portfolio-wide default rate. We point out that the maximum-likelihood (ML) approach to fitting the mixture distribution not only gives an optimum but even an exact fit if we allow the mixture components to vary but keep their density ratio fix. From this observation we can conclude that the standard default rate forecast based on last year's conditional default rates will always be located between last year's portfolio-wide default rate and the ML forecast for next year. As an application example, then cost quantification is discussed. We also discuss how the mixture model based estimation methods can be used to forecast total loss. This involves the reinterpretation of an individual classification problem as a collective quantification problem.

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  • Dirk Tasche, 2014. "Exact fit of simple finite mixture models," Papers 1406.6038,, revised Jul 2014.
  • Handle: RePEc:arx:papers:1406.6038

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    References listed on IDEAS

    1. Daniel J. Hopkins & Gary King, 2010. "A Method of Automated Nonparametric Content Analysis for Social Science," American Journal of Political Science, John Wiley & Sons, vol. 54(1), pages 229-247, January.
    2. Hofer, Vera & Krempl, Georg, 2013. "Drift mining in data: A framework for addressing drift in classification," Computational Statistics & Data Analysis, Elsevier, vol. 57(1), pages 377-391.
    3. Dirk Tasche, 2012. "The art of probability-of-default curve calibration," Papers 1212.3716,, revised Nov 2013.
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