Electronic Journal of Statistics

Kernel density estimation with doubly truncated data

Carla Moreira and Jacobo de Uña-Álvarez

Full-text: Open access

Abstract

In some applications with astronomical and survival data, doubly truncated data are sometimes encountered. In this work we introduce kernel-type density estimation for a random variable which is sampled under random double truncation. Two different estimators are considered. As usual, the estimators are defined as a convolution between a kernel function and an estimator of the cumulative distribution function, which may be the NPMLE [2] or a semiparametric estimator [9]. Asymptotic properties of the introduced estimators are explored. Their finite sample behaviour is investigated through simulations. Real data illustration is included.

Article information

Source
Electron. J. Statist., Volume 6 (2012), 501-521.

Dates
First available in Project Euclid: 30 March 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1333113100

Digital Object Identifier
doi:10.1214/12-EJS683

Mathematical Reviews number (MathSciNet)
MR2988417

Zentralblatt MATH identifier
1274.62271

Subjects
Primary: 62G07: Density estimation
Secondary: 62N02: Estimation

Keywords
Biased sampling double truncation semiparametric estimator smoothing methods

Citation

Moreira, Carla; de Uña-Álvarez, Jacobo. Kernel density estimation with doubly truncated data. Electron. J. Statist. 6 (2012), 501--521. doi:10.1214/12-EJS683. https://projecteuclid.org/euclid.ejs/1333113100


Export citation

References

  • [1] Devroye, L.P. and Wagner, T.J. (1979). The, L1 convergence of kernel density estimates. The Annals of Statistics 7 1136–1139.
  • [2] Efron, B. and Petrosian, V. (1999). Nonparametric Methods for Doubly Truncated Data., Journal of the American Statistical Association 76 824–834.
  • [3] Lynden-Bell, D. (1971). A method for Allowing for Known Observational Selection in Small Samples Applied to 3CR Quasars., Monthly Notices of the Royal Astronomical Society 155 95–118.
  • [4] Marron, J.S. and de Uña-Álvarez, J. (2004). SiZer for length-biased, censored density and hazard estimation., Journal of Statistical Planning and Inference 121 149–161.
  • [5] Martin, E.C. and Betensky, R. A. (2005). Testing quasi-independence of failure and truncation times via conditional Kendall’s tau., Journal of the American Statistical Association 100 484–492.
  • [6] Wand, M. P. and Jones, M. C. (1985)., Kernel Smoothing. Monographs on Statistics and Applied Probability. Wiley, London.
  • [7] Moreira, C. (2010) “The Statistical Analysis of Doubly Truncated Data: new Methods, Software Development, and Biomedical Applications, Ph.D. dissertation, Universidade de Vigo, Spain.
  • [8] Moreira, C. and de Uña-Álvarez, J. (2010). Bootstrappping the NPMLE for Doubly Truncated Data., Journal of Nonparametric Statistics 22 567–583.
  • [9] Moreira, C. and de Uña-Álvarez, J. (2010). A semiparametric estimator of survival for doubly truncated data., Statistics in Medicine 29 3147–3159.
  • [10] Moreira, C. and de Uña-Álvarez, J. and Crujeiras, R.C. (2010). DTDA: an R package to analyze randomly truncated data., Journal of Statistical Software 37 1–20.
  • [11] Parzen, E. (1962). On estimation of a probability density function and mode., Annals of Mathematical Statistics 33 1065–1076.
  • [12] Shen, P.S. (2010). Nonparametric Analysis of Doubly Truncated Data., Annals of the Institute of Statistical Mathematics 62 835–853.
  • [13] Shen, P.S. (2010). Semiparametric Analysis of Doubly Truncated Data., Communications in Statistics – Theory and Methods 39 3178–3190.
  • [14] Stute, W. (1993). Almost Sure Representations of the Product-Limit Estimator for Truncated Data., The Annals of Statistics 21 146–156.
  • [15] Tsai, W.Y. and 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.
  • [16] Turnbull, B. W. (1976). The Empirical Distribution Function with Arbitrarily Grouped, Censored and Truncated Data., Journal of the Royal Statistical Society. Series B. Methodological 38 290–295.
  • [6] Wand, M. P. and Jones, M. C. (1985)., Kernel Smoothing. Monographs on Statistics and Applied Probability. Wiley, London.
  • [18] Wang, M.-C. (1991). Nonparametric Estimation from Cross-Sectional Survival Data., Journal of the American Statistical Association 86 130–143.
  • [19] Woodroofe, M. (1985). Estimating a Distribution Function with Truncated Data., The Annals of Statistics 13 163–177.
  • [20] Zhou, Y. and Yip, P. S. F. (1985). A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data., Journal of Multivariate Analysis 69 261–280.