Common structure of smoothing techniques in statistics

Summary A wide variety of statistical techniques involve the choice of a prescription and smoothing parameter for modifying an unsmoothed estimator towards some ultrasmooth solution. The basic common structure is illustrated by multinomial smoothing, nonparametric density estimation, linear and nonparametric regression, the solution of integral equations and image processing. Possible cross-fertilization among the different areas of application is suggested.

[1]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[2]  P. Whittle.  Curve and Periodogram Smoothing , 1957 .

[3]  P. Whittle.  On the Smoothing of Probability Density Functions , 1958 .

[4]  David L. Phillips,et al.  A Technique for the Numerical Solution of Certain Integral Equations of the First Kind , 1962, JACM.

[5]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[6]  S. Twomey,et al.  On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature , 1963, JACM.

[7]  M. B. Priestley,et al.  The Role of Bandwidth in Spectral Analysis , 1965 .

[8]  S. Twomey.  The application of numerical filtering to the solution of integral equations encountered in indirect sensing measurements , 1965 .

[9]  L. Ahrens,et al.  Observations on the Fe-Si-Mg relationship in chondrites , 1965 .

[10]  C. Reinsch.  Smoothing by spline functions , 1967 .

[11]  J. Dickey.  Smoothed Estimates for Multinomial Cell Probabilities , 1968 .

[12]  R. Kronmal,et al.  The Estimation of Probability Densities and Cumulatives by Fourier Series Methods , 1968 .

[13]  Otto Neall Strand,et al.  Statistical Estimation of the Numerical Solution of a Fredholm Integral Equation of the First Kind , 1968, JACM.

[14]  Edward J. Wegman,et al.  Maximum Likelihood Estimation of a Unimodal Density Function , 1970 .

[15]  E. Wegman.  Maximum likelihood estimation of a unimodal density. II , 1970 .

[16]  David G. Kendall,et al.  Spline Transformations: Three New Diagnostic Aids for the Statistical Data‐Analyst , 1971 .

[17]  Valentin F. Turchin,et al.  The use of mathematical-statistics methods in the solution of incorrectly posed problems , 1971 .

[18]  R. A. Gaskins,et al.  Nonparametric roughness penalties for probability densities , 2022 .

[19]  D. Lindley,et al.  Bayes Estimates for the Linear Model , 1972 .

[20]  M. Priestley,et al.  Non‐Parametric Function Fitting , 1972 .

[21]  E. Wegman.  Nonparametric Probability Density Estimation: I. A Summary of Available Methods , 1972 .

[22]  P. Holland,et al.  Simultaneous Estimation of Multinomial Cell Probabilities , 1973 .

[23]  B. Efron,et al.  Combining Possibly Related Estimation Problems , 1973 .

[24]  Tom Leonard,et al.  A Bayesian method for histograms , 1973 .

[25]  B. R. Hunt,et al.  The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer , 1973, IEEE Transactions on Computers.

[26]  P. Bloomfield,et al.  Numerical differentiation procedures for non-exact data , 1974 .

[27]  P. Bloomfield,et al.  A Time Series Approach To Numerical Differentiation , 1974 .

[28]  Subhash C. Narula.  Predictive Mean Square Error and Stochastic Regressor Variables , 1974 .

[29]  M. Stone.  Cross-validation and multinomial prediction , 1974 .

[30]  R. Tapia,et al.  Nonparametric Maximum Likelihood Estimation of Probability Densities by Penalty Function Methods , 1975 .

[31]  Seymour Geisser,et al.  The Predictive Sample Reuse Method with Applications , 1975 .

[32]  M. J. Fryer.  Some Errors Associated with the Non-parametric Estimation of Density Functions , 1976 .

[33]  J. Aitchison,et al.  Multivariate binary discrimination by the kernel method , 1976 .

[34]  Robert P. W. Duin,et al.  On the Choice of Smoothing Parameters for Parzen Estimators of Probability Density Functions , 1976, IEEE Transactions on Computers.

[35]  M. Stone,et al.  Cross‐Validatory Choice and Assessment of Statistical Predictions , 1976 .

[36]  M. Stone.  Asymptotics for and against cross-validation , 1977 .

[37]  H. D. Brunk,et al.  Discussion: Consistent Nonparametric Regression , 1977 .

[38]  Tom Leonard,et al.  A Bayesian Approach to Some Multinomial Estimation and Pretesting Problems , 1977 .

[39]  M. J. Fryer.  A Review of Some Non-parametric Methods of Density Estimation , 1977 .

[40]  L. Breiman,et al.  Variable Kernel Estimates of Multivariate Densities , 1977 .

[41]  Lloyd A. Currie.  Model uncertainty and bias in the evaluation of nuclear spectra , 1977 .

[42]  G. Wahba.  Optimal Smoothing of Density Estimates , 1977 .

[43]  G. Wahba.  Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression , 1978 .

[44]  W. Wertz.  Statistical density estimation: A survey , 1978 .

[45]  R. Tapia,et al.  Nonparametric Probability Density Estimation , 1978 .

[46]  B. Silverman,et al.  Choosing the window width when estimating a density , 1978 .

[47]  B. Silverman,et al.  Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives , 1978 .

[48]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[49]  Lynn Roy LaMotte,et al.  Bayes Linear Estimators , 1978 .

[50]  Norman R. Draper,et al.  Ridge Regression and James-Stein Estimation: Review and Comments , 1979 .

[51]  G. Walter,et al.  Probability Density Estimation Using Delta Sequences , 1979 .

[52]  Adrian Bowman.  A note on consistency of the kernel method for the analysis of categorical data , 1980 .

[53]  G. Wahba,et al.  Some New Mathematical Methods for Variational Objective Analysis Using Splines and Cross Validation , 1980 .

[54]  D. M. Titterington,et al.  A Comparative Study of Kernel-Based Density Estimates for Categorical Data , 1980 .

[55]  J. Skilling,et al.  Deconvolution by maximum entropy, as illustrated by application to the jet of M87 , 1980 .

[56]  D. W. Scott,et al.  Nonparametric Probability Density Estimation by Discrete Maximum Penalized- Likelihood Criteria , 1980 .

[57]  G. Wahba.  Ill Posed Problems: Numerical and Statistical Methods for Mildly, Moderately and Severely Ill Posed Problems with Noisy Data. , 1980 .

[58]  I. Good,et al.  Density Estimation and Bump-Hunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data , 1980 .

[59]  F. Utreras Diaz,et al.  Sur le choix du paramètre d'ajustement dans le lissage par fonctions spline , 1980 .

[60]  B. Silverman,et al.  Using Kernel Density Estimates to Investigate Multimodality , 1981 .

[61]  J. V. Ryzin,et al.  A class of smooth estimators for discrete distributions , 1981 .

[62]  G. Collomb.  Estimation Non-paramétrique de la Régression: Revue Bibliographique@@@Estimation Non-parametrique de la Regression: Revue Bibliographique , 1981 .

[63]  David W. Scott,et al.  Monte Carlo Study of Three Data-Based Nonparametric Probability Density Estimators , 1981 .

[64]  P. Hall.  On nonparametric multivariate binary discrimination , 1981 .

[65]  Grace Wahba,et al.  Constrained Regularization for Ill Posed Linear Operator Equations, with Applications in Meteorology and Medicine. , 1982 .

[66]  J. Rice.  An Approach to Peak Area Estimation. , 1982, Journal of research of the National Bureau of Standards.

[67]  V. Klonias.  Consistency of Two Nonparametric Maximum Penalized Likelihood Estimators of the Probability Density Function , 1982 .

[68]  B. Silverman,et al.  On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method , 1982 .

[69]  P. Read.  The analysis of umkehr observations of stratospheric ozone by a ‘Maximum Entropy’ method , 1982 .

[70]  M. C. Jones,et al.  The Statistical Analysis of Compositional Data , 1986 .

[71]  Peter Hall,et al.  Cross-validation in density estimation , 1982 .

[72]  P. Hall.  Large Sample Optimality of Least Squares Cross-Validation in Density Estimation , 1983 .

[73]  C. Morris.  Parametric Empirical Bayes Inference: Theory and Applications , 1983 .

[74]  Peter Hall,et al.  Orthogonal Series Distribution Function Estimation, with Applications , 1983 .

[75]  G. Wahba.  Bayesian "Confidence Intervals" for the Cross-validated Smoothing Spline , 1983 .

[76]  I. W. Wright.  Splines in Statistics , 1983 .

[77]  M. Rosenblatt,et al.  Smoothing Splines: Regression, Derivatives and Deconvolution , 1983 .

[78]  T. Fearn.  A Misuse of Ridge Regression in the Calibration of a Near Infrared Reflectance Instrument , 1983 .

[79]  B. W. Silverman,et al.  Probability, Statistics and Analysis: Some properties of a test for multimodality based on kernel density estimates , 1983 .

[80]  Jeffrey S. Simonoff,et al.  A Penalty Function Approach to Smoothing Large Sparse Contingency Tables , 1983 .

[81]  S. Geman,et al.  Consistent Cross-Validated Density Estimation , 1983 .

[82]  W. Wong.  A Note on the Modified Likelihood for Density Estimation , 1983 .

[83]  John Skilling,et al.  Image restoration by a powerful maximum entropy method , 1982, Comput. Vis. Graph. Image Process..

[84]  J. Copas.  Regression, Prediction and Shrinkage , 1983 .

[85]  G. S. Watson.  Smoothing and interpolation by kriging and with splines , 1984 .

[86]  D. M. Titterington,et al.  Cross-validation in nonparametric estimation of probabilities and probability densities , 1984 .

[87]  A. Bowman.  An alternative method of cross-validation for the smoothing of density estimates , 1984 .

[88]  C. J. Stone,et al.  An Asymptotically Optimal Window Selection Rule for Kernel Density Estimates , 1984 .

[89]  M. Hutchinson,et al.  Smoothing noisy data with spline functions , 1985 .

[90]  A. Bowman.  A comparative study of some kernel-based nonparametric density estimators , 1985 .

[91]  J. Marron.  An Asymptotically Efficient Solution to the Bandwidth Problem of Kernel Density Estimation , 1985 .

[92]  D. Titterington,et al.  A comparative study of smoothing procedures for ordered categorical data , 1985 .