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Nonparametric estimation when data on derivatives are available. (English) Zbl 1114.62049

Summary: We consider settings where data are available on a nonparametric function and various partial derivatives. Such circumstances arise in practice, for example in the joint estimation of cost and input functions in economics. We show that when derivative data are available, local averages can be replaced in certain dimensions by nonlocal averages, thus reducing the nonparametric dimension of the problem. We derive optimal rates of convergence and conditions under which dimension reduction is achieved. Kernel estimators and their properties are analyzed, although other estimators, such as local polynomial, spline and nonparametric least squares, may also be used. Simulations and an application to the estimation of electricity distribution costs are included.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62P20 Applications of statistics to economics
91B38 Production theory, theory of the firm

Software:

KernSmooth
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Full Text: DOI arXiv

References:

[1] Bickel, P. and Ritov, Y. (2003). Nonparametric estimators which can be “plugged-in”. Ann. Statist. 31 1033–1053. · Zbl 1058.62031 · doi:10.1214/aos/1059655904
[2] Breiman, L., Friedman, J. H., Olshen, R. A. and Stone, C. J. (1984). Classification and Regression Trees . Wadsworth, Belmont, CA. · Zbl 0541.62042
[3] Buja, A., Hastie, T. and Tibshirani, R. (1989). Linear smoothers and additive models (with discussion). Ann. Statist. 17 453–555. · Zbl 0689.62029 · doi:10.1214/aos/1176347115
[4] Donoho, D. L. and Johnstone, I. M. (1989). Projection-based approximation and a duality with kernel methods. Ann. Statist. 17 58–106. · Zbl 0699.62067 · doi:10.1214/aos/1176347004
[5] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and its Applications. Chapman and Hall, London. · Zbl 0873.62037
[6] Fan, J., Härdle, W. and Mammen, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26 943–971. · Zbl 1073.62527 · doi:10.1214/aos/1024691083
[7] Florens, J.-P., Ivaldi, M. and Larribeau, S. (1996). Sobolev estimation of approximate regressions. Econometric Theory 12 753–772. JSTOR:
[8] Friedman, J. H. (1991). Multivariate adaptive regression splines (with discussion). Ann. Statist. 19 1–141. · Zbl 0765.62064 · doi:10.1214/aos/1176347963
[9] Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. J. Amer. Statist. Assoc. 76 817–823. JSTOR: · doi:10.2307/2287576
[10] Fuss, M. and McFadden, D., eds. (1978). Production Economics: A Dual Approach to Theory and Applications 1 . The Theory of Production. North-Holland, Amsterdam. · Zbl 0432.90001
[11] Hastie, T. and Tibshirani, R. (1986). Generalized additive models (with discussion). Statist. Sci. 1 297–318. · Zbl 0645.62068 · doi:10.1214/ss/1177013604
[12] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models (with discussion). J. Roy. Statist. Soc. Ser. B 55 757–796. JSTOR: · Zbl 0796.62060
[13] Huber, P. J. (1985). Projection pursuit (with discussion). Ann. Statist. 13 435–525. · Zbl 0595.62059 · doi:10.1214/aos/1176349519
[14] Jorgenson, D. (1986). Econometric methods for modeling producer behavior. In Handbook of Econometrics 3 (Z. Griliches and M. D. Intriligator, eds.) 1841–1915. North-Holland, Amsterdam. · Zbl 0613.62144
[15] Linton, O. and Nielsen, J. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82 93–100. JSTOR: · Zbl 0823.62036 · doi:10.1093/biomet/82.1.93
[16] Murray-Smith, R. and Sbarbaro, D. (2002). Nonlinear adaptive control using non-parametric Gaussian process prior models. In 15th IFAC World Congress on Automatic Control 21–26. Barcelona.
[17] Silberberg, E. and Suen, W. (2001). The Structure of Economics: A Mathematical Analysis . McGraw-Hill, New York.
[18] Simonoff, J. S. (1996). Smoothing Methods in Statistics . Springer, New York. · Zbl 0859.62035
[19] Solak, E., Murray-Smith, R., Leithead, W. E., Leith, D. J. and Rasmussen, C. E. (2003). Derivative observations in Gaussian Process models of dynamic systems. In Advances in Neural Information Processing Systems 15 1033–1040. MIT Press, Cambridge, MA.
[20] Stone, C. J. (1980). Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 1348–1360. · Zbl 0451.62033 · doi:10.1214/aos/1176345206
[21] Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Ann. Statist. 10 1040–1053. · Zbl 0511.62048 · doi:10.1214/aos/1176345969
[22] Stone, C. J. (1985). Additive regression and other nonparametric models. Ann. Statist. 13 689–705. · Zbl 0605.62065 · doi:10.1214/aos/1176349548
[23] Varian, H. (1992). Microeconomic Analysis. Norton, New York.
[24] Yatchew, A. (2003). Semiparametric Regression for the Applied Econometrician . Cambridge Univ. Press. · Zbl 1067.62041 · doi:10.1017/CBO9780511615887
[25] Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing . Chapman and Hall, London. · Zbl 0854.62043
[26] Zhang, H. (2004). Recursive partitioning and tree-based methods. In Handbook of Computational Statistics (J. E. Gentle, W. Härdle and Y. Mori, eds.) 813–840. Springer, Berlin.
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