zbMATH — the first resource for mathematics

Improved density and distribution function estimation. (English) Zbl 1437.62134
The authors study the estimation of the density function or the distribution function of generalised residuals in semi-parametric models defined by a finite number of moment restrictions. The paper gives conditions for the consistency and derives the asymptotic mean squared error properties of the kernel density and distribution function estimators proposed in the paper. Simulation studies are presented to evaluate the small sample performance of these estimators.
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI Euclid
[1] Ahmad, I. A. (1992). Residuals density estimation in nonparametric regression., Statist. Probab. Lett.14(2), 133-139. · Zbl 0785.62035
[2] Antoine, B., Bonnal, H. and Renault, E. (2007). On the efficient use of the informational content of estimating equations: Implied probabilities and Euclidean empirical likelihood., J. Econometrics138(2), 461-487. · Zbl 1418.62101
[3] Back, K. and Brown, D. P. (1993). Implied probabilities in GMM estimators., Econometrica61(4), 971-975. · Zbl 0785.62031
[4] Bartlett, M. S. (1963). Statistical estimation of density functions., Sankhyā Ser. A25(3), 245-254. · Zbl 0129.32302
[5] Bhattacharya, R. N. and Ghosh, J. K. (1978). On the validity of the formal Edgeworth expansion., Ann. Statist.6(2), 434-451. · Zbl 0396.62010
[6] Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates., Ann. Statist.1(6), 1071-1095. · Zbl 0275.62033
[7] Bochner, S. (1955)., Harmonic analysis and the theory of probability. University of California Press. · Zbl 0068.11702
[8] Bott, A.-K., Devroye, L. and Kohler, M. (2013). Estimation of a distribution from data with small measurement errors., Electron. J. Stat.7, 2457-2476. · Zbl 1293.62068
[9] Brown, B. W. and Newey, W. K. (1998). Efficient semiparametric estimation of expectations., Econometrica66(2), 453-464. · Zbl 1008.62571
[10] Brown, B. W. and Newey, W. K. (2002). Generalized method of moments, efficient bootstrapping, and improved inference., J. Bus. Econom. Statist.20(4), 507-517.
[11] Brown, S., Greene, W. H., Harris, M. N. and Taylor, K. (2015). An inverse hyperbolic sine heteroskedastic latent class panel tobit model: An application to modelling charitable donations., Economic Modelling50, 228-236. doi: 10.1016/j.econmod.2015.06.018
[12] Burbidge, J. B., Magee, L. and Robb, A. L. (1988). Alternative transformations to handle extreme values of the dependent variable., J. Amer. Statist. Assoc.83(401), 123-127.
[13] Cao, R. and Lugosi, G. (2005). Goodness-of-fit tests based on the kernel density estimator., Scand. J. Statist.32(4), 599-616. · Zbl 1091.62031
[14] Chamberlain, G. (1987). Asymptotic efficiency in estimation with conditional moment restrictions., J. Econometrics34(3), 305-334. · Zbl 0618.62040
[15] Chen, J. and Qin, J. (1993). Empirical likelihood estimation for finite populations and the effective usage of auxiliary information., Biometrika80(1), 107-116. · Zbl 0769.62006
[16] Chen, S. X. (1997). Empirical likelihood-based kernel density estimation., Austral. J. Statist.39(1), 47-56. · Zbl 0877.62034
[17] Chen, S. X. and Cui, H. (2007). On the second-order properties of empirical likelihood with moment restrictions., J. Econometrics141(2), 492-516. · Zbl 1407.62157
[18] Cheng, F. (2004). Weak and strong uniform consistency of a kernel error density estimator in nonparametric regression., J. Statist. Plann. Inference119(1), 95-107. · Zbl 1031.62025
[19] Cheng, F. (2005). Asymptotic distributions of error density estimators in first-order autoregressive models., Sankhyā67(3), 553-567. · Zbl 1193.62017
[20] Chernozhukov, V., Fernández-Val, I. and Galichon, A. (2009). Improving point and interval estimators of monotone functions by rearrangement., Biometrika96(3), 559-575. · Zbl 1170.62025
[21] Corcoran, S. A. (1998). Bartlett adjustment of empirical discrepancy statistics., Biometrika85(4), 967-972. doi: 10.1093/biomet/85.4.967 · Zbl 1101.62330
[22] Cox, D. R. and Snell, E. J. (1968). A general definition of residuals., J. Roy. Statist. Soc. Ser. B30(2), 248-275. · Zbl 0164.48903
[23] Cressie, N. and Read, T. R. C. (1984). Multinomial goodness-of-fit tests., J. Roy. Statist. Soc. Ser. B46(3), 440-464. · Zbl 0571.62017
[24] Fan, Y. (1994). Testing the goodness of fit of a parametric density function by kernel method., Econometric Theory10(2), 316-356.
[25] Fan, Y. (1998). Goodness-of-fit tests based on kernel density estimators with fixed smoothing parameters., Econometric Theory14(5), 604-621.
[26] Glad, I. K., Hjort, N. L. and Ushakov, N. G. (2003). Correction of density estimators that are not densities., Scand. J. Statist.30(2), 415-427. · Zbl 1051.60037
[27] Györfi, L. and Walk, H. (2012). Strongly consistent density estimation of the regression residual., Statist. Probab. Lett.82(11), 1923-1929. · Zbl 1312.62047
[28] Hall, A. R. (2005)., Generalized method of moments. Oxford University Press. · Zbl 1076.62118
[29] Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators., Econometrica50(4), 1029-1054. · Zbl 0502.62098
[30] Hansen, L. P., Heaton, J. and Yaron, A. (1996). Finite-sample properties of some alternative GMM estimators., J. Bus. Econom. Statist.14(3), 262-280. doi: 10.2307/1392442
[31] Imbens, G. W., Spady, R. H. and Johnson, P. (1998). Information theoretic approaches to inference in moment condition models., Econometrica66(2), 333-357. · Zbl 1055.62512
[32] Jensen, J. L. (1989). Validity of the formal Edgeworth expansion when the underlying distribution is partly discrete., Probab. Theory Related Fields81(4), 507-519. · Zbl 0658.62017
[33] Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation., Biometrika36(1-2), 149-176. · Zbl 0033.07204
[34] Kitamura, Y. and Stutzer, M. (1997). An information-theoretic alternative to generalized method of moments estimation., Econometrica65(4), 861-874. · Zbl 0894.62011
[35] Kiwitt, S., Nagel, E. and Neumeyer, N. (2008). Empirical likelihood estimators for the error distribution in nonparametric regression models., Math. Methods Statist.17(3), 241-260. · Zbl 1231.62067
[36] Kundhi, G. and Rilstone, P. (2012). Edgeworth expansions for GEL estimators., J. Multivariate Anal.106, 118-146. · Zbl 1235.62053
[37] Loynes, R. M. (1969). On Cox and Snell’s general definition of residuals., J. Roy. Statist. Soc. Ser. B31(1), 103-106. url: www.jstor.org/stable/2984331 · Zbl 0211.51103
[38] MacKinnon, J. G. and Magee, L. (1990). Transforming the dependent variable in regression models., International Economic Review31(2), 315-339. doi: 10.2307/2526842 · Zbl 0711.62104
[39] Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error., Ann. Statist.20(2), 712-736. · Zbl 0746.62040
[40] Mátyás, L., ed. (1999)., Generalized method of moments estimation. Cambridge University Press.
[41] Muhsal, B. and Neumeyer, N. (2010). A note on residual-based empirical likelihood kernel density estimation., Electron. J. Stat.4, 1386-1401. · Zbl 1329.62190
[42] Nadaraya, E. A. (1964). Some new estimates for distribution functions., Theory Probab. Appl.9(3), 497-500. · Zbl 0152.17605
[43] Newey, W. K. and Smith, R. J. (2004). Higher order properties of GMM and generalized empirical likelihood estimators., Econometrica72(1), 219-255. · Zbl 1151.62313
[44] Oryshchenko, V. (2019). Exact mean integrated squared error and bandwidth selection for kernel distribution function estimators., Comm. Statist. Theory Methods, To Appear. doi: 10.1080/03610926.2018.1563182
[45] Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional., Biometrika75(2), 237-249. · Zbl 0641.62032
[46] Owen, A. (1990). Empirical likelihood ratio confidence regions., Ann. Statist.18(1), 90-120. · Zbl 0712.62040
[47] Pagan, A. and Ullah, A. (1999)., Nonparametric econometrics. Cambridge University Press.
[48] Parente, P. M. and Smith, R. J. (2014). Recent developments in empirical likelihood and related methods., Annual Review of Economics6, 77-102. doi: 10.1146/annurev-economics-080511-110925
[49] Parzen, E. (1962). On estimation of a probability density function and mode., Ann. Math. Statist.33(3), 1065-1076. · Zbl 0116.11302
[50] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations., Ann. Statist.22(1), 300-325. · Zbl 0799.62049
[51] Ramirez, O. A., Moss, C. B. and Boggess, W. G. (1994). Estimation and use of the inverse hyperbolic sine transformation to model non-normal correlated random variables., J. Appl. Stat.21(4), 289-304. doi: 10.1080/757583872
[52] Rao, B. L. S. P. (1983)., Nonparametric functional estimation. Academic Press. · Zbl 0542.62025
[53] Robinson, P. M. (1991). Best nonlinear three-stage least squares estimation of certain econometric models., Econometrica59(3), 755-786. · Zbl 0729.62106
[54] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function., Ann. Math. Statist.27(3), 832-837. · Zbl 0073.14602
[55] Schennach, S. M. (2007). Point estimation with exponentially tilted empirical likelihood., Ann. Statist.35(2), 634-672. · Zbl 1117.62024
[56] Silverman, B. W. (1986)., Density estimation for statistics and data analysis. Chapman & Hall. · Zbl 0617.62042
[57] Smith, R. J. (1997). Alternative semi-parametric likelihood approaches to generalised method of moments estimation., The Economic Journal107(441), 503-519. doi: 10.1111/j.0013-0133.1997.174.x
[58] Smith, R. J. (2011). GEL criteria for moment condition models., Econometric Theory27(6), 1192-1235. · Zbl 1228.62040
[59] Stacy, E. W. (1962). A generalization of the gamma distribution., Ann. Math. Statist.33(3), 1187-1192. · Zbl 0121.36802
[60] Tsai, A. C., Liou, M., Simak, M. and Cheng, P. E. (2017). On hyperbolic transformations to normality., Comput. Statist. Data Anal.115, 250-266. · Zbl 1466.62202
[61] Tsybakov, A. B. (2009)., Introduction to nonparametric estimation, (Springer Series in Statistics). Springer. · Zbl 1176.62032
[62] Van Ryzin, J. (1969). On strong consistency of density estimates., Ann. Math. Statist.40(5), 1765-1772. · Zbl 0198.23502
[63] Wand, M. P. and Jones, M. C. (1995)., Kernel smoothing. Chapman & Hall. · Zbl 0854.62043
[64] Wand, M. P. and Schucany, W. R. (1990). Gaussian-based kernels., Canad. J. Statist.18(3), 197-204.
[65] Watson, G. S. and Leadbetter, M. R. (1964). Hazard analysis II., Sankhyā Ser. A26(1), 101-116. · Zbl 0138.13906
[66] Yamato, H. (1973). Uniform convergence of an estimator of a distribution function., Bulletin of Mathematical Statistics15(3-4), 69-78. url: http://ci.nii.ac.jp/naid/120001036895/ · Zbl 0277.62032
[67] Yuan, A., Xu, J. and Zheng, G. (2014). On empirical likelihood statistical functions., J. Econometrics178(3), 613-623. · Zbl 1293.62085
[68] Zhang, B. (1995). \(M\)-estimation and quantile estimation in the presence of auxiliary information., J. Statist. Plann. Inference44(1), 77-94. · Zbl 0816.62031
[69] Zhang, B. (1998). A note on kernel density estimation with auxiliary information., Comm. Statist. Theory Methods27(1), 1-11. · Zbl 0887.62042
[70] Zygmund, A. (2003)., Trigonometric series, 3rd edn. Cambridge University Press. · JFM 58.0296.09
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.