AStA Advances in Statistical Analysis

, Volume 97, Issue 1, pp 33–47 | Cite as

Nearest neighbor hazard estimation with left-truncated duration data

  • Rafael Weißbach
  • Wladislaw Poniatowski
  • Walter Krämer
Original Paper

Abstract

Duration data often suffer from both left-truncation and right-censoring. We show how both deficiencies can be overcome at the same time when estimating the hazard rate nonparametrically by kernel smoothing with the nearest-neighbor bandwidth. Smoothing Turnbull’s estimator of the cumulative hazard rate, we derive strong uniform consistency of the estimate from Hoeffding’s inequality, applied to a generalized empirical distribution function. We also apply our estimator to rating transitions of corporate loans in Germany.

Keywords

Kernel smoothing Hazard rate Left-truncation Right-censoring 

References

  1. Bluhm, C., Overbeck, L., Wagner, C.: An Introduction to Credit Risk Modeling. Chapman & Hall, London (2002) CrossRefGoogle Scholar
  2. Einmahl, U., Mason, D.: Uniform in bandwidth consistency of kernel-type function estimators. Ann. Stat. 33, 1380–1403 (2005) MathSciNetMATHCrossRefGoogle Scholar
  3. Gefeller, O., Dette, H.: Nearest neighbour kernel estimation of the hazard function from censored data. J. Stat. Comput. Simul. 43, 93–101 (1992) CrossRefGoogle Scholar
  4. Gefeller, O., Weißbach, R., Bregenzer, T.: The implementation of a data-driven selection procedure for the smoothing parameter in nonparametric hazard rate estimation using SAS/IML software. In: Friedl, H., Berghold, A., Kauermann, G. (eds.) Proceedings of the 13th SAS European Users Group International Conference, Stockholm, pp. 1288–1300. SAS Institute, Carry (1996) Google Scholar
  5. Goto, F.: Achieving semiparametric efficiency bounds in left-censored duration models. Econometrica 64(2), 439–442 (1996) MathSciNetMATHCrossRefGoogle Scholar
  6. Grillenzoni, C.: Robust nonparametric estimation of the intensity function of point data. AStA Adv. Stat. Anal. 92, 117–134 (2008) MathSciNetCrossRefGoogle Scholar
  7. Hewitt, E., Savage, L.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc. 80, 470–501 (1955) MathSciNetMATHCrossRefGoogle Scholar
  8. Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58, 13–30 (1963) MathSciNetMATHCrossRefGoogle Scholar
  9. Kiefer, N., Larson, C.: A simulation estimator for testing the time homogeneity of credit rating transitions. J. Empir. Finance 14, 818–835 (2007) CrossRefGoogle Scholar
  10. Kim, Y.-D., James, L., Weißbach, R.: Bayesian analysis of multi-state event history data: Beta–Dirichlet process prior. Biometrika 99, 127–140 (2012) MathSciNetMATHCrossRefGoogle Scholar
  11. Knüppel, L., Hermsen, O.: Median split, k-group split, and optimality in continuous populations. AStA Adv. Stat. Anal. 94, 53–74 (2010) MathSciNetCrossRefGoogle Scholar
  12. Li, D., Li, Q.: Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters. J. Econom. 157, 179–190 (2010) CrossRefGoogle Scholar
  13. Merton, R.: On the pricing of corporate debt: the risk structure of interest rates. J. Finance 29, 449–470 (1974) Google Scholar
  14. Schäfer, H.: Local convergence of empirical measures in the random censorship situation with application to density and rate estimators. Ann. Stat. 14, 1240–1245 (1986) MATHCrossRefGoogle Scholar
  15. Shorack, G., Wellner, J.: Empirical Processes with Application to Statistics. Wiley, New York (1986) Google Scholar
  16. Silverman, B.: Density Estimation. Chapman & Hall, London (1986) MATHGoogle Scholar
  17. Stute, W.: Almost sure representations of the product-limit estimator for truncated data. Ann. Stat. 21, 146–156 (1993) MathSciNetMATHCrossRefGoogle Scholar
  18. Turnbull, B.W.: The empirical distribution function with arbitrarily grouped, censored and truncated data. J. R. Stat. Soc., Ser. B, Stat. Methodol. 38, 290–295 (1976) MathSciNetMATHGoogle Scholar
  19. Weißbach, R., Mollenhauer, T.: Modelling rating transitions. J. Korean Stat. Soc. 4, 469–485 (2011) CrossRefGoogle Scholar
  20. Weißbach, R., Pfahlberg, A., Gefeller, O.: Double-smoothing in kernel hazard rate estimation. Methods Inf. Med. 47, 167–173 (2008) Google Scholar
  21. Weißbach, R., Tschiersch, P., Lawrenz, C.: Testing time-homogeneity of rating transitions after origination of debt. Empir. Econ. 36, 575–596 (2009) CrossRefGoogle Scholar
  22. Weißbach, R., Walter, R.: A likelihood ratio test for stationarity of rating transitions. J. Econom. 155, 188–194 (2010) CrossRefGoogle Scholar
  23. Weißbach, R.: A general kernel functional estimator with general bandwidth—strong consistency and applications. J. Nonparametr. Stat. 18, 1–12 (2006) MathSciNetMATHCrossRefGoogle Scholar
  24. Wied, D., Weißbach, R.: Consistency of the kernel density estimator—a survey. Stat. Pap. 53, 1–21 (2012) MATHCrossRefGoogle Scholar
  25. Woodroofe, M.: Estimating a distribution function with truncated data. Ann. Stat. 13, 163–177 (1985) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Rafael Weißbach
    • 1
  • Wladislaw Poniatowski
    • 2
  • Walter Krämer
    • 3
  1. 1.Lehrstuhl Statistik, Institut für Volkswirtschaftslehre, Fakultät für Wirtschafts- und SozialwissenschaftenUniversität RostockRostockGermany
  2. 2.Deutsche TelekomBonnGermany
  3. 3.Fakultät StatistikTechnische Universität DortmundDortmundGermany

Personalised recommendations