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**Statistical independence and fractional age assumptions. With discussion and a reply by the author.**
*(English)*
Zbl 1080.62549

Summary: This paper considers in some detail the issue of statistical independence of the curtate future lifetime and the fractional part of the future lifetime of a general status. Statistical independence is often employed in actuarial contexts, primarily because it leads to simple relationships between quantities of interest and statistical information that is of a discrete nature, such as a life table. The uniform distribution of deaths (UDD) assumption is the most commonly used because of its simplicity and intuitive appeal, but it can be somewhat restrictive. For example, all deaths or withdrawals may be assumed to be at a particular point in the year such as the middle; assumptions of this type are often made in a multiple decrement context. This paper attempts to unify these assumptions and extend their applicability in an actuarial context.

The conditions for independence need to be stated carefully, and the last-survivor status is cited as an example in which failure to do so can lead to erroneous conclusions.

The fractional independence (FI) assumption is defined, and it is demonstrated that many of the formulas for life table functions that hold under the more restrictive UDD assumption are extended more easily to the general FI case. The simple relationship under UDD between insurances payable on other than an annual mode and those payable at the end of the year of death is extended to the FI case as well. These results are then used to obtain results for annuities and reserves, again generalizing UDD relationships. It is then demonstrated that many contingent probabilities in the multiple life context are exactly the same under the FI assumption as under the more restrictive UDD assumption. Finally, a very general result that holds in the multiple decrement context is shown to hold under the FI assumption.

The conditions for independence need to be stated carefully, and the last-survivor status is cited as an example in which failure to do so can lead to erroneous conclusions.

The fractional independence (FI) assumption is defined, and it is demonstrated that many of the formulas for life table functions that hold under the more restrictive UDD assumption are extended more easily to the general FI case. The simple relationship under UDD between insurances payable on other than an annual mode and those payable at the end of the year of death is extended to the FI case as well. These results are then used to obtain results for annuities and reserves, again generalizing UDD relationships. It is then demonstrated that many contingent probabilities in the multiple life context are exactly the same under the FI assumption as under the more restrictive UDD assumption. Finally, a very general result that holds in the multiple decrement context is shown to hold under the FI assumption.

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

62E15 | Exact distribution theory in statistics |

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\textit{G. E. Willmot}, N. Am. Actuar. J. 1, No. 1, 84--99 (1997; Zbl 1080.62549)

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### References:

[1] | Bowers N., Actuarial Mathematics (1986) · Zbl 0634.62107 |

[2] | Hogg R., Loss Distributions (1984) |

[3] | Johnson N., Continuous Univariate Distributions 2, 2. ed. (1995) · Zbl 0821.62001 |

[4] | Jordan C., Life Contingencies (1967) |

[5] | Shiu E., Transactions of the Society of Actuaries pp 571– (1982) |

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