Hostname: page-component-6898cfcb55-zkb6p Total loading time: 0 Render date: 2026-11-11T08:18:41.989Z Has data issue: false hasContentIssue false
  • English
  • Français

The Estimation of Nonparametric Functions in a Hilbert Space

Published online by Cambridge University Press:  18 October 2010

A. R. Bergstrom
A. R. Bergstrom*
Affiliation:
University of Essex, England
Get accessRights & Permissions [Opens in a new window]

Abstract

This paper is concerned with the estimation of a nonlinear regression function which is not assumed to belong to a prespecified parametric family of functions. An orthogonal series estimator is proposed, and Hilbert space methods are used in the derivation of its properties and the proof of several convergence theorems. One of the main objectives of the paper is to provide the theoretical basis for a practical stopping rule which can be used for determining the number of Fourier coefficients to be estimated from a given sample.

Information

Type
Research Article
Information
Econometric Theory , Volume 1 , Issue 1 , April 1985, pp. 7 - 26
Copyright
Copyright © Cambridge University Press 1985 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

REFERENCES

1. Amemiya, T. (1974). The nonlinear two stage least squares estimator. Journal of Econometrics, 2, 105–110.10.1016/0304-4076(74)90033-5Google Scholar
2. Amemiya, T. (1977). The maximum likelihood and nonlinear three stage least squares estimator in the general nonlinear simultaneous equations model. Econometrica, 45, 955–968.Google Scholar
3. Benedetti, J. K. (1977). On the nonparametric estimation of regression functions. Journal of the Royal Statistical Society, Series B, 39, 248–253.Google Scholar
4. Bergstrom, A. R. (1983). Gaussian estimation of structural parameters in higher order continuous time dynamic models. Econometrica, 51, 117–152.10.2307/1912251Google Scholar
5. Bergstrom, A. R., & Wymer, C. R. (1976). A Model of Disequilibrium Neoclassical Growth and its Application to the United Kingdom. Chapter 10 and pages 267–327 in Bergstrom, A. R. (ed.) Statistical Inference in Continuous Time Economic Models, North-Holland: Amsterdam.Google Scholar
6. Bierens, H. J. (1983). Uniform consistency of kernel estimators of a regression function under generalized conditions. Journal of the American Statistical Association, 78, 699–707.10.1080/01621459.1983.10478031Google Scholar
7. Cencor, N. N. (1962). Evaluation of an Unknown Density from Observations. Soviet Mathematics, 3, 1559–1562.Google Scholar
8. Christensen, L. R., Jorgenson, D. W., & Lau, L. J. (1971). Conjugate duality and the transcendental logarithmic function. Econometrica, 39, 255–256.Google Scholar
9. Christensen, L. R., Jorgenson, D. W., & Lau, L. J. (1973). Transcendental logarithmic production frontiers, Review of Economics and Statistics, 55, 28–45.Google Scholar
10. Deaton, A., & Muellbauer, J. (1980). An almost ideal demand system. American Economic Review, 70, 312–326.Google Scholar
11. Gallant, A. R. (1981). On the bias in flexible functional forms and an essentially unbiased form: The Fourier flexible form. Journal of Econometrics, 15, 211–246.10.1016/0304-4076(81)90115-9Google Scholar
12. Geweke, J., & Meese, R. (1981). Estimating regression models of finite but unknown order. International Economic Review, 22, 55–70.Google Scholar
13. Ibragimov, I. A., & Khas'minskii, R. Z. (1982). Bounds for the risks of nonparametric regression estimates. Theory of Probability and its Applications, 27, 84–99.10.1137/1127008Google Scholar
14. Jennrich, R. I. (1969). Asymptotic properties of nonlinear least squares estimators. Annals of Mathematical Statistics, 40, 633–643.Google Scholar
15. Kendall, M. G., & Stuart, A. (1979). The Advanced Theory of Statistics (4th ed.) (vol. 2). London: Griffin.Google Scholar
16. Kolmogorov, A. N., & Fomin, S. V. (1961). Elements of the Theory of Functions and Functional Analysis (Vol. 2). New York: Graylock Press.Google Scholar
17. Kronmal, R., & Tarter, M. (1968). The estimation of probability densities and cumulations by Fourier series methods. Journal of the American Statistical Association, 63, 925–952.Google Scholar
18. Malinvaud, E. (1970). The consistency of nonlinear regressions. Annals of Mathematical Statistics, 41, 956–969.10.1214/aoms/1177696972Google Scholar
19. Malinvaud, E. (1980). Statistical Methods of Econometrics (3rd ed.) Amsterdam: North Holland.Google Scholar
20. Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and its Applications, 9, 141–142.10.1137/1109020Google Scholar
21. Parzen, E. (1962). On the estimation of a probability density function and mode. Annals of Mathematical Statistics, 33, 1065–1076.Google Scholar
22. Phillips, P.C.B. (1976). The iterated minimum distance estimator and the quasi-maximum likelihood estimator. Econometrica, 44, 449–460.10.2307/1913973Google Scholar
23. Phillips, P.C.B. (1982). On the consistency of nonlinear FIML. Econometrica, 50, 1307–1325.Google Scholar
24. Priestley, M. B., & Chao, M. T. (1972). Nonparametric function fitting. Journal of the Royal Statistical Society, Series B, 34, 385–392.Google Scholar
25. Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function Annals of Mathematical Statistics, 27, 832–837.Google Scholar
26. Schwartz, T. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.Google Scholar
27. Souza, G., & Gallant, A. R. (1979). Statistical inference based on M-estimators for the multivariate nonlinear regression model in implicit form. Institute of Statistics Mimeograph Series, No. 1229.Google Scholar
28. Stone, C. J. (1977). Consistent nonparametric regression. Annuals of Statistics, 5, 595–654.Google Scholar
29. Stone, C. J. (1982). Optimal global rates of convergence for nonparametric regression. Annals of Statistics, 10, 1040–1053.Google Scholar
30. Titchmarsh, E. C. (1939). The Theory of Functions. London: Oxford University Press.Google Scholar
31. Watson, G. S. (1964). Smooth regression analysis. Sankya, Series A, 26, 359–372.Google Scholar
32. Whittle, P. (1958). On the smoothing of probability density functions. Journal of the Royal Statistical Society, Series B, 20, 334–343.Google Scholar
33. Zygmund, A. (1979). Trigonometric Series. Cambridge, England: Cambridge University Press.Google Scholar