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Positivity properties of the ARFIMA(0,d,0) specifications and credibility analysis of frequency risks

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  • Pinquet, Jean

Abstract

Allowing for the seniority of claims and of risk exposure in the prediction of frequency risks necessitates dynamic random effects in Poisson mixtures. Non-life insurance data show evidence of long memory in stationary random effects. This paper proves that the ARFIMA(0,d,0) mixtures of Poisson distributions ensure nonnegative credibilities per period in the affine prediction of frequency risks. This is true regardless of the risk exposure. This property is maintained if the random effect is the product of a time-invariant component (which provides the highest level of memory in the data) and of a component that follows an ARFIMA(0,d,0) specification. The proof uses approximations of the ARFIMA(0,d,0) time series by AR(p) time series, which result from truncations of the filtering equations that define the former ones. Every given ARFIMA(0,d,0) specification inherits the positivity properties of the truncations because the supremum of the spectral densities of these truncations is integrable on the frequency domain. These semiparametric specifications are easily estimated from longitudinal count data, with the generalized method of moments.

Suggested Citation

  • Pinquet, Jean, 2020. "Positivity properties of the ARFIMA(0,d,0) specifications and credibility analysis of frequency risks," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 159-165.
    • Handle: RePEc:eee:insuma:v:95:y:2020:i:c:p:159-165
    DOI: 10.1016/j.insmatheco.2020.10.001
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    References listed on IDEAS

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    Cited by:

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    More about this item

    Keywords

    ARFIMA(0; d; 0) specifications; Poisson mixtures; Semiparametric analysis; Linear credibility; Spectral measure of stationary random effects;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C18 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Methodolical Issues: General
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models
    • C35 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions

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