×

zbMATH — the first resource for mathematics

A revisit to correlation analysis for distortion measurement error data. (English) Zbl 1278.62065
Summary: We consider the estimation problem of a correlation coefficient between unobserved variables of interest. These unobservable variables are distorted in a multiplicative fashion by an observed confounding variable. Two estimators, the moment-based estimator and the direct plug-in estimator, are proposed, and we show their asymptotic normality. Moreover, the direct plug-in estimator is shown to be asymptotically efficient. Furthermore, we suggest a bootstrap procedure and an empirical likelihood-based statistic to construct the confidence intervals. The empirical likelihood statistic is shown to be asymptotically chi-squared. Simulation studies are conducted to examine the performance of the proposed estimators. These methods are applied to analyze the Boston housing price data as an illustration.

MSC:
62G20 Asymptotic properties of nonparametric inference
62F12 Asymptotic properties of parametric estimators
62F40 Bootstrap, jackknife and other resampling methods
62G08 Nonparametric regression and quantile regression
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Carroll, R. J.; Delaigle, A.; Hall, P., Testing and estimating shape-constrained nonparametric density and regression in the presence of measurement error, J. Amer. Statist. Assoc., 106, 191-202, (2011) · Zbl 1396.62084
[2] Carroll, R. J.; Fan, J.; Gijbels, I.; Wand, M. P., Generalized partially linear single-index models, J. Amer. Statist. Assoc., 92, 477-489, (1997) · Zbl 0890.62053
[3] Carroll, R. J.; Ruppert, D.; Crainiceanu, C. M.; Tosteson, T. D.; Karagas, M. R., Nonlinear and nonparametric regression and instrumental variables, J. Amer. Statist. Assoc., 99, 736-750, (2004) · Zbl 1117.62306
[4] Carroll, R. J.; Ruppert, D.; Stefanski, L. A.; Crainiceanu, C. M., Nonlinear measurement error models, A modern perspective, (2006), Chapman and Hall New York · Zbl 1119.62063
[5] Cui, X.; Guo, W.; Lin, L.; Zhu, L., Covariate-adjusted nonlinear regression, Ann. Statist., 37, 1839-1870, (2009) · Zbl 1168.62035
[6] Delaigle, A.; Fan, J.; Carroll, R. J., A design-adaptive local polynomial estimator for the errors-in-variables problem, J. Amer. Statist. Assoc., 104, 348-359, (2009) · Zbl 1388.62106
[7] Escanciano, J. C., A consistent diagnostic test for regression models using projections, Econom. Theory, 22, 1030-1051, (2006) · Zbl 1170.62318
[8] Ferguson, T. S., (A Course in Large Sample Theory, Texts in Statistical Science Series, (1996), Chapman & Hall London)
[9] Fuller, W. A., (Measurement Error Models, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, (1987), John Wiley & Sons Inc. New York)
[10] Harrison, D. J.; Rubinfeld, D. L., Hedonic housing prices and demand for Clean air, J. Environ. Econ. Manag., 5, 81-102, (1978) · Zbl 0375.90023
[11] Kaysen, G. A.; Dubin, J. A.; Müller, H.-G.; Mitch, W. E.; Rosales, L. M.; Levin the Hemo Study Group, Nathan W., Relationships among inflammation nutrition and physiologic mechanisms establishing albumin levels in hemodialysis patients, Kidney Int., 61, 2240-2249, (2002)
[12] Li, T.; Hsiao, C., Robust estimation of generalized linear models with measurement errors, J. Econometrics, 118, 51-65, (2004), Contributions to econometrics, time-series analysis, and systems identification: a Festschrift in honor of Manfred Deistler · Zbl 1033.62068
[13] Li, F.; Lin, L.; Cui, X., Covariate-adjusted partially linear regression models, Comm. Statist. Theory Methods, 39, 1054-1074, (2010) · Zbl 1284.62422
[14] Li, G.; Lin, L.; Zhu, L., Empirical likelihood for a varying coefficient partially linear model with diverging number of parameters, J. Multivariate Anal., 105, 85-111, (2012) · Zbl 1236.62020
[15] Lian, H., Empirical likelihood confidence intervals for nonparametric functional data analysis, J. Statist. Plann. Inference, 142, 1669-1677, (2012) · Zbl 1238.62057
[16] Liang, H.; Härdle, W.; Carroll, R. J., Estimation in a semiparametric partially linear errors-in-variables model, Ann. Statist., 27, 1519-1535, (1999) · Zbl 0977.62036
[17] Liang, H.; Li, R., Variable selection for partially linear models with measurement errors, J. Amer. Statist. Assoc., 104, 234-248, (2009) · Zbl 1388.62208
[18] Liang, H.; Qin, Y.; Zhang, X.; Ruppert, D., Empirical likelihood-based inferences for generalized partially linear models, Scandinavian J. Statist., 36, 433-443, (2009) · Zbl 1197.62092
[19] Liang, H.; Ren, H., Generalized partially linear measurement error models, J. Comput. Graph. Statist., 14, 237-250, (2005)
[20] Liang, H.; Wang, N., Partially linear single-index measurement error models, Statist. Sinica, 15, 99-116, (2005) · Zbl 1061.62098
[21] Nguyen, D. V.; Şentürk, D., Multicovariate-adjusted regression models, J. Stat. Comput. Simul., 78, 813-827, (2008) · Zbl 1431.62167
[22] Nguyen, D. V.; Şentürk, D.; Carroll, R. J., Covariate-adjusted linear mixed effects model with an application to longitudinal data, J. Nonparametr. Stat., 20, 459-481, (2008) · Zbl 1145.62032
[23] Owen, A. B., Empirical likelihood, (2001), Chapman and Hall/CRC London · Zbl 0989.62019
[24] Qin, J.; Lawless, J., Empirical likelihood and general estimating equations, Ann. Statist., 22, 300-325, (1994) · Zbl 0799.62049
[25] Schafer, D. W., Semiparametric maximum likelihood for measurement error model regression, Biometrics, 57, 53-61, (2001) · Zbl 1209.62051
[26] Schennach, S. M., Estimation of nonlinear models with measurement error, Econometrica, 72, 33-75, (2004) · Zbl 1151.91726
[27] Schennach, S. M., Instrumental variable estimation of nonlinear errors-in-variables models, Econometrica, 75, 201-239, (2007) · Zbl 1201.62052
[28] Şentürk, D.; Müller, H.-G., Covariate adjusted correlation analysis via varying coefficient models, Scandinavian J. Statist., 32, 365-383, (2005) · Zbl 1089.62068
[29] Şentürk, D.; Müller, H.-G., Covariate-adjusted regression, Biometrika, 92, 75-89, (2005) · Zbl 1068.62082
[30] Şentürk, D.; Müller, H.-G., Inference for covariate adjusted regression via varying coefficient models, Ann. Statist., 34, 654-679, (2006) · Zbl 1095.62045
[31] Şentürk, D.; Müller, H.-G., Covariate-adjusted generalized linear models, Biometrika, 96, 357-370, (2009) · Zbl 1163.62053
[32] Silverman, B. W., (Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability, (1986), Chapman & Hall London) · Zbl 0617.62042
[33] Stute, W.; González Manteiga, W.; Presedo Quindimil, M., Bootstrap approximations in model checks for regression, J. Amer. Statist. Assoc., 93, 141-149, (1998) · Zbl 0902.62027
[34] Tang, N.; Zhao, P., Empirical likelihood-based inference in nonlinear regression models with missing responses at random, Statistics, 47, 6, 1141-1159, (2012) · Zbl 1440.62262
[35] Tang, N.; Zhao, P., Empirical likelihood semiparametric nonlinear regression analysis for longitudinal data with responses missing at random, Ann. Inst. Statist. Math., 65, 4, 639-665, (2013) · Zbl 1273.62091
[36] Taupin, M.-L., Semi-parametric estimation in the nonlinear structural errors-in-variables model, Ann. Statist., 29, 66-93, (2001) · Zbl 1029.62039
[37] Wang, L.; Hsiao, C., Method of moments estimation and identifiability of semiparametric nonlinear errors-in-variables models, J. Econometrics, 165, 30-44, (2011) · Zbl 1441.62899
[38] Wang, X.; Li, G.; Lin, L., Empirical likelihood inference for semi-parametric varying-coefficient partially linear EV models, Metrika, 73, 171-185, (2011) · Zbl 1206.62080
[39] Wei, Z.; Zhu, L., Evaluation of value at risk: an empirical likelihood approach, Statist. Sinica, 20, 455-468, (2010) · Zbl 1180.62153
[40] Wu, C.-F. J., Jackknife, bootstrap and other resampling methods in regression analysis, Ann. Statist., 14, 1261-1350, (1986) · Zbl 0618.62072
[41] Zhang, J.; Feng, S.; Li, G.; Lian, H., Empirical likelihood inference for partially linear panel data models with fixed effects, Econom. Lett., 113, 165-167, (2011) · Zbl 1238.62049
[42] Zhang, J.; Gai, Y.; Wu, P., Estimation in linear regression models with measurement errors subject to single-indexed distortion, Comput. Statist. Data Anal., 59, 103-120, (2013)
[43] Zhang, J.; Yu, Y.; Zhu, L.; Liang, H., Partial linear single index models with distortion measurement errors, Ann. Inst. Statist. Math., 65, 2, 237-267, (2013) · Zbl 1440.62141
[44] Zhang, J.; Zhu, L.; Liang, H., Nonlinear models with measurement errors subject to single-indexed distortion, J. Multivariate Anal., 112, 1-23, (2012) · Zbl 1274.62304
[45] Zhang, J.; Zhu, L.; Zhu, L., On a dimension reduction regression with covariate adjustment, J. Multivariate Anal., 104, 39-55, (2012) · Zbl 1231.62076
[46] Zhou, Y.; Liang, H., Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates, Ann. Statist., 37, 427-458, (2009) · Zbl 1156.62036
[47] Zhu, L.; Lin, L.; Cui, X.; Li, G., Bias-corrected empirical likelihood in a multi-link semiparametric model, J. Multivariate Anal., 101, 850-868, (2010) · Zbl 1181.62039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.