Estimation of a change point in the variance function based on the χ²-distribution

References

• Bhattacharya, P.K., Brockwell, P.J. (1976). The minimum of an additive process with applications to signal estimation and storage theory. Z. Wahrscheinlichkeitstheorie Verw. Geb. 37:51–75.
   Web of Science ®Google Scholar
• Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
   Google Scholar
• Carroll, R.J., Fan, J., Gijbels, I., Wand, M.P. (1997). Generalized partially linear single-index models. J. Am. Stat. Assoc. 92:477–489.
   Web of Science ®Google Scholar
• Fan, J., Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. London: Chapman and Halls.
   Google Scholar
• Fan, J., Heckman, N.E., Wand, M.P. (1995). Local polynomial kernel regression for generalized linear models and quasi-likelihood functions. J. Am. Stat. Assoc. 90:141–150.
   Web of Science ®Google Scholar
• Gasser, T., Sroka, L., Jennen-Steinmetz, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73:625–634.
   Web of Science ®Google Scholar
• Gijbels, I., Lambert, A., Qiu, P. (2007). Jump-preserving regression and smoothing using local linear fitting: a compromise. Ann. Inst. Stat. Math. 59:235–272.
   Web of Science ®Google Scholar
• Grégoire, G., Hamrouni, Z. (2002). Change point estimation by local linear smoothing. J. Multivar. Anal. 83:56–83.
   Web of Science ®Google Scholar
• Hall, P., Carroll, R.J. (1989). Variance function estimation in regression: the effect of estimating the mean. J. Roy. Stat. Soc. Ser. B 51:3–14.
   Google Scholar
• Hall, P., Kay, J.W., Titterington, D.M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77:521–528.
   Web of Science ®Google Scholar
• Hall, P., Titterington, D.M. (1992). Edge-preserving and peak-preserving smoothing. Technometrics 34:429–440.
   Web of Science ®Google Scholar
• Huh, J. (2010). Detection of a change point based on local-likelihood. J. Multivar. Anal. 101:1681–1700.
   Web of Science ®Google Scholar
• Huh, J. (2012). Nonparametric estimation of the regression function having a change point in generalized linear models. Stat. Lett. 82:843–851.
   Web of Science ®Google Scholar
• Huh, J., Carrière, K.C. (2002). Estimation of regression functions with a discontinuity in a derivative with local polynomial fits. Stat. Probab. Lett. 56:329–343.
   Web of Science ®Google Scholar
• Huh, J., Park, B.U. (2004). Detection of change point with local polynomial fits for random design case. Aust. N. Z. J. Stat. 46:425–441.
   Web of Science ®Google Scholar
• Joo, J., Qiu, P. (2009). Jump detection in a regression curve and its derivative. Technometrics 51:289–305.
   Web of Science ®Google Scholar
• Jose, C.T., Ismail, B. (1999). Change points in nonparametric regression functions. Commun. Stat.-Theory Meth. 28:1883–1902.
   Web of Science ®Google Scholar
• Kang, K.H., Huh, J. (2006). Nonparametric estimation of the variance function with a change point. J. Kor. Stat. Soc. 35:1–24.
   Google Scholar
• Loader, C.R. (1996). Change point estimation using nonparametric regression. Ann. Stat. 24:1667–1678.
   Web of Science ®Google Scholar
• Mack, Y.P., Silverman, B.W. (1982). Weak and strong uniform consistency of kernel regression estimates. Z. Wahrscheinlichkeitstheorie verw. Gebiete 61:405–415.
   Web of Science ®Google Scholar
• Müller, H.G. (1992). Change-points in nonparametric regression analysis. Ann. Stat. 20:737–761.
   Web of Science ®Google Scholar
• Müller, H.G., Stadtmüller, U. (1987). Estimation of heteroscedasticity in regression analysis. Ann. Stat. 15:610–625.
   Web of Science ®Google Scholar
• Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Economet. Theory 7:186–199.
   Web of Science ®Google Scholar
• Qiu, P. (1994). Estimation of the number of jumps of the jump regression functions. Commun. Stat. Theory Meth. 23:2141–2155.
   Web of Science ®Google Scholar
• Qiu, P. (2003). A jump-preserving curve fitting procedure based on local piecewise-linear kernel estimation. J. Nonpara. Stat. 15:437–453.
   Web of Science ®Google Scholar
• Qiu, P., Asano, C., Li, X. (1991). Estimation of jump regression functions. Bull. Inf. Cybern. 24:197–212.
   Google Scholar
• Qiu, P., Yandell, B. (1998). A local polynomial jump-detection algorithm in nonparametric regression. Technometrics 40:142–152.
   Google Scholar
• Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Stat. 12:1215–1230.
   Web of Science ®Google Scholar
• Ruppert, D., Wand, M.P., Holst, U., Hössjer, O. (1997). Local polynomial variance-function estimation. Technometrics 39:262–273.
   Web of Science ®Google Scholar
• Wu, J.S., Chu, C.K. (1993). Kernel-type estimators of jump points and values of a regression function. Ann. Stat. 21:1545–1566.
   Web of Science ®Google Scholar
• Yin, Y.Q. (1988). Detection of the number, locations and magnitudes of jumps. Commun. Stat. Stoch. Models 4:445–455.
   Google Scholar
• Yu, K., Jones, M.C. (2004). Likelihood-based local linear estimation of the conditional variance function. J. Am. Stat. Assoc. 99:139–144.
   Web of Science ®Google Scholar
• Zhang, B., Su, Z., Qiu, P. (2009). On jump detection in regression curves using local polynomial kernel estimation. Pak. J. Stat. 25:505–528.
   Web of Science ®Google Scholar