Electronic Journal of Statistics

The Signed-rank estimator for nonlinear regression with responses missing at random

Huybrechts F. Bindele

Full-text: Open access

Abstract

This paper is concerned with the study of the signed-rank estimator of the regression coefficients under the assumption that some responses are missing at random in the regression model. Strong consistency and asymptotic normality of the proposed estimator are established under mild conditions. To demonstrate the performance of the signed-rank estimator, a simulation study is conducted under different settings of model error’s distributions, and shows that the proposed estimator is more efficient than the least squares estimator whenever the error distribution is heavy-tailed or contaminated. When the model error follows a normal distribution, the simulation experiment shows that the signed-rank estimator is more efficient than its least squares counterpart whenever a large proportion of the responses are missing.

Article information

Source
Electron. J. Statist., Volume 9, Number 1 (2015), 1424-1448.

Dates
Received: June 2014
First available in Project Euclid: 29 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1435584427

Digital Object Identifier
doi:10.1214/15-EJS1042

Mathematical Reviews number (MathSciNet)
MR3366482

Zentralblatt MATH identifier
1327.62391

Subjects
Primary: 62J02: General nonlinear regression 62G05: Estimation
Secondary: 62F12: Asymptotic properties of estimators 62G20: Asymptotic properties

Keywords
Signed-rank norm strong consistency asymptotic normality imputation missing at random

Citation

Bindele, Huybrechts F. The Signed-rank estimator for nonlinear regression with responses missing at random. Electron. J. Statist. 9 (2015), no. 1, 1424--1448. doi:10.1214/15-EJS1042. https://projecteuclid.org/euclid.ejs/1435584427


Export citation

References

  • [1] Abebe, A., McKean, J. W. and Bindele, H. F. (2012). On the consistency of a class of nonlinear regression estimators., PJSOR 8 543–555.
  • [2] Bindele, H. F. (2014). Asymptotics of the signed-rank estimator under dependent observations., Journal of Statistical Planning and Inference 146 49–55.
  • [3] Bindele, H. F. and Abebe, A. (2012). Bounded influence nonlinear signed-rank regression., Canadian Journal of Statistics 40 172–189.
  • [4] Brunner, E. and Denker, M. (1994). Rank statistics under dependent observations and applications to factorial designs., Journal of Statistical Planning and Inference 42 353–378.
  • [5] Chen, J. and Shao, J. (2001). Jackknife variance estimation for nearest-neighbor imputation., Journal of the American Statistical Association 96 260–269.
  • [6] Chen, S. X. and Hall, P. (1993). Smoothed Empirical Likelihood Confidence Intervals for Quantiles., The Annals of Statistics 21 pp. 1166–1181.
  • [7] Cheng, P. E. (1994). Nonparametric Estimation of Mean Functionals with Data Missing at Random., Journal of the American Statistical Association 89 pp. 81–87.
  • [8] Ciuperca, G. (2011). Empirical likelihood for nonlinear models with missing responses., Journal of Statistical Computation and Simulation 0 1–20.
  • [9] Culp, M. and Michailidis, G. (2008). An iterative algorithm for extending learners to a semi-supervised setting., Journal of Computational and Graphical Statistics 17 545–571.
  • [10] Delecroix, M., Hristache, M. and Patilea, V. (2006). On semiparametric-estimation in single-index regression., Journal of Statistical Planning and Inference 136 730–769.
  • [11] Deville, J.-C. and Särndal, C.-E. (1994). Variance estimation for the regression imputed Horvitz-Thompson estimator., Journal of Official Statistics-Stockholm- 10 381–381.
  • [12] Einmahl, U. and Mason, D. M. (2005). Uniform in Bandwidth Consistency of Kernel-Type Function Estimators., The Annals of Statistics 33 pp. 1380–1403.
  • [13] Hall, P. and La Scala, B. (1990). Methodology and Algorithms of Empirical Likelihood., International Statistical Review 58 109–127.
  • [14] Haupt, H. and Oberhofer, W. (2009). On asymptotic normality in nonlinear regression., Statist. Probab. Lett. 79 848–849.
  • [15] Healy, M. and Westmacott, M. (1956). Missing Values in Experiments Analysed on Automatic Computers., Journal of the Royal Statistical Society. Series C (Applied Statistics) 5 pp. 203–206.
  • [16] Hettmansperger, T. P. and McKean, J. W. (1998)., Robust nonparametric statistical methods. Kendall’s Library of Statistics 5. Edward Arnold, London.
  • [17] Jureckova, J. (1971). Nonparametric Estimate of Regression Coefficients., The Annals of Mathematical Statistics 42 pp. 1328–1338.
  • [18] Jurečková, J. (2008). Regression rank scores in nonlinear models. In, Beyond parametrics in interdisciplinary research: Festschrift in honor of Professor Pranab K. Sen. Inst. Math. Stat. Collect. 1 173–183. Inst. Math. Statist., Beachwood, OH.
  • [19] Kalton, G. (1983). Compensating for missing survey, data.
  • [20] Kim, J. K. and Shao, J. (2014)., Statistical methods for handling incomplete data. Chapman & Hall/CRC, Boca Raton, FL.
  • [21] Kitamura, Y. (1997). Empirical Likelihood Methods with Weakly Dependent Processes., The Annals of Statistics 25 pp. 2084–2102.
  • [22] Lessler, J. T. and Kalsbeek, W. D. (1992)., Nonsampling error in surveys. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Inc., New York A Wiley-Interscience Publication.
  • [23] Little, R. J. A. and Rubin, D. B. (1987)., Statistical analysis with missing data. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons Inc., New York.
  • [24] Little, R. J. A. and Rubin, D. B. (2002)., Statistical analysis with missing data, second ed. Wiley Series in Probability and Statistics. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ.
  • [25] Mukherjee, K. (1999). Asymptotics of Quantiles and Rank Scores in Nonlinear Time Series., Journal of Time Series Analysis 20 173–192.
  • [26] Müller, U. U. (2009). Estimating linear functionals in nonlinear regression with responses missing at random., Ann. Statist. 37 2245–2277.
  • [27] Nordholt, E. S. (1998). Imputation: methods, simulation experiments and practical examples., International Statistical Review 66 157–180.
  • [28] Oberhofer, W. (1982). The Consistency of Nonlinear Regression Minimizing the L1-Norm., The Annals of Statistics 10 pp. 316–319.
  • [29] Owen, A. (1990). Empirical Likelihood Ratio Confidence Regions., The Annals of Statistics 18 pp. 90–120.
  • [30] Peng, L. (2004). Empirical-Likelihood-Based Confidence Interval for the Mean with a Heavy-Tailed Distribution., The Annals of Statistics 32 pp. 1192–1214.
  • [31] Qin, Y., Li, L. and Lei, Q. (2009). Empirical likelihood for linear regression models with missing responses., Statistics & Probability Letters 79 1391–1396.
  • [32] Rao, J. N. and Sitter, R. (1995). Variance estimation under two-phase sampling with application to imputation for missing data., Biometrika 82 453–460.
  • [33] Rao, J. N. K. and Shao, J. (1992). Jackknife variance estimation with survey data under hot deck imputation., Biometrika 79 811–822.
  • [34] Robins, J. M., Rotnitzky, A. and Zhao, L. P. (1994). Estimation of Regression Coefficients When Some Regressors Are Not Always Observed., Journal of the American Statistical Association 89 pp. 846–866.
  • [35] Rubin, D. B. (1976). Inference and missing data., Biometrika 63 581–592.
  • [36] Rubin, D. B. (2004)., Multiple imputation for nonresponse in surveys. Wiley Classics Library. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ Reprint of the 1987 edition [John Wiley & Sons, Inc., New York; MR899519].
  • [37] Schafer, J. L. (1997)., Analysis of incomplete multivariate data. Monographs on Statistics and Applied Probability 72. Chapman & Hall, London.
  • [38] Schafer, J. L. and Graham, J. W. (2002). Missing data: our view of the state of the art., Psychological methods 7 147.
  • [39] Shao, J. and Sitter, R. R. (1996). Bootstrap for imputed survey data., Journal of the American Statistical Association 91 1278–1288.
  • [40] Shao, J. and Steel, P. (1999). Variance estimation for survey data with composite imputation and nonnegligible sampling fractions., Journal of the American Statistical Association 94 254–265.
  • [41] Skinner, C. J. and Rao, J. (2002). Jackknife variance estimation for multivariate statistics under hot-deck imputation from common donors., Journal of Statistical Planning and Inference 102 149–167.
  • [42] Sun, Z., Wang, Q. and Dai, P. (2009). Model checking for partially linear models with missing responses at random., J. Multivar. Anal. 100 636–651.
  • [43] Van Zwet, W. R. (1980). A Strong Law for Linear Functions of Order Statistics., The Annals of Probability 8 pp. 986–990.
  • [44] Wang, C. Y., Wang, S., Zhao, L.-P. and Ou, S.-T. (1997). Weighted Semiparametric Estimation in Regression Analysis With Missing Covariate Data., Journal of the American Statistical Association 92 pp. 512–525.
  • [45] Wang, Q., Linton, O. and Härdle, W. (2004). Semiparametric Regression Analysis with Missing Response at Random., Journal of the American Statistical Association 99 pp. 334–345.
  • [46] Wang, Q. and Rao, J. N. K. (2002). Empirical Likelihood-Based Inference under Imputation for Missing Response Data., The Annals of Statistics 30 pp. 896–924.
  • [47] Wang, Q. and Sun, Z. (2007). Estimation in partially linear models with missing responses at random., Journal of Multivariate Analysis 98 1470–1493.
  • [48] Wu, C.-F. (1981). Asymptotic theory of nonlinear least squares estimation., Ann. Statist. 9 501–513.
  • [49] Xue, L. (2009). Empirical Likelihood Confidence Intervals for Response Mean with Data Missing at Random., Scandinavian Journal of Statistics 36 671–685.
  • [50] Xue, L. and Zhu, L. (2007). Empirical Likelihood Semiparametric Regression Analysis for Longitudinal Data., Biometrika 94 921–937.
  • [51] Xue, L. and Zhu, L. (2007). Empirical Likelihood for a Varying Coefficient Model With Longitudinal Data., Journal of the American Statistical Association 102 642–654.
  • [52] Xue, L.-G. and Zhu, L. (2006). Empirical likelihood for single-index models., Journal of Multivariate Analysis 97 1295–1312.
  • [53] Yung, W. and Rao, J. (2000). Jackknife variance estimation under imputation for estimators using poststratification information., Journal of the American Statistical Association 95 903–915.
  • [54] Zhao, L. P., Lipsitz, S. and Lew, D. (1996). Regression Analysis with Missing Covariate Data Using Estimating Equations., Biometrics 52 pp. 1165–1182.