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**Fitting combinations of exponentials to probability distributions.**
*(English)*
Zbl 1142.60321

In this paper two techniques for approximating distributions on the positive half-line by combinations of exponentials are described. One is based on Jacobi polynomial expansions, and the other on the logbeta distribution. The class of approximating distributions considered in this paper consists of those with a distribution function \(F\) which can be written as \(1-F(t)=\sum_{j=1}^{n}a_{j}e^{-\lambda_{j}t}\), \(t\geq0, \lambda_{j}>0\), \(j=1,\dots,n\), \(1\leq n<\infty\). The author defines the Jacobi polynomials, gives some of their properties and shows how these polynomials may be used to obtain a convergent series of exponentials that represents a continuous distribution function exactly. Two versions of the method are described. The first one yields a convergent sequence of combinations of exponentials which are not precisely density functions, while the second version produces true density functions. It is shown that sequences of logbeta distributions converge to the Dirac mass, and that this can be used to find a different sequence of combinations of exponentials which converges to any distribution on a finite interval. The proposed methods are applied to approximation of some distributions: degenerate, uniform, Pareto, lognormal, Makeham and others. An error bound is given in the case of logbeta approximations.

Reviewer: A. D. Borisenko (Kyïv)

### MSC:

60E99 | Distribution theory |

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\textit{D. Dufresne}, Appl. Stoch. Models Bus. Ind. 23, No. 1, 23--48 (2007; Zbl 1142.60321)

### References:

[1] | Matrix-Geometric Solutions in Stochastic Models. The Johns Hopkins University Press: Baltimore, 1981. |

[2] | . Renewal theory and queueing algorithms for matrix-exponential distributions. In Matrix-Analytic Methods in Stochastic Models, (eds). Marcel Dekker Inc.: New York, 1997. · Zbl 0872.60064 |

[3] | Phase-type distributions and related point processes: fitting and recent advances. In Matrix-Analytic Methods in Stochastic Models, (eds). Marcel Dekker Inc.: New York, 1997. · Zbl 0865.62057 |

[4] | Feldmann, Performance Evaluation 31 pp 245– (1998) |

[5] | Dufresne, Australian Actuarial Journal 7 pp 755– (2001) |

[6] | An Introduction to Probability Theory and its Applications, II (2nd edn). Wiley: New York, 1971. |

[7] | Yor, Advances in Applied Probability 24 pp 509– (1992) |

[8] | Bessel processes and a functional of Brownian motion. In Numerical Methods in Finance, (eds). Kluwer Academic Publisher: Dordrecht, 2005. |

[9] | Dufresne, North American Actuarial Journal (2005) |

[10] | The Special Functions and their Approximations. Academic Press: New York, 1969. |

[11] | Completeness and Basis Properties of Sets of Special Functions. Cambridge University Press: Cambridge, MA, 1977. |

[12] | Special Functions and their Applications. Dover: New York, 1972. |

[13] | , , , . Actuarial Mathematics (2nd edn). Society of Actuaries: Itasca, Illinois, 1997. |

[14] | Dufresne, Advances in Applied Probability 36 pp 747– (2004) · Zbl 1063.60115 |

[15] | de Prony, Journal de l’École Polytechnique 1 pp 24– (1795) |

[16] | . Handbook of Mathematical Functions (Tenth printing). Dover: New York, 1972. |

[17] | Probability and Measure (2nd edn). Wiley: New York, 1987. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.