Journal of Applied Probability

Exact boundaries in sequential testing for phase-type distributions

Hansjörg Albrecher, Peiman Asadi, and Jevgenijs Ivanovs


Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.

Article information

J. Appl. Probab., Volume 51A (2014), 347-358.

First available in Project Euclid: 2 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62L12: Sequential estimation
Secondary: 91B30: Risk theory, insurance

Sequential probability ratio test Markov additive process scale function two-sided exit problem Esscher transform


Albrecher, Hansjörg; Asadi, Peiman; Ivanovs, Jevgenijs. Exact boundaries in sequential testing for phase-type distributions. J. Appl. Probab. 51A (2014), 347--358. doi:10.1239/jap/1417528485.

Export citation


  • Asmussen, S. (1989). Exponential families generated by phase-type distributions and other Markov lifetimes. Scand. J. Statist. 16, 319–334.
  • Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities (Adv. Ser. Statist. Sci. Appl. Prob. 14), 2nd edn. World Scientific, Hackensack, NJ.
  • Franx, G. J. (2001). A simple solution for the M/D/c waiting time distribution. Operat. Res. Lett. 29, 221–229.
  • Gerber, H. U. (1988). Mathematical fun with ruin theory. Insurance Math. Econom. 7, 15–23.
  • Ivanovs, J. (2013). A note on killing with applications in risk theory. Insurance Math. Econom. 52, 29–34.
  • Ivanovs, J. and Palmowski, Z. (2012). Occupation densities in solving exit problems for Markov additive processes and their reflections. Stoch. Process. Appl. 122, 3342–3360.
  • Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
  • Lotov, V. I. (1987). Asymptotic expansions for a sequential likelihood ratio test. Teor. Veroyat. i Primen. 32, 62–72.
  • Segerdahl, C.-O. (1959). A survey of results in the collective theory of risk. In Probability and Statistics: The Harald Cramér Volume, Almqvist & Wiksell, Stockholm, pp. 276–299.
  • Shiryaev, A. N. (1973). Statistical Sequential Analysis. American Mathematical Society.
  • Teugels, J. L. and Van Assche, W. (1986). Sequential testing for exponential and Pareto distributions. Sequent. Anal. 5, 223–236.
  • Wald, A. (1947). Sequential Analysis. John Wiley, New York.
  • Wald, A. and Wolfowitz, J. (1948). Optimum character of the sequential probability ratio test. Ann. Math. Statist. 19, 326–339.
  • Weiss, L. (1956). On the uniqueness of Wald sequential tests. Ann. Math. Statist. 27, 1178–1181.
  • Wijsman, R. A. (1960). A monotonicity property of the sequential probability ratio test. Ann. Math. Statist. 31, 677–684.
  • Wijsman, R. A. (1963). Existence, uniqueness and monotonicity of sequential probability ratio tests. Ann. Math. Statist. 34, 1541–1548.