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Estimation and hypotheses testing in boundary regression models. (English) Zbl 1442.62083

Summary: Consider a nonparametric regression model with one-sided errors and regression function in a general Hölder class. We estimate the regression function via minimization of the local integral of a polynomial approximation. We show uniform rates of convergence for the simple regression estimator as well as for a smooth version. These rates carry over to mean regression models with a symmetric and bounded error distribution. In such a setting, one obtains faster rates for irregular error distributions concentrating sufficient mass near the endpoints than for the usual regular distributions. The results are applied to prove asymptotic \(\sqrt{n}\)-equivalence of a residual-based (sequential) empirical distribution function to the (sequential) empirical distribution function of unobserved errors in the case of irregular error distributions. This result is remarkably different from corresponding results in mean regression with regular errors. It can readily be applied to develop goodness-of-fit tests for the error distribution. We present some examples and investigate the small sample performance in a simulation study. We further discuss asymptotically distribution-free hypotheses tests for independence of the error distribution from the points of measurement and for monotonicity of the boundary function as well.

MSC:

62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
62E20 Asymptotic distribution theory in statistics
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References:

[1] Akritas, M.G. and Van Keilegom, I. (2001). Non-parametric estimation of the residual distribution. Scand. J. Stat.28 549-567. · Zbl 0980.62027 · doi:10.1111/1467-9469.00254
[2] Anevski, D. and Fougères, A.-L. (2007). Limit properties of the monotone rearrangement for density and regression function estimation. Available at arXiv:0710.4617v1. · Zbl 1442.62075
[3] Birke, M. and Neumeyer, N. (2013). Testing monotonicity of regression functions – an empirical process approach. Scand. J. Stat.40 438-454. · Zbl 1364.62104 · doi:10.1111/j.1467-9469.2012.00820.x
[4] Birke, M., Neumeyer, N. and Volgushev, S. (2017). The independence process in conditional quantile location-scale models and an application to testing for monotonicity. Statist. Sinica27 1815-1839. · Zbl 1392.62107
[5] Chernozhukov, V., Fernández-Val, I. and Galichon, A. (2009). Improving point and interval estimators of monotone functions by rearrangement. Biometrika96 559-575. · Zbl 1170.62025 · doi:10.1093/biomet/asp030
[6] Daouia, A., Noh, H. and Park, B.U. (2016). Data envelope fitting with constrained polynomial splines. J. R. Stat. Soc., B78 3-30. · Zbl 1411.62098
[7] Darling, D.A. (1955). The Cramér-Smirnov test in the parametric case. Ann. Math. Stat.26 1-20. · Zbl 0064.13701 · doi:10.1214/aoms/1177728589
[8] Einmahl, J.H.J. and Van Keilegom, I. (2008). Specification tests in nonparametric regression. J. Econometrics143 88-102. · Zbl 1418.62156 · doi:10.1016/j.jeconom.2007.08.008
[9] Färe, R. and Grosskopf, S. (1983). Measuring output efficiency. European J. Oper. Res.13 173-179. · Zbl 0513.90008 · doi:10.1016/0377-2217(83)90080-2
[10] Gijbels, I. (2005). Monotone regression. In The Encyclopedia of Statistical Sciences, 2nd ed. (N. Balakrishnan, S. Kotz, C.B. Read and B. Vadakovic, eds.). Hoboken, NJ: Wiley.
[11] Gijbels, I., Mammen, E., Park, B. and Simar, L. (1999). On estimation of monotone and concave frontier functions. J. Amer. Statist. Assoc.94 220-228. · Zbl 1043.62105 · doi:10.1080/01621459.1999.10473837
[12] Gijbels, I. and Peng, L. (1999). Estimation of a support curve via order statistics. Extremes3 251-277. · Zbl 0979.62035 · doi:10.1023/A:1011407111136
[13] Girard, S., Guillou, A. and Stupfler, G. (2013). Frontier estimation with kernel regression on high order moments. J. Multivariate Anal.116 172-189. · Zbl 1277.62111 · doi:10.1016/j.jmva.2012.12.001
[14] Girard, S. and Jacob, P. (2008). Frontier estimation via kernel regression on high power-transformed data. J. Multivariate Anal.99 403-420. · Zbl 1206.62070 · doi:10.1016/j.jmva.2006.11.006
[15] Hall, P. and Park, B.U. (2004). Bandwidth choice for local polynomial estimation of smooth boundaries. J. Multivariate Anal.91 240-261. · Zbl 1056.62050 · doi:10.1016/j.jmva.2003.10.002
[16] Hall, P., Park, B.U. and Stern, S.E. (1998). On polynomial estimators of frontiers and boundaries. J. Multivariate Anal.66 71-98. · Zbl 1127.62358 · doi:10.1006/jmva.1998.1738
[17] Hall, P. and Van Keilegom, I. (2009). Nonparametric “regression” when errors are positioned at end-points. Bernoulli15 614-633. · Zbl 1200.62036 · doi:10.3150/08-BEJ173
[18] Härdle, W., Park, B.U. and Tsybakov, A.B. (1995). Estimation of non-sharp support boundaries. J. Multivariate Anal.55 205-218. · Zbl 0863.62030 · doi:10.1006/jmva.1995.1075
[19] Jirak, M., Meister, A. and Reiß, M. (2014). Adaptive function estimation in nonparametric regression with one-sided errors. Ann. Statist.42 1970-2002. · Zbl 1305.62172 · doi:10.1214/14-AOS1248
[20] 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 · doi:10.1214/009053604000000274
[21] 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 · doi:10.1214/08-AOS680
[22] Knight, K. (2006). Asymptotic theory for \(M\)-estimators of boundaries. In The Art of Semiparametrics (S. Sperlich, W. Härdle and G. Aydinli, eds.) Contrib. Statist. 1-21. Heidelberg: Physica-Verlag/Springer. · Zbl 1270.62052
[23] Koul, H.L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models, 2nd ed. New York: Springer. · Zbl 1007.62047
[24] Meister, A. and Reiß, M. (2013). Asymptotic equivalence for nonparametric regression with non-regular errors. Probab. Theory Related Fields155 201-229. · Zbl 1257.62045 · doi:10.1007/s00440-011-0396-x
[25] Müller, U.U. and Wefelmeyer, W. (2010). Estimation in nonparametric regression with non-regular errors. Comm. Statist. Theory Methods39 1619-1629. · Zbl 1318.62135 · doi:10.1080/03610920802339833
[26] Neumeyer, N., Dette, H. and Nagel, E.-R. (2006). Bootstrap tests for the error distribution in linear and nonparametric regression models. Aust. N. Z. J. Stat.48 129-156. · Zbl 1108.62032 · doi:10.1111/j.1467-842X.2006.00431.x
[27] Neumeyer, N. and Van Keilegom, I. (2009). Change-point tests for the error distribution in nonparametric regression. Scand. J. Stat.36 518-541. Online supporting information available at http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9469.2009.00639.x/suppinfo. · Zbl 1195.62054 · doi:10.1111/j.1467-9469.2009.00639.x
[28] Picard, D. (1985). Testing and estimating change-points in time series. Adv. in Appl. Probab.17 841-867. · Zbl 0585.62151 · doi:10.2307/1427090
[29] Reiß, M. and Selk, L. (2017). Efficient estimation of functionals in nonparametric boundary models. Bernoulli23 1022-1055. · Zbl 1380.62177 · doi:10.3150/15-BEJ768
[30] Reiss, R.-D. (1989). Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics. New York: Springer. · Zbl 0682.62009
[31] Resnick, S.I. (2007). Heavy-Tail Phenomena. Springer Series in Operations Research and Financial Engineering. New York: Springer. · Zbl 1152.62029
[32] Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. New York: Wiley. · Zbl 1170.62365
[33] Simar, L. and Wilson, P.W. (1998). Sensitivity analysis of efficiency scores: How to bootstrap in nonparametric frontier models. Manage. Sci.44 49-61. · Zbl 1012.62501 · doi:10.1287/mnsc.44.1.49
[34] Stephens, M.A. (1976). Asymptotic results for goodness-of-fit statistics with unknown parameters. Ann. Statist.4 357-369. · Zbl 0325.62014 · doi:10.1214/aos/1176343411
[35] van der Vaart, A.W. (2000). Asymptotic Statistics. Cambridge: Cambridge Univ. Press. · Zbl 0910.62001
[36] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. New York: Springer. · Zbl 0862.60002
[37] Wilson, P.W. (2003). Testing independence in models of productive efficiency. J. Product. Anal.20 361-390.
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