Skip to main content
SpringerLink
Log in

Density Estimation with Replicate Heteroscedastic Measurements

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

Abstract

We present a deconvolution estimator for the density function of a random variable from a set of independent replicate measurements. We assume that measurements are made with normally distributed errors having unknown and possibly heterogeneous variances. The estimator generalizes well-known deconvoluting kernel density estimators, with error variances estimated from the replicate observations. We derive expressions for the integrated mean squared error and examine its rate of convergence as n → ∞ and the number of replicates is fixed. We investigate the finite-sample performance of the estimator through a simulation study and an application to real data.

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

Similar content being viewed by others

References

  • Andrews G., Askey R., Roy R. (1999) Special functions. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Bailey W.N. (1928) Products of generalized hypergeometric series. Proceedings of the London Mathematical Society 2: 242–254

    Article  Google Scholar 

  • Carroll R.J., Hall P. (1988) Optimal rates of convergence for deconvolving a density. Journal of the American Statistical Association 83: 1184–1186

    Article  MATH  MathSciNet  Google Scholar 

  • Cook J.R., Stefanski L.A. (1994) Simulation–extrapolation estimation in parametric measurement error models. Journal of the American Statistical Association 89: 1314–1328

    Article  MATH  Google Scholar 

  • Delaigle A. (2008) An alternative view of the deconvolution problem. Statistica Sinica 18: 1025–1045

    MATH  MathSciNet  Google Scholar 

  • Delaigle A., Gijbels I. (2004) Bootstrap bandwidth selection in kernel density estimation from a contaminated sample. Annals of the Institute of Statistical Mathematics 56: 19–47

    Article  MATH  MathSciNet  Google Scholar 

  • Delaigle A., Hall P. (2008) Using SIMEX for smoothing-parameter choice in errors-in-variables problems. Journal of the American Statistical Association 103: 280–287

    Article  MATH  MathSciNet  Google Scholar 

  • Delaigle A., Meister A. (2007) Nonparametric regression estimation in the heteroscedastic errors-in-variables problem. Journal of the American Statistical Association 102: 1416–1426

    Article  MATH  MathSciNet  Google Scholar 

  • Delaigle A., Meister A. (2008) Density estimation with heteroscedastic error. Bernoulli 14: 562–569

    Article  MATH  MathSciNet  Google Scholar 

  • Delaigle A., Hall P., Müller H. (2007) Accelerated convergence for nonparametric regression with coarsened predictors. Annals of Statistics 35: 2639–2653

    Article  MATH  MathSciNet  Google Scholar 

  • Delaigle A., Hall P., Meister A. (2008) On Deconvolution with repeated measurements. Annals of Statistics 36: 665–685

    Article  MATH  MathSciNet  Google Scholar 

  • Devroye L. (1989) Consistent deconvolution in density estimation. The Canadian Journal of Statistics 17: 235–239

    Article  MATH  MathSciNet  Google Scholar 

  • Diggle P.J., Hall P. (1993) A Fourier approach to nonparametric deconvolution of a density estimate. Journal of the Royal Statistical Society, Series B 55: 523–531

    MATH  MathSciNet  Google Scholar 

  • Erdelyi A.E. (1953) Higher transcendental functions (Vol. 2). McGraw-Hill, New York

    Google Scholar 

  • Fan J. (1991a) Asymptotic normality for deconvolution kernel density estimators. Sankhya, Series A, Indian Journal of Statistics 53: 97–110

    MATH  Google Scholar 

  • Fan J. (1991b) On the optimal rates of convergence for nonparametric deconvolution problems. Annals of Statistics 19: 1257–1272

    Article  MATH  MathSciNet  Google Scholar 

  • Fan J. (1992) Deconvolution with supersmooth distributions. The Canadian Journal of Statistics 20: 155–169

    Article  MATH  Google Scholar 

  • Li T., Vuong Q. (1998) Nonparametric estimation of the measurement error model using multiple indicators. Journal of Multivariate Analysis 65: 139–165

    Article  MATH  MathSciNet  Google Scholar 

  • Marron J.S. (1992) Bootstrap bandwidth selection. In: LePage R., Billard L. (eds) Exploring the limits of bootstrap. Wiley, New York

    Google Scholar 

  • McIntyre, J. (2003). Density deconvolution with replicate measurements and auxiliary data. Ph.D. Thesis, North Carolina State University.

  • Meister A. (2006) Density estimation with normal measurement error with unknown variance. Statistica Sinica 16: 195–211

    MATH  MathSciNet  Google Scholar 

  • Neumann M.H. (2007) Deconvolution from panel data with unknown error distribution. Journal of Multivariate Analysis 98: 1955–1968

    Article  MATH  MathSciNet  Google Scholar 

  • Patil P. (1996) A note on deconvolution density estimation. Statistics & Probability Letters 29: 79–84

    Article  MATH  MathSciNet  Google Scholar 

  • Silverman B.W. (1986) Density estimation for statistics and data analysis. Chapman & Hall, New York

    MATH  Google Scholar 

  • Staudenmayer J., Ruppert D., Buonaccorsi J.P. (2008) Density estimation in the presence of heteroskedastic measurement error. Journal of the American Statistical Association 103: 726–736

    Article  MATH  MathSciNet  Google Scholar 

  • Stefanski L.A. (1989) Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models. Communications in Statistics, Series A 18: 4335–4558

    Article  MATH  MathSciNet  Google Scholar 

  • Stefanski L.A. (1990) Rates of convergence of some estimators in a class of deconvolution problems. Statistics & Probability Letters 9: 229–235

    Article  MATH  MathSciNet  Google Scholar 

  • Stefanski L.A., Carroll R.J. (1990) Deconvoluting kernel density estimators. Statistics 21: 169–184

    Article  MATH  MathSciNet  Google Scholar 

  • Stefanski L.A., Cook J.R. (1995) Simulation–extrapolation: the measurement error jackknife. Journal of the American Statistical Association 90: 1247–1256

    Article  MATH  MathSciNet  Google Scholar 

  • Stefanski L.A., Novick S.J., Devanarayan V. (2005) Estimating a nonlinear function of a normal mean. Biometrika 92: 732–736

    Article  MATH  MathSciNet  Google Scholar 

  • Wand M.P. (1998) Finite sample performance of deconvolving density estimators. Statistics & Probability Letters 37: 131–139

    Article  MATH  MathSciNet  Google Scholar 

  • Watson G.S. (1983) Statistics on spheres. Wiley, New York

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, 99775, USA

    Julie McIntyre

  2. Department of Statistics, North Carolina State University, Raleigh, NC, 27695-8203, USA

    Leonard A. Stefanski

Authors
  1. Julie McIntyre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonard A. Stefanski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julie McIntyre.

About this article

Cite this article

McIntyre, J., Stefanski, L.A. Density Estimation with Replicate Heteroscedastic Measurements. Ann Inst Stat Math 63, 81–99 (2011). https://doi.org/10.1007/s10463-009-0220-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-009-0220-x

Keywords

Search

Navigation