The Annals of Applied Statistics

Jump detection in generalized error-in-variables regression with an application to Australian health tax policies

Yicheng Kang, Xiaodong Gong, Jiti Gao, and Peihua Qiu

Full-text: Open access


Without measurement errors in predictors, discontinuity of a nonparametric regression function at unknown locations could be estimated using a number of existing approaches. However, it becomes a challenging problem when the predictors contain measurement errors. In this paper, an error-in-variables jump point estimator is suggested for a nonparametric generalized error-in-variables regression model. A major feature of our method is that it does not impose any parametric distribution on the measurement error. Its performance is evaluated by both numerical studies and theoretical justifications. The method is applied to studying the impact of Medicare Levy Surcharge on the private health insurance take-up rate in Australia.

Article information

Ann. Appl. Stat., Volume 9, Number 2 (2015), 883-900.

Received: October 2014
Revised: February 2015
First available in Project Euclid: 20 July 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Bandwidth selection demand for private health insurance exponential family generalized regression kernel smoothing measurement errors


Kang, Yicheng; Gong, Xiaodong; Gao, Jiti; Qiu, Peihua. Jump detection in generalized error-in-variables regression with an application to Australian health tax policies. Ann. Appl. Stat. 9 (2015), no. 2, 883--900. doi:10.1214/15-AOAS814.

Export citation


  • Buchmueller, T., Didardo, J. and Valletta, R. (2011). The effect of an employer health insurance mandate on health insurance coverage and the demand for labor: Evidence from Hawaii. American Economic Journal Economic Policy 3 25–51.
  • Butler, J. R. G. (2002). Policy change and private health insurance: Did the cheapest policy do the trick? Aust. Health Rev. 25 33–41.
  • Carroll, R. J., Maca, J. D. and Ruppert, D. (1999). Nonparametric regression in the presence of measurement error. Biometrika 86 541–554.
  • Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective, 2nd ed. Chapman & Hall, Boca Raton, FL.
  • Cheng, K. F. and Lin, P. E. (1981). Nonparametric estimation of a regression function. Z. Wahrsch. Verw. Gebiete 57 223–233.
  • Comte, F. and Taupin, M.-L. (2007). Adaptive estimation in a nonparametric regression model with errors-in-variables. Statist. Sinica 17 1065–1090.
  • Cook, J. R. and Stefanski, L. A. (1994). Simulation–extrapolation estimation in parametric measurement error models. J. Amer. Statist. Assoc. 89 1314–1328.
  • Delaigle, A. (2008). An alternative view of the deconvolution problem. Statist. Sinica 18 1025–1045.
  • Delaigle, A. and Meister, A. (2007). Nonparametric regression estimation in the heteroscedastic errors-in-variables problem. J. Amer. Statist. Assoc. 102 1416–1426.
  • Fan, J. and Masry, E. (1992). Multivariate regression estimation with errors-in-variables: Asymptotic normality for mixing processes. J. Multivariate Anal. 43 237–271.
  • Fan, J. and Truong, Y. K. (1993). Nonparametric regression with errors in variables. Ann. Statist. 21 1900–1925.
  • Finkelstein, A. (2002). The effect of tax subsidies to employer-provided supplementary health insurance: Evidence from Canada. J. Public Econ. 84 305–339.
  • Frech, H., Hopkins, S. and MacDonald, G. (2003). The Australian private health insurance boom: Was it subsidies or liberalized regulation? Oxf. Econ. Pap. 22 58–64.
  • Gijbels, I. and Goderniaux, A.-C. (2004). Bandwidth selection for changepoint estimation in nonparametric regression. Technometrics 46 76–86.
  • Gruber, J. and Poterba, D. (1994). Tax incentives and the decision to purchase health insurance: Evidence from the self-employed. Q. J. Econ. 123 831–862.
  • Hall, P. and Meister, A. (2007). A ridge-parameter approach to deconvolution. Ann. Statist. 35 1535–1558.
  • Joo, J.-H. and Qiu, P. (2009). Jump detection in a regression curve and its derivative. Technometrics 51 289–305.
  • Kang, Y., Gong, X., Gao, J. and Qiu, P. (2015). Supplement to “Jump detection in generalized error-in-variables regression with an application to Australian health tax policies.” DOI:10.1214/15-AOAS814SUPP.
  • Lee, D. and Lemieux, T. (2010). Regression discontinuity designs in economics. J. Econ. Lit. 48 281–355.
  • Liang, H. and Wang, N. (2005). Partially linear single-index measurement error models. Statist. Sinica 15 99–116.
  • Meister, A. (2009). Deconvolution Problems in Nonparametric Statistics. Springer, Berlin.
  • Mitrinović, D. S., Pečarić, J. E. and Fink, A. M. (1993). Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht.
  • Müller, H.-G. (1992). Change-points in nonparametric regression analysis. Ann. Statist. 20 737–761.
  • Müller, C. H. (2002). Robust estimators for estimating discontinuous functions. Metrika 55 99–109 (electronic).
  • Palangkaraya, A. and Yong, J. (2005). Effects of recent carrot-and-stick policy initiatives on private health insurance coverage in Australia. Econ. Rec. 81 262–272.
  • Palangkaraya, A., Yong, J., Webster, E. and Dawkins, P. (2009). The income distributive implications of recent private health insurance policy reforms in Australia. Eur. J. Health Econ. 10 135–148.
  • Qiu, P. H. (1991). Estimation of a kind of jump regression function. Systems Sci. Math. Sci. 4 1–13.
  • Qiu, P. H. (1994). Estimation of the number of jumps of the jump regression functions. Comm. Statist. Theory Methods 23 2141–2155.
  • Qiu, P. (2005). Image Processing and Jump Regression Analysis. Wiley, Hoboken, NJ.
  • Qiu, P. and Yandell, B. (1998). A local polynomial jump-detection algorithm in nonparametric regression. Technometrics 40 141–152.
  • Rodríguez, M. and Stoyanova, A. (2004). The effect of private insurance access on the choice of GP/specialist and public/private provider in Spain. Health Econ. 13 689–703.
  • Staudenmayer, J. and Ruppert, D. (2004). Local polynomial regression and simulation–extrapolation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 66 17–30.
  • Stefanski, L. A. (2000). Measurement error models. J. Amer. Statist. Assoc. 95 1353–1358.
  • Stefanski, L. A. and Cook, J. R. (1995). Simulation–extrapolation: The measurement error jackknife. J. Amer. Statist. Assoc. 90 1247–1256.
  • Taupin, M.-L. (2001). Semi-parametric estimation in the nonlinear structural errors-in-variables model. Ann. Statist. 29 66–93.
  • Wu, J. S. and Chu, C. K. (1993). Kernel-type estimators of jump points and values of a regression function. Ann. Statist. 21 1545–1566.

Supplemental materials

  • Supplement to “Jump detection in generalized error-in-variables regression with an application to Australian health tax policies”. This supplemental file mainly gives the proof of Theorem 1.