Skip to main content
SpringerLink
Log in

Asymptotic normality of conditional density estimation with left-truncated and dependent data

  • Regular Article
  • Published:
Statistical Papers Aims and scope Submit manuscript

Abstract

Based on the idea of the local polynomial smoother, we construct the Nadaraya–Watson type and local linear estimators of conditional density function for a left-truncation model. Asymptotic normality of the estimators is established under the lifetime observations are assumed to be a sequence of stationary \(\alpha \)-mixing random variables. Finite sample behavior of the estimators is investigated via simulations too.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  • Aït-Sahalia Y (1999) Transition densities for interest rate and other non-linear diffusions. J Financ 54:1361–1395

    Article  Google Scholar 

  • Cai T, Wei LJ, Wilcox M (2000) Semiparametric regression analysis for clusetered survival data. Biometrika 87:867–878

    Article  MathSciNet  MATH  Google Scholar 

  • De Gooijer JG, Zerom D (2003) On conditional density estimation. Stat Neerl 57:159–176

    Article  MATH  Google Scholar 

  • Doukhan P (1994) Mixing: properties and examples. Lecture notes in statistics, vol 85. Springer, Berlin

  • Emura T, Konno Y (2012) Multivariate normal distribution approaches for dependently truncated data. Stat Pap 53(1):133–149

    Article  MathSciNet  MATH  Google Scholar 

  • Fan J, Yim TH (2004) A crossvalidation method for estimating conditional densities. Biometrika 91:819–834

    Article  MathSciNet  MATH  Google Scholar 

  • Feigelson ED, Babu GJ (1992) Statistical challenges in modern astronomy. Springer, Berlin

    Book  Google Scholar 

  • Hall P, Heyde CC (1980) Martingale limit theory and its application. Academic Press, New York

    MATH  Google Scholar 

  • Hall P, Racine J, Li Q (2004) Cross-validation and the estimation of conditional probability densities. J Am Stat Assoc 99:1015–1026

    Article  MathSciNet  MATH  Google Scholar 

  • Hosseinioun N, Doosti H, Nirumand HA (2012) Nonparametric estimation of the derivatives of a density by the method of wavelet for mixing sequences. Stat Pap 53(1):195–203

    Article  MathSciNet  MATH  Google Scholar 

  • Kang SS, Koehler KJ (1997) Modification of the Greenwood formula for correlated response times. Biometrics 53:885–899

    Article  MATH  Google Scholar 

  • Liang HY, de Uña-Álvarez J, Iglesias-Pérez MC (2011) Local polynomial estimation of a conditional mean function with dependent truncated data. Test 20(3):653–677

    Article  MathSciNet  MATH  Google Scholar 

  • Liebscher E (1996) Strong convergence of sums of \(\alpha \)-mixing random variables with applications to density estimation. Stoch Process Appl 65(1):69–80

    Article  MathSciNet  MATH  Google Scholar 

  • Liebscher E (2001) Estimation of the density and the regression function under mixing conditions. Stat Decis 19(1):9–26

    MathSciNet  MATH  Google Scholar 

  • Liu W, Lu X (2011) Empirical likelihood for density-weighted average derivatives. Stat Pap 52(2):391–412

    Article  MATH  Google Scholar 

  • Lynden-Bell D (1971) A method of allowing for known observational selection in small samples applied to 3CR quasars. Mon Not R Astron Soc 155:95–118

    Article  Google Scholar 

  • Ould-Saïd E, Tatachak A (2007) Asymptotic properties of the kernel estimator of the conditional mode for the left truncated model. C R Acad Sci Paris Ser I 344:651–656

    Article  MATH  Google Scholar 

  • Ould-Saïd E, Yahia D, Necir A (2009) A strong uniform convergence rate of a kernel conditional quantile estimator under random left-truncation and dependent data. Electron J Stat 3:426–445

    Article  MathSciNet  MATH  Google Scholar 

  • Volkonskii VA, Rozanov YA (1959) Some limit theorems for random functions. Theory Probab Appl 4:178–197

    Article  MathSciNet  Google Scholar 

  • Wang MC, Jemell NP, Tsai WY (1986) Asymptotic properties of the product-limit estimate under random truncation. Ann Stat 14:1597–1605

    Article  MATH  Google Scholar 

  • Wied D, Wei\(\beta \)bach R (2012) Consistency of the kernel density estimator: a survey. Stat Pap 53(1):1–21

  • Withers CS (1981) Conditions for linear processes to be strong mixing. Z Wahrsch Verw Geb 57:477–480

    Article  MathSciNet  MATH  Google Scholar 

  • Woodroofe M (1985) Estimating a distribution function with truncated data. Ann Stat 13:163–177

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The research of Han-Ying Liang was partially supported by the National Natural Science Foundation of China (11271286) and the Specialized Research Fund for the Doctor Program of Higher Education of China (20120072110007), the research of Jong-IL Baek was partially supported by a Wonkwang University Grant in 2014.

Author information

Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China

    Han-Ying Liang

  2. School of Mathematics & Informational Statistics, and Institute of Basic Natural Science, Wonkwang University, Ik-San, 570-749, South Korea

    Jong-Il Baek

Authors
  1. Han-Ying Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jong-Il Baek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jong-Il Baek.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liang, HY., Baek, JI. Asymptotic normality of conditional density estimation with left-truncated and dependent data. Stat Papers 57, 1–20 (2016). https://doi.org/10.1007/s00362-014-0635-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00362-014-0635-1

Keywords

Mathematics Subject Classification

Search

Navigation