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Feasible Multivariate Nonparametric Estimation Using Weak Separability

  • Joris Pinkse

    (University of British Columbia)

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    One of the main practical problems of nonparametric regression estimation is the curse of dimensionality. The curse of dimensionality arises because nonparametric regression estimates are dependent variable averages local to the point at which the regression function is to be estimated. The number of observations `local' to the point of estimation decreases exponentially with the number of dimensions. The consequence is that the variance of unconstrained nonparametric regression estimators of multivariate regression functions is often so great that the unconstrained nonparametric regression estimates are of no practical use. In this paper I propose a new estimation method of weakly separable multivariate nonparametric regression functions. Weak separability is a weaker condition than required by other dimension--reduction techniques, although similar asymptotic variance reductions obtain. Indeed, weak separability is weaker than generalized additivity (see Hardle and Linton, 1996 and Horowitz, 1998). The proposed estimator is relatively easy to compute. Theoretical results in this paper include (i) a uniform law of large numbers for marginal integration estimators, (ii) a uniform law of large numbers for marginal summation estimators, (iii) a uniform law of large numbers for my new nonparametric regression estimator for weakly separable regression functions, (iv) both a uniform strong and weak law of large numbers for U-statistics, and (v) three central limit theorems for my nonparametric regression estimator for weakly separable regression functions.

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    File URL: http://fmwww.bc.edu/RePEc/es2000/1241.pdf
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    Paper provided by Econometric Society in its series Econometric Society World Congress 2000 Contributed Papers with number 1241.

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    Date of creation: 01 Aug 2000
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    Handle: RePEc:ecm:wc2000:1241
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    1. Ahn, Hyungtaik, 1997. "Semiparametric Estimation of a Single-Index Model with Nonparametrically Generated Regressors," Econometric Theory, Cambridge University Press, vol. 13(01), pages 3-31, February.
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    7. BLACKORBY, Ch. & SCHWORM, W. & FISHER, T., 1986. "Testing for the existence of input aggregates in an economy production function," CORE Discussion Papers 1986046, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. Diewert, W E & Wales, T J, 1988. "Normalized Quadratic Systems of Consumer Demand Functions," Journal of Business & Economic Statistics, American Statistical Association, vol. 6(3), pages 303-12, July.
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    26. Hall, Peter, 1984. "Central limit theorem for integrated square error of multivariate nonparametric density estimators," Journal of Multivariate Analysis, Elsevier, vol. 14(1), pages 1-16, February.
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