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Non‐parametric confidence bands in deconvolution density estimation

Author

Listed:
  • Nicolai Bissantz
  • Lutz Dümbgen
  • Hajo Holzmann
  • Axel Munk

Abstract

Summary. Uniform confidence bands for densities f via non‐parametric kernel estimates were first constructed by Bickel and Rosenblatt. In this paper this is extended to confidence bands in the deconvolution problem g=f*ψ for an ordinary smooth error density ψ. Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (L∞‐distance) between a deconvolution kernel estimator and f. Further consistency of the simple non‐parametric bootstrap is proved. For our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth. For practical purposes we propose a new data‐driven bandwidth selector that is based on heuristic arguments, which aims at minimizing the L∞‐distance between and f. Although not constructed explicitly to undersmooth the estimator, a simulation study reveals that the bandwidth selector suggested performs well in finite samples, in terms of both area and coverage probability of the resulting confidence bands. Finally the methodology is applied to measurements of the metallicity of local F and G dwarf stars. Our results confirm the ‘G dwarf problem’, i.e. the lack of metal poor G dwarfs relative to predictions from ‘closed box models’ of stellar formation.

Suggested Citation

  • Nicolai Bissantz & Lutz Dümbgen & Hajo Holzmann & Axel Munk, 2007. "Non‐parametric confidence bands in deconvolution density estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(3), pages 483-506, June.
  • Handle: RePEc:bla:jorssb:v:69:y:2007:i:3:p:483-506
    DOI: 10.1111/j.1467-9868.2007.599.x
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    Cited by:

    1. Adusumilli, Karun & Kurisu, Daisuke & Otsu, Taisuke & Whang, Yoon-Jae, 2020. "Inference on distribution functions under measurement error," Journal of Econometrics, Elsevier, vol. 215(1), pages 131-164.
    2. repec:jss:jstsof:39:i10 is not listed on IDEAS
    3. Dong, Hao & Otsu, Taisuke & Taylor, Luke, 2021. "Average Derivative Estimation Under Measurement Error," Econometric Theory, Cambridge University Press, vol. 37(5), pages 1004-1033, October.
    4. Joel L. Horowitz, 2013. "Ill-posed inverse problems in economics," CeMMAP working papers CWP37/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. Kato, Kengo & Sasaki, Yuya, 2019. "Uniform confidence bands for nonparametric errors-in-variables regression," Journal of Econometrics, Elsevier, vol. 213(2), pages 516-555.
    6. Dong, Hao & Taylor, Luke, 2022. "Nonparametric Significance Testing In Measurement Error Models," Econometric Theory, Cambridge University Press, vol. 38(3), pages 454-496, June.
    7. Joel L. Horowitz, 2013. "Ill-posed inverse problems in economics," CeMMAP working papers 37/13, Institute for Fiscal Studies.
    8. Hao Dong & Yuya Sasaki, 2022. "Estimation of average derivatives of latent regressors: with an application to inference on buffer-stock saving," Departmental Working Papers 2204, Southern Methodist University, Department of Economics.
    9. Ma, Jun & Marmer, Vadim & Yu, Zhengfei, 2023. "Inference on individual treatment effects in nonseparable triangular models," Journal of Econometrics, Elsevier, vol. 235(2), pages 2096-2124.
    10. Kato, Kengo & Kurisu, Daisuke, 2020. "Bootstrap confidence bands for spectral estimation of Lévy densities under high-frequency observations," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1159-1205.
    11. Kato, Kengo & Sasaki, Yuya, 2018. "Uniform confidence bands in deconvolution with unknown error distribution," Journal of Econometrics, Elsevier, vol. 207(1), pages 129-161.
    12. Birke, Melanie & Bissantz, Nicolai & Holzmann, Hajo, 2008. "Confidence bands for inverse regression models with application to gel electrophoresis," Technical Reports 2008,16, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    13. Huang, Danyang & Hu, Wei & Jing, Bingyi & Zhang, Bo, 2023. "Grouped spatial autoregressive model," Computational Statistics & Data Analysis, Elsevier, vol. 178(C).
    14. Kurisu, Daisuke & Otsu, Taisuke, 2022. "On linearization of nonparametric deconvolution estimators for repeated measurements model," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    15. van Es, Bert & Gugushvili, Shota, 2008. "Weak convergence of the supremum distance for supersmooth kernel deconvolution," Statistics & Probability Letters, Elsevier, vol. 78(17), pages 2932-2938, December.
    16. Hao Dong & Daniel L. Millimet, 2020. "Propensity Score Weighting with Mismeasured Covariates: An Application to Two Financial Literacy Interventions," JRFM, MDPI, vol. 13(11), pages 1-24, November.
    17. Adusumilli, Karun & Kurisu, Daisies & Otsu, Taisuke & Whang, Yoon-Jae, 2020. "Inference on distribution functions under measurement error," LSE Research Online Documents on Economics 102692, London School of Economics and Political Science, LSE Library.
    18. Birke, Melanie & Bissantz, Nicolai, 2007. "Shape constrained estimators in inverse regression models with convolution-type operator," Technical Reports 2007,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    19. Wang, Xiao-Feng & Fan, Zhaozhi & Wang, Bin, 2010. "Estimating smooth distribution function in the presence of heteroscedastic measurement errors," Computational Statistics & Data Analysis, Elsevier, vol. 54(1), pages 25-36, January.
    20. Kurisu, Daisuke & Otsu, Taisuke, 2022. "On linearization of nonparametric deconvolution estimators for repeated measurements model," LSE Research Online Documents on Economics 112676, London School of Economics and Political Science, LSE Library.

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