Abstract
The main objective of this work is the nonparametric estimation of the regression function with correlated errors when observations are missing in the response variable. Two nonparametric estimators of the regression function are proposed. The asymptotic properties of these estimators are studied; expresions for the bias and the variance are obtained and the joint asymptotic normality is established. A simulation study is also included.
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References
Aerts M., Claeskens G., Hens N., Molenberghs G. (2002). Local multile imputation. Biometrika, 89(2): 375–388
Beveridge S. (1992). Least squares estimation of missing values in time series. Communications in Statistics. Theory and Methods, 21(12): 3479–3496
Brockwell P. J., Davis R. A. (1991). Time series. Theory and methods, 2 edn. Heidelberg, Springer
Chen J. H., Shao J. (2000). Nearest neighbor imputation for survey data. Journal of Official Statistics, 16, 113–131
Cheng P.E. (1994). Nonparametric estimation of mean functionals with data missing at random. Journal of the American Statistical Association, 89(425): 81–87
Chow G. C., Lin A. L. (1976). Best linear unbiased estimation of missing observations in an economic time series. Journal of the American Statistical Association, 71(355): 719–721
Chu C. K., Cheng P. E. (1995). Nonparametric regression estimation with missing data. Journal of Statistical Planning and Inference, 48, 85–99
Fan J., Gijbels I. (1996). Local polynomial modelling and its applications. London, Chapman and Hall
Francisco-Fernández M., Vilar-Fernández J. M. (2001). Local polynomial regression estimation with correlated errors. Communications in Statistics. Theory and Methods, 30(7): 1271–1293
Glasser M. (1964). Linear regression analysis with missing observations among the independent variables. Journal of the American Statistical Association, 59, 834–844
González-Manteiga W., Pérez-González A. (2004). Nonparametric mean estimation with missing data. Communications in Statistics. Theory and Methods, 33(2): 277–303
Hall P., Van Keilegom I. (2003). Using differences based methods for inference in nonparametric regression with time-series errors. Journal of the Royal Statistical Society: Series B, 65, 443–456
Härdle W., Tsybakov A. (1997). Local polynomial estimators of the volatility function in nonparametric autoregression. Journal of Econometrics, 81(1): 223–242
Härdle W., Tsybakov A., Yang L. (1998). Nonparametric vector autoregression. Journal of Statistical Planning and Inference, 68(2): 221–245
Herrmann E., Gasser T., Kneip A. (1992). Choice of bandwidth for kernel regression when residuals are correlated. Biometrika, 79(4): 783–795
Ibrahim J. G. (1990). Incomplete data in generalized linear models. Journal of the American Statistical Association, 85(411): 765–769
Jones R. H. (1980). Maximum likelihood fitting of ARMA models to time series with missing observations. Technometrics, 22(3): 389–395
Little R. J. A. (1992). Regression with missing X’s: a review. Journal of the American Statistical Association, 87, 1227–1237
Little R.J.A., Rubin D.B. (1987). Statistical analysis with missing data. New York, Wiley
Masry E. (1996a). Multivariate local polynomial regression for time series: uniform strong consistency and rates. Journal of Time Series Analysis, 17(6): 571–599
Masry E. (1996b). Multivariate regression estimation—local polynomial fitting for time series. Stochastic Processes and their Applications, 65(1): 81–101
Masry E., Fan J. (1997). Local polynomial estimation of regression function for mixing processes. Scandinavian Journal of Statistics. Theory and Applications, 24, 165–179
Meng X. L. (2000). Missing data: Dial M for ???. Journal of the American Statistical Association, 95(452): 1325–1330
Müller H., Stadtmüller U. (1988). Detecting dependencies in smooth regression models. Biometrika, 75(4): 639–650
Nieuwenhuis G. (1992). Central limits theorems for sequences with m(n)-dependent main part. Journal of Statistical Planning and Inference, 32, 229–241
Peña D., Tiao G. C. (1991). A note on likelihood estimation of missing values in time series. The American Statistician 45(3): 212–213
Robinson P. M. (1984). Kernel estimation and interpolation for time series containing missing observations. Annals of the Institute of Statistical Mathematics, 36(1): 403–417
Rubin, D. B. (1987). Multiple imputation for nonresponse in surveys. In Wiley series in probability and mathematical statistics: applied probability and statistics. New York: Wiley.
Vilar J., Vilar J. (1998). Recursive estimation of regression function by local polynomial fitting. Annals of the Institute of Statistical Mathematics, 50(4): 729–754
Wang Q., Linton O., Härdle W. (2004). Semiparametric regression analysis with missing response at random. Journal of the American Statistical Association, 99(466): 334–345
Wang W., Rao J. N. K. (2002). Empirical likelihood-based inference under imputation for missing response data. Annals of Statistics, 30(3): 896–924
Yates F. (1933). The analysis of replicated experiments when the field results are incomplete. Empire Journal of Experimental Agriculture, 1, 129–142
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Pérez-González, A., Vilar-Fernández, J.M. & González-Manteiga, W. Asymptotic properties of local polynomial regression with missing data and correlated errors. Ann Inst Stat Math 61, 85–109 (2009). https://doi.org/10.1007/s10463-007-0136-2
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DOI: https://doi.org/10.1007/s10463-007-0136-2