Abstract
In this paper, we study the linear quantile regression model when response data are missing at random. Based on the inverse probability weight method, we establish an estimation equation on quantile regression and define standard quantile regression estimator of unknown parameter. At the same time, we construct the empirical likelihood (EL) ratio function for the unknown parameter, and define the maximum EL estimator of the unknown parameter. Under suitable assumptions, we investigate the asymptotic normality of the proposed estimators and prove the EL ratio statistics has a standard chi-squared limiting distribution. A simulation study is done to investigate the finite sample performance of the estimators and compare the difference of the proposed EL method and bootstrap approach.
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References
Chen SX, Van Keilegom I (2009) A review on empirical likelihood methods for regression. Test 18:415–447
Chen SX, Van Keilegom I (2013) Estimation in semiparametric models with missing data. Ann Inst Stat Math 65:785–805
Chen X, Wan ATK, Zhou Y (2015) Efficient quantile regression analysis with missing observations. J Am Stat Assoc 110(510):723–741
Ciuperca G (2013) Empirical likelihood for nonlinear models with missing responses. J Stat Comput Simul 83:739–758
Fan GL, Liang HY, Wang JF (2013) Empirical likelihood for heteroscedastic partially linear errors-in-variables model with \(\alpha \)-mixing errors. Stat Pap 54:85–112
Healy M, Westmacott M (1956) Missing values in experiments analysis on automatic computers. Appl Stat 5:203–206
Karimi O, Mohammadzadeh M (2012) Bayesian spatial regression models with closed skew normal correlated errors and missing observations. Stat Pap 53:205–218
Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46:33–50
Koenker R (2005) Quantile regression. Econometric society monographs 38. Cambridge University Press, Cambridge
Koenker R, Zhao Q (1996) Conditional quantile estimation and inference for ARCH models. Econ Theory 12:793–813
Liang H, Wang S, Carroll RJ (2007) Partially linear models with missing response variables and error-prone covariates. Biometrika 94:185–198
Little R, Rubin DB (1987) Statistical analysis with missing data. Wiley, New York
Lv XF, Li R (2013) Smoothed empirical likelihood analysis of partially linear quantile regression models with missing response variables. AStA Adv Stat Anal 97:317–347
Müller UU, Van Keilegom I (2012) Efficient parameter estimation in regression with missing responses. Electron J Stat 6:1200–1219
Otsu T (2008) Conditional empirical likelihood estimation and inference for quantile regression models. J Econ 142:508–538
Owen AB (1988) Empirical likelihood ratio confidence intervals for a single function. Biometrika 75:237–249
Owen AB (1990) Empirical likelihood ratio confidence regions. Ann Stat 18:90–120
Owen AB (2001) Empirical likelihood. Chapman and Hall/CRC, New york
Qin GS, Tsao M (2003) Empirical likelihood inference for median regression models for censored survival data. J Multivar Anal 85:416–430
Robins JM, Rotnitzky A, Zhao LP (1994) Estimation of regression coefficients when some regressors are not always observed. J Am Stat Assoc 89:846–866
Ruppert D, Carroll RJ (1980) Trimmed least squares estimation in the linear model. J Am Stat Assoc 75:828–838
Sherwood B, Wang L, Zhou XH (2013) Weighted quantile regression for analyzing health care cost data with missing covariates. Stat Med 32:4967–4979
Tang CY, Leng CL (2012) An empirical likelihood approach to quantile regression with auxiliary information. Stat Probab Lett 82:29–36
Wang CY, Wang SJ, Zhao LP, Ou ST (1997) Weighted semiparametric estimation in regression analysis with missing covariate data. J Am Stat Assoc 92:512–525
Wang HJ, Wang L (2009) Locally weighted censored quantile regression. J Am Stat Assoc 104:1117–1128
Wang HJ, Zhu ZY (2011) Empirical likelihood for quantile regression models with longitudinal data. J Stat Plan Inference 141:1603–1615
Wang QH, Linton O, Härdle W (2004) Semiparametric regression analysis with missing response at random. J Am Stat Assoc 99:334–345
Wang QH, Sun ZH (2007) Estimation in partially linear models with missing responses at random. J Multivar Anal 98:1470–1493
Wei Y, Ma YY, Carroll RJ (2012) Multiple imputation in quantile regression. Biometrika 99:423–438
Whang YJ (2006) Smoothed Empirical likelihood methods for quantile regression models. Econ Theory 22:173–205
Wu T, Li G, Tang C (2015) Empirical likelihood for censored linear regression and variable selection. Scand J Stat 42(3):798–812
Yi G, He WQ (2009) Median regression models for longitudinal data with dropouts. Biometrics 65:618–625
Zhang T, Zhu Z (2011) Empirical likelihood inference for longitudinal data with missing response variables and error-prone covariates. Commun Stat Theory Methods 40:3230–3244
Zhao W, Zhang R, Liu Y, Liu J (2015) Empirical likelihood based modal regression. Stat Pap 56:411–430
Zheng J, Shen J, He S (2014) Adjusted empirical likelihood for right censored lifetime data. Stat Pap 55:827–839
Zhou Y, Wan A, Wang X (2008) Estimating equations inference with missing data. J Am Stat Assoc 103:1187–1199
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Authors were supported by the National Natural Science Foundation of China (11271286) and the Specialized Research Fund for the Doctor Program of Higher Education of China (20120072110007).
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Shen, Y., Liang, HY. Quantile regression and its empirical likelihood with missing response at random. Stat Papers 59, 685–707 (2018). https://doi.org/10.1007/s00362-016-0784-5
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DOI: https://doi.org/10.1007/s00362-016-0784-5