Finance and Stochastics

, Volume 14, Issue 4, pp 569–591 | Cite as

A global consistency result for the two-dimensional Pareto distribution in the presence of misspecified inflation

  • Peter Grandits
  • Grigory Temnov


A global consistency result for the ML estimator of a misspecified two-parameter Pareto distribution is proved. The misspecification is due to the assumption of a wrong inflation rate, which violates the i.i.d. assumption in the model. We also investigate how far away from the true parameters one finds the ML estimator of the misspecified model (asymptotically for a small misspecification r). Finally, for the case where the misspecification depends on the number of observations n, i.e., r=r n , and where \(r_{n}\stackrel{n\to \infty}{\longrightarrow}0\), we prove a central limit theorem for the ML estimator.


ML estimation Global consistency Pareto distribution Misspecified model Inflation 

Mathematics Subject Classification (2000)

62F12 91B70 

JEL Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bronstein, I., Semendjajew, K.: Taschenbuch der Mathematik. Verlag Harri Deutsch, Hamburg (2000) Google Scholar
  2. 2.
    Cramér, H.: A contribution to the theory of statistical estimation. Skand. Aktuarietidskr. 24, 85–94 (1946) Google Scholar
  3. 3.
    Embrechts, P., Klüppelberg, C., Mikosch, T.: Modelling Extremal Events for Insurance and Finance. Springer, Berlin (1997) zbMATHGoogle Scholar
  4. 4.
    Grandits, P., Kainhofer, R., Temnov, G.: On the impact of hidden trends for a compound Poisson model with Pareto-type claims. Preprint, TU Vienna (2009). Available at
  5. 5.
    Hoadley, B.: Asymptotic properties of maximum likelihood estimators for the independent not identically distributed case. Ann. Math. Stat. 42, 1977–1991 (1971) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Huber, P.J.: The behavior of maximum likelihood estimates under nonstandard conditions. In: Le Cam, L.M., Neyman, J. (eds.) Proceedings of the Fifth Berkeley Symposium on Mathematics, Statistics and Probability, pp. 221–233. University of California Press, Berkeley (1967) Google Scholar
  7. 7.
    Lehman, E.L.: Elements of Large-Sample Theory. Springer, Berlin (1999) CrossRefGoogle Scholar
  8. 8.
    Rockette, H., Antle, C., Klimko, L.A.: Maximum likelihood estimation with the Weibull model. J. Am. Stat. Assoc. 69, 246–249 (1974) zbMATHCrossRefGoogle Scholar
  9. 9.
    Smith, R.L.: Estimating tails of probability distributions. Ann. Stat. 15, 1174–1207 (1987) zbMATHCrossRefGoogle Scholar
  10. 10.
    Wald, A.: Note on the consistency of the maximum likelihood estimate. Ann. Math. Stat. 20, 595–601 (1949) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    White, H.: Maximum likelihood estimation of misspecified models. Econometrica 50, 1–25 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institute for Mathematical Methods in Economics, Financial and Actuarial MathematicsVienna University of TechnologyViennaAustria
  2. 2.Edgeworth Centre for Financial MathematicsUniversity College CorkCorkIreland

Personalised recommendations