Abstract
The missing response problem in single-index models is studied, and a bias-correction method to infer the index coefficients is developed. Two weighted empirical log-likelihood ratios with asymptotic chisquare are derived, and the corresponding empirical likelihood confidence regions for the index coefficients are constructed. In addition, the estimators of the index coefficients and the link function are defined, and their asymptotic normalities are proved. A simulation study is conducted to compare the empirical likelihood and the normal approximation based method in terms of coverage probabilities and average lengths of confidence intervals. A real example illustrates our methods.
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References
Arnold S F. The Theory of Linear Models and Multivariate Analysis. New York: John Wiley & Sons, 1981
Chang Z, Xue L G, Zhu L X. On an asymptotically more efficient estimation of the single-index model. J Mult Anal, 2010, 101: 1898–1901
Cheng P E. Nonparametric estimation of mean functionals with data missing at random. J Amer Statist Assoc, 1994, 89: 81–87
Delecroix M, Hristache M, Patilea V. On semiparametric M-estimation in single-index regression. J Statist Plan Infer, 2006, 136: 730–769
Fan J, Gijbels I. Local Polynomial Modeling and Its Applications. London: Chapman and Hall, 1996
Härdle W, Hall P, Ichimura, H. Optimal smoothing in single-index models. Ann Statist, 1993, 21: 157–178
Hjort N L, McKeague I W, Van Keilegom I. Extending the scope of empirical likelihood. Ann Statist, 2009, 37: 1079–1111
Ichimura H. Estimation of single index models. PhD thesis. Department of Economics, MIT, 1987
Lai P, Wang Q H. Partially linear single-index model with missing responses at random. J Statist Plan Infer, 2011, 141: 1047–1058
Lee A J, Scott A J. Ultrasound in ante-natal diagnosis. In: Brook R J, Arnold G C, Hassard T H, et al., eds. The Fascination of Statistics. New York: Marcel Dekker, 1986, 277–293
Li K C. Sliced inverse regression for dimension reduction (with discussion). J Amer Statist Assoc, 1993, 86: 316–342
Li W K, Tong H, Xia Y, et al. A goodness-of-fit test for singleindex models. Statist Sinica, 2004, 14: 1–28
Masry E, Tjøstheim D. Nonparametric estimation and identification of nonlinear ARCH time series: Strong convergence and asymptotic normality. Econometric Theory, 1995, 11: 258–289
Müller U U, Schick A, Wefelmeyer W. Imputing responses that are not missing. In: Nikulin M, Commenges D, Huber C, eds. Probability, Statistics and Modelling in Public Health. New York: Springer, 2006, 350–363
Owen A B. Empirical likelihood ratio confidence intervals for a single function. Biometrika, 1988, 75: 237–249
Owen A B. Empirical likelihood ratio confidence regions. Ann Statist, 1990, 18: 90–120
Peng H, Schick A. An empirical likelihood approach to goodness of fit testing. Bernoulli, 2013, 19: 954–981
Qin J, Zhang B. Empirical-likelihood-based inference in missing response problems and its application in observational studies. J Roy Statist Soc Ser B, 2007, 69: 101–122
Seber G A F, Wild C J. Nonlinear Regression. New York: John Wiley & Sons, 1989, 103–110
Serfling R J. Approximation Theorems of Mathematical Statistics. New York: John Wiley & Sons, 1980
Simonoff J S, Tsai C L. Score tests for the single index model. J Smer Statist Assoc, 2002, 44: 142–151
Stute W, Xue L G, Zhu L X. Empirical likelihood inference in nonlinear error in covariables models with validation data. J Amer Statist Assoc, 2007, 102: 332–346
Stute W, Zhu L X. Nonparametric checks for single-index models. Ann Statist, 2005, 33: 1048–1083
Wang J L, Xue L G, Zhu L X, et al. Estimation for a partial-linear single-index model. Ann Statist, 2010, 38: 246–274
Wang Q H, Rao J N K. Empirical likelihood-based inference under imputation for missing response data. Ann Statist, 2002, 30: 896–924
Wang Y H, Shen J S, He S Y, et al. Estimation of single index model with missing response at random. J Statist Plan Infer, 2010, 140: 1671–1690
Xia Y, Tong H, Li W K, et al. An adaptive estimation of dimension reduction space. J Roy Statist Soc Ser B, 2002, 64: 363–410
Xue L G. Empirical likelihood for linear models with missing responses. J Mult Anal, 2009, 100: 1353–1366
Xue L G. Empirical likelihood confidence intervals for response mean with data missing at random. Scandinavian J Statist, 2009, 36: 671–685
Xue L G. Empirical likelihood local polynomial regression analysis of clustered data. Scandinavian J Statist, 2010, 37: 644–663
Xue L G. Estimation and empirical likelihood for single-index models with missing data in the covariates. Comput Statist Data Anal, 2013, 60: 82–97
Xue L G, Xue D. Empirical likelihood for semiparametric regression model with missing response data. J Mult Anal, 2011, 102: 723–740
Xue L G, Zhu L X. Empirical likelihood for single-index model. J Mult Anal, 2006, 97: 1295–1312
Xue L G, Zhu L X. Empirical Likelihood for a varying coefficient model with longitudinal data. J Amer Statist Assoc, 2007, 102: 642–654
Xue L G, Zhu L X. Empirical likelihood semiparametric regression analysis for longitudinal data. Biometrika, 2007, 94: 921–937
Yin X, Cook D R. Direction estimation in single-index regressions. Biometrika, 2005, 92: 371–384
Zhu L X, Xue L G. Empirical likelihood confidence regions in a partially linear single-index model. J Roy Statist Soc Ser B, 2006, 68: 549–570
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Xue, L., Lian, H. Empirical likelihood for single-index models with responses missing at random. Sci. China Math. 59, 1187–1207 (2016). https://doi.org/10.1007/s11425-015-5097-y
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DOI: https://doi.org/10.1007/s11425-015-5097-y