This is a preview of subscription content, log in via an institution to check access.
Notes
1The ‘fixed design’ estimator
$${\widehat{m}_{_{ND}}}\left( {{X_i}} \right) = {{{1 \over {nh}}\sum\nolimits_{j = 1}^n K \left( {{{{X_i} - {X_j}} \over h}} \right){Y_j}} \over {f\left( {{X_i}} \right)}},$$where K(·) is a second order kernel and h = h(n) is a bandwidth sequence, is a well known example for which (1) is not satisfied. We have
$${T_{n3}} = {{{1 \over {nh}}\sum\nolimits_{j = 1}^n K \left( {{{{X_i} - {X_j}} \over h}} \right) - f\left( {{X_i}} \right)} \over {f\left( {{X_i}} \right)}}m\left( {{X_i}} \right),$$which itself has unconditional variance of order 1/nh and bias of order h2, i.e., of the same magnitude as Tn1 and Tn2. In fact, the unconditional variance of \({\widehat{m}_{ND}}\left( {{X_i}} \right)\) is strictly greater than that of the Nadaraya-Watson estimate, which does satisfy (1), even asymptotically.
2For example, if the estimator is the standard kernel with symmetric kernel function, then the symmetrized estimator is the average of the kernel and the ‘internal’ kernel estimate and has weights
$${1 \over {2nh}}K\left( {{{{X_i} - {X_j}} \over h}} \right){Y_j}\left\{ {{1 \over {\hat f\left( {{X_i}} \right)}} + {1 \over {\hat f\left( {{X_j}} \right)}}} \right\},$$and has bias which is the average of the bias of these two estimators.
3This criterion has the interpretation of minimizing the average variance of the difference of Wy and W0y. Thus the modified smoother is as close to the original one as possible.
Suppose that we want to find \(w_{ij}^0\) to minimize the weighted criterion function \(\sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^n {{{\left\{ {{w_{ij}} - w_{ij}^0} \right\}}^2}} } {\gamma _{ij}}\) for some weights {γij}, then we proceed as below by taking \({w_{ij}}\sqrt {{\gamma _{ij}}} \) and dividing the answer by \(\sqrt {{\gamma _{ij}}} \).
References
Cohen, A. (1966). All admissible linear estimates of the mean vector. Annals of Mathematical Statistics 37, 458–463.
Deming, W.E. and F.F. Stephan (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Annals of Mathematical Statistics 11, 427–444.
Fienberg, S.E. (1970). An iterative procedure for estimation in contingency tables. Annals of Mathematical Statistics 41, 907–917.
Golub, G.H., and C.F. van Loan (1989): Matrix Computations. The Johns Hopkins University Press: Baltimore.
Härdle, W., and O.B. Linton (1994). Applied nonparametric methods. The Handbook of Econometrics, vol. IV, eds. D.F. McFadden and R.F. Engle III. North Holland.
Hastie, T. and R. Tibshirani (1990). Generalized Additive Models. Chapman and Hall, London.
Linton, O.B. (1995). Second Order Approximation in the Partially Linear Regression Model. Econometrica 63, 1079–1112.
Robinson, P. M. (1987). Asymptotically Efficient Estimation in the Presence of Heteroscedasticity of Unknown form. Econometrica 56, 875–891.
Robinson, P. M. (1988). Root-N-Consistent Semiparametric Regression. Econometrica, 56, 931–954.
Acknowledgements
I would like to thank John Hartigan for the reference to Arthur Cohen’s work. I would also like to thank an anonymous referee, Chris Jones and Berwin Turlach for helpful comments. This research was supported by the National Science Foundation and the North Atlantic Treaty Organization.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Linton, O. Symmetrizing and unitizing transformations for linear smoother weights. Computational Statistics 16, 153–164 (2001). https://doi.org/10.1007/s001800100056
Published:
Issue Date:
DOI: https://doi.org/10.1007/s001800100056