Preaveraged Localized Orthogonal Polynomial Estimators for Surface Smoothing and Partial Differentiation

References

• Alemi , M. H. , Azari , A. S. and Nielsen , D. R. 1988 . Kriging and Univariate Modeling of Spatially Correlated Data . Soil Technology , 1 : 133 – 147 .
   Google Scholar
• Azari , A. S. , Mack , Y. P. and Müller , H. G. Orthogonal Polynomial, Kernel and Hybrid Estimators for Nonparametric Regression . Sankhyā , 54 (in press), Ser. A
   Google Scholar
• Cacoullos , R. 1966 . Estimation of a Multivariate Density . Annals of the Institute of Statistical Mathematics , 18 : 179 – 189 .
   Web of Science ®Google Scholar
• Cencov , N. N. 1964 . Evaluation of an Unknown Distribution Density From Observations . Soviet Mathematics , 3 : 1559 – 1562 .
   Google Scholar
• Cleveland , W. S. and Devlin , S. J. 1988 . Locally Weighted Regression: An Approach to Regression Analysis by Local Fitting . Journal of the American Statistical Association , 83 : 596 – 610 .
   Web of Science ®Google Scholar
• Cressie , N. 1989 . Geostatistics . The American Statistician , 43 : 197 – 202 .
   Web of Science ®Google Scholar
• Cressie , N. 1991 . Statistics for Spatial Data , New York : Wiley .
   Google Scholar
• Epanechnikov , V. A. 1969 . Nonparametric Estimation of a Multivariate Probability Density . Theory of Probability and its Applications , 14 : 153 – 158 .
   Web of Science ®Google Scholar
• Gasser , T. and Müller , H. G. 1979 . “ Kernel Estimation of Regression Functions ” . In Lecture Notes in Mathematics , Vol. 757 , 23 – 68 . New York : Springer .
   Google Scholar
• Georgiev , A. A. 1987 . A Note on the Weak Pointwise Convergence of the Priestley–Chao Multiple Regression Estimate . Journal of Statistical Planning and Inference , 16 : 261 – 262 .
   Google Scholar
• Härdle , W. and Scott , D. W. 1990 . Smoothing by Weighted Averaging of Rounded Points , Rice University, Dept. of Statistics . Technical Report 90–5
   Google Scholar
• Hastie , T. and Tibshirani , R. 1986 . Generalized Additive Models . Statistical Science , 1 : 297 – 318 .
   Google Scholar
• Hastie , T. and Tibshirani , R. 1990 . Generalized Additive Models , New York : Chapman and Hall .
   Google Scholar
• Jones , R. H. 1989 . Fitting a Stochastic Partial Differential Equation to Aquifer Head Data . Stochastic Hydrology and Hydraulics , 3 : 85 – 96 .
   Web of Science ®Google Scholar
• Karlin , S. , Micchelli , C. A. and Rinott , Y. 1986 . Multivariate Splines: A Probabilistic Approach . Journal of Multivariate Analysis , 20 : 69 – 90 .
   Web of Science ®Google Scholar
• Kronmal , R. and Tarter , M. 1968 . The Estimation of Probability Densities and Cumulative Fourier Series Method . Journal of the American Statistical Association , 63 : 925 – 952 .
   Web of Science ®Google Scholar
• Müller , H. G. 1988 . Nonparametric Regression Analysis of Longitudinal Data , New York : Springer-Verlag .
   Google Scholar
• Nussbaum , M. 1986 . Nonparametric Estimation of a Smooth Regression Function Defined on a Domain of Rk . Theory of Probability and Applications , 31 : 108 – 115 .
   Google Scholar
• Ripley , B. D. 1981 . Spatial Statistics , New York : Wiley .
   Google Scholar
• Rutkowski , L. 1982 . “ Orthogonal Series Estimates of a Regression Function With Applications in System Identification ” . In Probability and Statistical Inference , Edited by: Grossmann , W. , Pflug , C. G. and Wertz , W. 343 – 347 . Dordrecht , , Netherlands : Reidel .
   Google Scholar
• Schwartz , S. C. 1967 . Estimation of Probability Density by an Orthogonal Series . The Annals of Mathematical Statistics , 38 : 1262 – 1265 .
   Google Scholar
• Scott , D. W. 1985 . Averaged Shifted Histograms: Effective Nonparametric Density Estimators in Several Dimensions . The Annals of Statistics , 13 : 1024 – 1040 .
   Web of Science ®Google Scholar
• Scott , D. W. 1986 . “ Data Analysis in Three and Four Dimensions with Non-parametric Density Estimation ” . In Statistical Image Processing and Graphics , Edited by: Wegman , E. J. and DePriest , D. J. 291 – 305 . New York : Marcel Dekker .
   Google Scholar
• Scott , D. W. 1992 . Multivariate Density Estimation , New York : Wiley .
   Google Scholar
• Singh , R. S. 1976 . Nonparametric Estimation of Mixed Partial Derivatives of a Multivariate Density . Journal of Multivariate Analysis , 6 : 111 – 122 .
   Web of Science ®Google Scholar
• Staniswalis , J. G. , Messer , K. and Finston , D. R. 1990 . Kernel Estimators for Multivariate Smoothing , University of Texas, El Paso, Dept. of Mathematics . technical report
   Google Scholar
• Szegö , G . 1975 . Orthogonal Polynomials, , 4th ed. , Providence , RI : American Mathematical Society Colloquium Publications .
   Google Scholar
• Utreras , F. I. 1987 . On Generalized Cross-Validation for Multivariate Smoothing Spline Functions . SIAM Journal on Scientific and Statistical Computing , 8 : 630 – 643 .
   Google Scholar
• Vecchia , A. V. 1989 . Estimation and Model Identification for Continuous Spatial Processes . Journal of the Royal Statistical Society , : 297 – 312 . Ser. B50
   Google Scholar
• Viera , S. R. , Hatfield , J. L. M. , Nielsen , D. R. and Biggar , J. W. 1983 . Geostatistical Theory and Application to Variability of Some Agronomical Properties . Hilgardia , 51 : 1 – 75 .
   Google Scholar
• Villalobos , M. and Wahba , G. 1987 . Inequality Constrained Multivariate Smoothing Splines With Application to the Estimation of Probabilities . Journal of the American Statistical Association , 82 : 239 – 248 .
   Web of Science ®Google Scholar
• Wahba , G. 1990 . Spline Models for Observational Data , Philadelphia : SIAM .
   Google Scholar
• Whittle , P. 1954 . On Stationary Processes in the Plane . Biometrika , 41 : 434 – 449 .
   Web of Science ®Google Scholar
• Yakowitz , S. J. and Szidarovsky , F. 1985 . A Comparison of Kriging With Nonparametric Regression Methods . Journal of Multivariate Analysis , 16 : 21 – 53 .
   Web of Science ®Google Scholar