The Annals of Statistics

On deconvolution with repeated measurements

Aurore Delaigle, Peter Hall, and Alexander Meister

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Abstract

In a large class of statistical inverse problems it is necessary to suppose that the transformation that is inverted is known. Although, in many applications, it is unrealistic to make this assumption, the problem is often insoluble without it. However, if additional data are available, then it is possible to estimate consistently the unknown error density. Data are seldom available directly on the transformation, but repeated, or replicated, measurements increasingly are becoming available. Such data consist of “intrinsic” values that are measured several times, with errors that are generally independent. Working in this setting we treat the nonparametric deconvolution problems of density estimation with observation errors, and regression with errors in variables. We show that, even if the number of repeated measurements is quite small, it is possible for modified kernel estimators to achieve the same level of performance they would if the error distribution were known. Indeed, density and regression estimators can be constructed from replicated data so that they have the same first-order properties as conventional estimators in the known-error case, without any replication, but with sample size equal to the sum of the numbers of replicates. Practical methods for constructing estimators with these properties are suggested, involving empirical rules for smoothing-parameter choice.

Article information

Source
Ann. Statist. Volume 36, Number 2 (2008), 665-685.

Dates
First available in Project Euclid: 13 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1205420515

Digital Object Identifier
doi:10.1214/009053607000000884

Mathematical Reviews number (MathSciNet)
MR2396811

Zentralblatt MATH identifier
1133.62026

Subjects
Primary: 62G07: Density estimation 62G08: Nonparametric regression
Secondary: 65R32: Inverse problems

Keywords
Bandwidth choice density estimation errors in variables Fourier inversion kernel methods nonparametric regression rates of convergence ridge parameter replication statistical smoothing

Citation

Delaigle, Aurore; Hall, Peter; Meister, Alexander. On deconvolution with repeated measurements. Ann. Statist. 36 (2008), no. 2, 665--685. doi:10.1214/009053607000000884. https://projecteuclid.org/euclid.aos/1205420515.


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References

  • Andersen, C. M., Bro, R. and Brockhoff, P. B. (2003). Effect of sampling errors on predictions using replicated measurements. J. Chemometrics 17 1–9.
  • Barry, J. and Diggle, P. (1995). Choosing the smoothing parameter in a Fourier approach to nonparametric deconvolution of a density function. J. Nonparametr. Statist. 4 223–232.
  • Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • Biemer, P., Groves, R., Lyberg, L., Mathiowetz, N. and Sudman, S., eds. (1991). Measurement Errors in Surveys. Wiley, New York.
  • Bland, J. M and Altman, D. G. (1986). Statistical methods for assessing agreement between two methods of clinical measurement. Lancet i 307–310.
  • Carroll, R. J., Eltinge, J. L. and Ruppert, D. (1993). Robust linear regression in replicated measurement error models. Statist. Probab. Lett. 19 169–175.
  • Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
  • Carroll, R. J., Ruppert, D. and Stefanski, L. A. (1995). Measurement Error in Nonlinear Models. Chapman and Hall, London.
  • Delaigle, A. and Gijbels, I. (2002), Estimation of integrated squared density derivatives from a contaminated sample. J. Roy. Statist. Soc. Ser. B 64 869–886.
  • Delaigle, A. and Gijbels, I. (2004). Practical bandwidth selection in deconvolution kernel density estimation. Comput. Statist. Data Anal. 45 249–267.
  • Delaigle, A. Hall, P. and Meister, A. (2006). On deconvolution with repeated measurements—long version. Available from the authors upon request.
  • Diggle, P. and Hall, P. (1993). A Fourier approach to nonparametric deconvolution of a density estimate. J. Roy. Statist. Soc. Ser. B 55 523–531.
  • Dunn, G. (1989). Design and Analysis of Reliability Studies. Arnold, London.
  • Dunn, G. (2004). Statistical Evaluation of Measurement Errors, Design and Analysis of Reliability Studies, 2nd ed. Arnold, London.
  • Eliasziw, M., Young, S. L., Woodbury, M. G. and Fryday-Field, K. (1994). Statistical methodology for the concurrent assessment of interrater and intrarater reliability: Using goniometric measurements as an example. Phys. Therapy 74 777–788.
  • Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • Fan, J. and Truong, Y. K. (1993). Nonparametric regression with errors in variables. Ann. Statist. 21 1900–1925.
  • Horowitz, J. L. and Markatou, M. (1996). Semiparametric estimation of regression models for panel data. Rev. Econom. Stud. 63 145–168.
  • Huwang, L. and Yang, J. (2000). Trimmed estimation in the measurement error model when the covariate has replicated observations. Proc. Nat. Sci. Council ROC(A) 24 405–412.
  • Jaech, J. (1985). Statistical Analysis of Measurement Errors. Wiley, New York.
  • Li, T. (2002). Robust and consistent estimation of nonlinear errors-in-variables, models. J. Econometrics 110 1–26.
  • Li, T. and Hsiao, C. (2004). Robust estimation of generalised linear models with measurement errors. J. Econometrics 118 51–65.
  • Li, T. and Vuong, Q. (1998). Nonparametric estimation of the measurement error model using multiple indicators. J. Multivar. Anal. 65 139–165.
  • Madansky, A. (1959). The fitting of straight lines when both variables are subject to error. J. Amer. Statist. Assoc. 54 173–205.
  • Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error. Ann. Statist. 20 712–736.
  • Neumann, M. H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Statist. 7 307–330.
  • Newey, W. K. and Powell, J. L. (2003). Instrumental variable estimation of nonparametric models. Econometrica 71 1565–1578.
  • Oman, S. D., Meir, N. and Haim, N. (1999). Comparing two measures of creatinine clearance: An application of errors-in-variables and bootstrap techniques. J. Roy. Statist. Soc. Ser. C 48 39–52.
  • Schennach, S. M. (2004a). Estimation of nonlinear models with measurement error. Econometrica 72 33–75.
  • Schennach, S. M. (2004b). Nonparametric regression in the presence of measurement error. Econometric Theory 20 1046–1093.
  • Stefanski, L. A. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169–184.
  • Susko, E. and Nadon, R. (2002). Estimation of a residual distribution with small numbers of repeated measurements. Canad. J. Statist. 30 383–400.
  • Turner, S. W., Toone, B. K. and Brett-Jones, J. R. (1986). Computerized tomographic scan changes in early schizophrenia—preliminary findings. Psychological Medicine 16 219–225.
  • Van Es, B. and Hu, H. W. (2005). Asymptotic normality of kernel-type deconvolution estimators. Scand. J. Statist. 32 467–483.