Confidence intervals of the premiums of optimal bonus malus systems

Dimitris Karlis*, George Tzougas, Nicholas Frangos

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


In view of the economic importance of motor third-party liability insurance in developed countries the construction of optimal BMS has been given considerable interest. However, a major drawback in the construction of optimal BMS is that they fail to account for the variability on premium calculations which are treated as point estimates. The present study addresses this issue. Specifically, nonparametric mixtures of Poisson laws are used to construct an optimal BMS with a finite number of classes. The mixing distribution is estimated by nonparametric maximum likelihood (NPML). The main contribution of this paper is the use of the NPML estimator for the construction of confidence intervals for the premium rates derived by updating the posterior mean claim frequency. Furthermore, we advance one step further by improving the performance of the confidence intervals based on a bootstrap procedure where the estimated mixture is used for resampling. The construction of confidence intervals for the individual premiums based on the asymptotic maximum likelihood theory is beneficial for the insurance company as it can result in accurate and effective adjustments to the premium rating policies from a practical point of view.

Original languageEnglish
Pages (from-to)129-144
Number of pages16
JournalScandinavian Actuarial Journal
Issue number2
Early online date25 Apr 2017
Publication statusPublished - 7 Feb 2018


  • asymptotic normality
  • Efron percentile bootstrap confidence intervals
  • nonparametric maximum likelihood
  • Optimal BMS
  • Wald-type two-sided confidence intervals

ASJC Scopus subject areas

  • Statistics and Probability
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty


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