Actuarial applications of multivariate phase-type distributions: Model calibration and credibility

Phase-type (PH) distributions are used as probability models for positive random variables. Their origin stems from the works of Neuts published in the early eighties. The first applications are found in operational research as models for waiting times in the field of queuing theory. Probability models in actuarial science are also fraught with positive variables such as losses and survival times which may explain the recent interest of actuaries in PH distributions. Statistical estimation of PH distributions with the EM algorithm was developed in the mid nineties by Asmussen and his coworkers. Actuaries have also applied this class of models to risk theory and ruin probabilities. Extensions to multivariate PH distributions were also developed in the eighties following the seminal work of Neuts. They can serve as models for the joint probability distribution of two or more positive random variables.;The second paper extends Jewell's theorem in credibility theory to a larger class of distributions, outside of exponential distributions or even the linear exponential family. Jewell's Theorem proves that exact credibility occurs in the univariate and multivariate linear exponential family of conditional distributions, when paired with the appropriate conjugate prior distribution. Here, exact credibility is discussed in a univariate and multivariate PH setting. Hidden Markov chains are used, embedding the unobservable risk parameters in the PH distributions.;KEY WORDS: Phase-type distributions; continuous Markov processes; hidden Markov chain; EM algorithm; parametric bootstrap; exact credibility; coxian distributions; Jewell's theorem.;The first paper treats of maximum likelihood estimation by the EM algorithm and goodness-of-fit tests by parametric bootstrap when the model is a bivariate PH distribution. Ahlstrom and his coworkers published in 1999 an EM algorithm for the parametric estimation of relapse time distributions in survival analysis. They used a bivariate PH distribution with one component greater than the other component with probability one. Although the EM method proposed in this thesis is similar, our model is more general. Moreover, we show how to evaluate with any desired degree of accuracy the Spearman or Kendall correlation coefficients of the fitted bivariate PH model. These correlation coefficients can then be compared with the non parametric Spearman or Kendall coefficients based on ranks. A close agreement is an indication of the validity of the model. A consistent goodness-of-fit testing procedure is proposed which compares the fitted (bivariate) parametric survival function with the empirical survival function using a statistic of the Cramer-von Mises type. A parametric bootstrap algorithm is also provided to obtain the critical region of the proposed test. The results are used to fit real data in the insurance industry relating losses (LOSS) and allocated loss adjustment expenses (ALAE). To our knowledge this is the first time that bivariate PH distributions are used to fit real data. The fitted bivariate PH distribution is used to obtain the quantiles and the mean of the conditional distribution of the variable ALAE for a given value of the other variable LOSS.