The Annals of Statistics

The strong law under random truncation

Shuyuan He and Grace L. Yang

Full-text: Open access

Abstract

The random truncation model is defined by the conditional probability distribution $H (x, y) =P[X\leq x,Y\leq y |X \geq Y] where $X$ and $Y$ are independent random variables. A problem of interest is the estimation of the distribution function $F$ of $X$ with data from the distribution $H$. Under random truncation, $F$ need not be fully identifiable from $H$ and only a part of it, say $F_0$ , is. We show that the nonparametric MLE $F_n$ of $F_0$ obeys the strong law of large numbers in the sense that for any nonnegative, measurable function $\phi(x)$, the integrals $\int\phi(x)dF_n(x)\to\int\phi(x)dF_0(x)$ almost surely as $n$ tends to infinity. Similar results were first obtained by Stute and Wang for the right censoring model. The results are useful in establishing the strong consistency of various estimates. Some of our results are derived from the weak consistency of $F_n$ obtained by Woodroofe.

Article information

Source
Ann. Statist. Volume 26, Number 3 (1998), 992-1010.

Dates
First available in Project Euclid: 21 June 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1024691085

Digital Object Identifier
doi:10.1214/aos/1024691085

Mathematical Reviews number (MathSciNet)
MR1635430

Zentralblatt MATH identifier
0929.62037

Subjects
Primary: 62G20

Keywords
Random truncation nonparametric estimation strong law of large numbers reverse supermartingale product limit

Citation

He, Shuyuan; Yang, Grace L. The strong law under random truncation. Ann. Statist. 26 (1998), no. 3, 992--1010. doi:10.1214/aos/1024691085. http://projecteuclid.org/euclid.aos/1024691085.


Export citation

References

  • Chao, M. T. and Lo, S.-H. (1988). Some representations of the nonparametric maximum likelihood estimators with truncated data. Ann. Statist. 16 661-668.
  • Chen, K., Chao, M. T. and Lo, S.-H. (1995). On strong uniform consistency of the Ly nden-Ball estimator for truncated data. Ann. Statist. 23 440-449.
  • Feigelson, E. D. and Babu, G. J., eds. (1992). Statistical Challenges in Modern Astronomy. Springer, New York.
  • He, S. and Yang, G. L. (1994). Estimating a lifetime distribution under different sampling plans. In Statistical Decision Theory and Related Topics (S. S. Gupta and J. O. Berger, eds.) 5 73-85. Springer, New York.
  • He, S. and Yang, G. L. (1998). Estimation of the truncation probability in the random truncation model. Ann. Statist. 26 1011-1027.
  • Major, P. and Rejt ¨o, L. (1988). Strong embedding of the estimator of the distribution function under random censorship. Ann. Statist. 16 1113-1132.
  • Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.
  • Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
  • Stute, W. and Wang, J. L. (1993). The strong law under random censorship. Ann. Statist. 21 1591-1607.
  • Tsai, W.-Y., Jewell, N. P. and Wang, M.-C. (1987). A note on the product-limit estimator under right censoring and left truncation. Biometrika 74 883-886.
  • Wang, M.-C., Jewell, N. P. and Tsai, W.-Y. (1986). Asy mptotic properties of the product limit estimate under random truncation. Ann. Statist. 14 1597-1605.
  • Woodroofe, M. (1985). Estimating a distribution function with truncated data. Ann. Statist. 13 163-177.