×

Goodness-of-fit testing of error distribution in linear measurement error models. (English) Zbl 1408.62085

Summary: This paper investigates a class of goodness-of-fit tests for fitting an error density in linear regression models with measurement error in covariates. Each test statistic is the integrated square difference between the deconvolution kernel density estimator of the regression model error density and a smoothed version of the null error density, an analog of the so-called P. J. Bickel and M. Rosenblatt test statistic [Ann. Stat. 1, 1071–1095 (1973; Zbl 0275.62033)]. The asymptotic null distributions of the proposed test statistics are derived for both the ordinary smooth and super smooth cases. The asymptotic power behavior of the proposed tests against a fixed alternative and a class of local nonparametric alternatives for both cases is also described. The finite sample performance of the proposed test is evaluated by a simulation study. The simulation study shows some superiority of the proposed test over some other tests. Finally, a real data is used to illustrate the proposed test.

MSC:

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 0275.62033
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] Bachmann, D. and Dette, H. (2005). A note on the Bickel–Rosenblatt test in autoregressive time series. Statist. Probab. Lett.74 221–234. · Zbl 1070.62067
[2] Battese, G. E., Fuller, W. and Hickman, R. D. (1976). Estimation of response variances from interview-reinterview surveys. J. Indian Soc. Agricultural Statist.28 1–14.
[3] Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist.1 1071–1095. · Zbl 0275.62033
[4] Butucea, C. (2004). Asymptotic normality of the integrated square error of a density estimator in the convolution model. SORT28 9–25. · Zbl 1274.62229
[5] Carroll, R. J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc.83 1184–1186. · Zbl 0673.62033
[6] Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective, 2nd ed. Monographs on Statistics and Applied Probability105. Chapman & Hall/CRC, Boca Raton, FL. · Zbl 1119.62063
[7] Cheng, C.-L. and Van Ness, J. W. (1999). Statistical Regression with Measurement Error. Kendall’s Library of Statistics6. Arnold, London.
[8] Delaigle, A. and Hall, P. (2006). On optimal kernel choice for deconvolution. Statist. Probab. Lett.76 1594–1602. · Zbl 1099.62035
[9] Fan, J. (1991). Asymptotic normality for deconvolution kernel density estimators. Sankhyā Ser. A53 97–110. · Zbl 0729.62034
[10] Fan, J. (1992). Deconvolution with supersmooth distributions. Canad. J. Statist.20 155–169. · Zbl 0754.62020
[11] Fuller, W. A. (1987). Measurement Error Models. Wiley, New York. · Zbl 0800.62413
[12] Gao, J. and Gijbels, I. (2008). Bandwidth selection in nonparametric kernel testing. J. Amer. Statist. Assoc.103 1584–1594. · Zbl 1286.62043
[13] Holzmann, H., Bissantz, N. and Munk, A. (2007). Density testing in a contaminated sample. J. Multivariate Anal.98 57–75. · Zbl 1102.62045
[14] Holzmann, H. and Boysen, L. (2006). Integrated square error asymptotics for supersmooth deconvolution. Scand. J. Stat.33 849–860. · Zbl 1164.62335
[15] Hong, Y. and Lee, Y.-J. (2013). A loss function approach to model specification testing and its relative efficiency. Ann. Statist.41 1166–1203. · Zbl 1293.62100
[16] Hušková, M. and Meintanis, S. G. (2007). Omnibus tests for the error distribution in the linear regression model. Statistics41 363–376. · Zbl 1126.62059
[17] Khmaladze, E. V. and Koul, H. L. (2004). Martingale transforms goodness-of-fit tests in regression models. Ann. Statist.32 995–1034. · Zbl 1092.62052
[18] Khmaladze, E. V. and Koul, H. L. (2009). Goodness-of-fit problem for errors in nonparametric regression: Distribution free approach. Ann. Statist.37 3165–3185. · Zbl 1369.62073
[19] Koul, H. L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models. Lecture Notes in Statistics166. Springer, New York. Second edition of Weighted Empiricals and Linear Models [Inst. Math. Statist., Hayward, CA, 1992; MR1218395]. · Zbl 1007.62047
[20] Koul, H. L. and Mimoto, N. (2012). A goodness-of-fit test for GARCH innovation density. Metrika75 127–149. · Zbl 1241.62067
[21] Koul, H. L. and Song, W. (2012). A class of goodness-of-fit tests in linear errors-in-variables model. J. SFdS153 52–70. · Zbl 1316.62052
[22] Koul, H. L., Song, W. and Zhu, X. (2018). Supplement to “Goodness-of-fit testing of error distribution in linear measurement error models.” DOI:10.1214/17-AOS1627SUPP.
[23] Laurent, B., Loubes, J.-M. and Marteau, C. (2011). Testing inverse problems: A direct or an indirect problem? J. Statist. Plann. Inference141 1849–1861. · Zbl 1394.62052
[24] Lee, S. and Na, S. (2002). On the Bickel–Rosenblatt test for first-order autoregressive models. Statist. Probab. Lett.56 23–35. · Zbl 0994.62082
[25] Loubes, J. M. and Marteau, C. (2014). Goodness-of-fit testing strategies from indirect observations. J. Nonparametr. Stat.26 85–99. · Zbl 1359.62147
[26] Loynes, R. M. (1980). The empirical d.f. of residuals from generalized regression. Ann. Statist.8 285–298. · Zbl 0451.62040
[27] Schennach, S. M. and Hu, Y. (2013). Nonparametric identification and semiparametric estimation of classical measurement error models without side information. J. Amer. Statist. Assoc.108 177–186. · Zbl 06158334
[28] Shao, J. and Tu, D. (1995). The Jackknife and Bootstrap. Springer, New York. · Zbl 0947.62501
[29] Stefanski, L. A. (1990). Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett.9 229–235. · Zbl 0686.62026
[30] Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics21 169–184. · Zbl 0697.62035
[31] Van Es, A. and Uh, H.-W. (2004). Asymptotic normality of nonparametric kernel type deconvolution density estimators: Crossing the Cauchy boundary. J. Nonparametr. Stat.16 261–277. · Zbl 1216.62074
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.