# Nearest neighbor hazard estimation with left-truncated duration data

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## 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## Notes

### Acknowledgement

Financial support by Deutsche Forschungsgemeinschaft is gratefully acknowledged (SFB 823 and Grant WE3573/2).

### References

- Bluhm, C., Overbeck, L., Wagner, C.: An Introduction to Credit Risk Modeling. Chapman & Hall, London (2002) CrossRefGoogle Scholar
- Einmahl, U., Mason, D.: Uniform in bandwidth consistency of kernel-type function estimators. Ann. Stat.
**33**, 1380–1403 (2005) MathSciNetMATHCrossRefGoogle Scholar - 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 - 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
- Goto, F.: Achieving semiparametric efficiency bounds in left-censored duration models. Econometrica
**64**(2), 439–442 (1996) MathSciNetMATHCrossRefGoogle Scholar - Grillenzoni, C.: Robust nonparametric estimation of the intensity function of point data. AStA Adv. Stat. Anal.
**92**, 117–134 (2008) MathSciNetCrossRefGoogle Scholar - Hewitt, E., Savage, L.: Symmetric measures on Cartesian products. Trans. Am. Math. Soc.
**80**, 470–501 (1955) MathSciNetMATHCrossRefGoogle Scholar - Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc.
**58**, 13–30 (1963) MathSciNetMATHCrossRefGoogle Scholar - 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 - 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 - Knüppel, L., Hermsen, O.: Median split,
*k*-group split, and optimality in continuous populations. AStA Adv. Stat. Anal.**94**, 53–74 (2010) MathSciNetCrossRefGoogle Scholar - 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 - Merton, R.: On the pricing of corporate debt: the risk structure of interest rates. J. Finance
**29**, 449–470 (1974) Google Scholar - 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 - Shorack, G., Wellner, J.: Empirical Processes with Application to Statistics. Wiley, New York (1986) Google Scholar
- Silverman, B.: Density Estimation. Chapman & Hall, London (1986) MATHGoogle Scholar
- Stute, W.: Almost sure representations of the product-limit estimator for truncated data. Ann. Stat.
**21**, 146–156 (1993) MathSciNetMATHCrossRefGoogle Scholar - 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 - Weißbach, R., Mollenhauer, T.: Modelling rating transitions. J. Korean Stat. Soc.
**4**, 469–485 (2011) CrossRefGoogle Scholar - Weißbach, R., Pfahlberg, A., Gefeller, O.: Double-smoothing in kernel hazard rate estimation. Methods Inf. Med.
**47**, 167–173 (2008) Google Scholar - Weißbach, R., Tschiersch, P., Lawrenz, C.: Testing time-homogeneity of rating transitions after origination of debt. Empir. Econ.
**36**, 575–596 (2009) CrossRefGoogle Scholar - Weißbach, R., Walter, R.: A likelihood ratio test for stationarity of rating transitions. J. Econom.
**155**, 188–194 (2010) CrossRefGoogle Scholar - Weißbach, R.: A general kernel functional estimator with general bandwidth—strong consistency and applications. J. Nonparametr. Stat.
**18**, 1–12 (2006) MathSciNetMATHCrossRefGoogle Scholar - Wied, D., Weißbach, R.: Consistency of the kernel density estimator—a survey. Stat. Pap.
**53**, 1–21 (2012) MATHCrossRefGoogle Scholar - Woodroofe, M.: Estimating a distribution function with truncated data. Ann. Stat.
**13**, 163–177 (1985) MathSciNetMATHCrossRefGoogle Scholar

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