Smoothing Methods in Statistics

Jeffrey S. Simonoff

New York University

Errata (first and second printings)

Page 40, lines -7 and -9: the kernel function should be positive on the interval [-1, 1), rather than (-1, 1].
Page 53, line -6: replace "explosion" with "accident."
Page 127, line -2: the exponent on the right-hand side of the equation is missing a minus sign.
Page 139, lines -15 to -14: the estimate of the qth derivative of m(x) is missing the multiplier (-1)^q out front.
Page 156, line 3: the range given should be [-.5, .5].
Page 221, line -3: in the formula for the mean sum of squared errors for the frequency estimator, N is the sample size (n is used elsewhere in the chapter).

Errata (first printing only)

Pages 58-59: the code used to produce Figure 3.14 contained an error, and the figure is not correct. You can download a Postscript version of the correct figure here. You can download a pdf (Portable Document Format) version of the correct figure here. The third and fourth sentences of the last full paragraph on page 58 should be replaced with the following two lines: "The estimate based on 10 nearest neighbors is too rough, but increasing the number of nearest neighbors does not remove all of the jaggedness of the estimate, as in plot (d). The estimate also does not capture the interesting features of the density (as the variable kernel does in Fig. 3.12), with the the bump at around 150 minutes being smoothed over in plots (b)-(d)."
Page 219, line -1: replace \overline{p}_i with \overline{p}_j.
Page 239, Figure 6.15: the vertical axis should read Density X 7000.

Updated and corrected references (first and second printings)

Gu, C. (1998) Model indexing and smoothing parameter selection in nonparametric function estimation (with discussion). Statistica Sinica, 8, 607-646 [given in book as Gu (1995a)].
Marx, B.D. and Eilers, P.H.C. (1998) Direct generalized additive modeling with penalized likelihood. Computational Statistics and Data Analysis, 28, 193-209 [given in book as Marx and Eilers (1994)].

Updated and corrected references (second printing only)

Jones, M.C., Samiuddin, M., Al-Harbey, A.H., and Maatouk, T.A.H. (1998) The edge frequency polygon. Biometrika, 85, 235-239.
Marron, J.S. (1998) Assessing bandwidth selectors with visual error criteria. Computational Statistics, 13, 511-528.

Updated and corrected references (first printing only)

Aerts, M., Augustyns, I., and Janssen, P. (1997a) Smoothing sparse multinomial data using local polynomial fitting. Journal of Nonparametric Statistics, 8, 127-147 [given in book as Aerts, Augustyns, and Janssen (1994)].
Aerts, M., Augustyns, I., and Janssen, P. (1997b) Local polynomial estimation of contingency table cell probabilities. Statistics, 30, 127-148 [given in book as Aerts, Augustyns, and Janssen (1995)].
Basu, A., Harris, I.R., and Basu, S. (1996) Tests of hypotheses in discrete models based on the penalized Hellinger distance. Statistics and Probability Letters, 27, 367-373.
Carroll, R.J., Fan, J., Gijbels, I., and Wand, M.P. (1997) Generally partially linear single-index models. Journal of the American Statistical Association, 92, 477-489 [given in book as Carroll, Fan, Gijbels, and Wand (1995)].
Cheng, M.-Y. (1997) A bandwidth selector for local linear density estimators. Annals of Statistics, 25, 1001-1013 [given in book as Cheng (1994)].
Cheng, M.-Y., Fan, J., and Marron, J.S. (1997) On automatic boundary corrections. Annals of Statistics, 25, 1691-1708 [given in book as Cheng, Fan, and Marron (1993)].
Cleveland, W.S. and Loader, C. (1996) Smoothing by local regression: principles and methods (with discussion). In Statistical Theory and Computational Aspects of Smoothing, eds. W. Hardle and M.G. Schimek, Physica-Verlag, Heidelberg, 10-49; 80-102; 113-120 [given in book as Cleveland and Loader (1995)].
Dong, J. and Ye, Q. (1996) A minimum variance kernel estimator and a discrete frequency polygon estimator for ordinal sparse contingency tables. Communications in Statistics - Theory and Methods, 25, 3217-3245 [given in book as Dong and Ye (1995)].
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., and Picard, D. (1996) Density estimation by wavelet thresholding. Annals of Statistics, 24, 508-539.
Efron, B. and Tibshirani, R. (1996) Using specially designed exponential families for density estimation. Annals of Statistics, 24, 2431-2461 [given in book as Efron and Tibshirani (1994)].
Eilers, P.H.C. and Marx, B.D. (1996) Flexible smoothing with B-splines and penalties (with discussion). Statistical Science, 11, 89-121 [given in book as Eilers and Marx (1994)].
Fan, J., Gasser, T., Gijbels, I., Brockmann, M., and Engel, J. (1997) Local polynomial fitting: optimal kernels and minimax efficiency. Annals of the Institute of Statistical Mathematics, 49, 79-99 [given in book as Fan, Gasser, Brockmann, and Engel (1993)].
Fan, J. and Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
Fan, J., Gijbels, I., Hu, T.-C., and Huang, L.-S. (1996) A study of variable bandwidth selection for local polynomial regression. Statistica Sinica, 6, 113-127.
Fan, J., Hall, P., Martin, M., and Patil, P. (1996) On local smoothing of nonparametric curve estimators. Journal of the American Statistical Association, 91, 258-266 [given in book as Fan, Hall, Martin, and Patil (1993)].
Gonzalez-Manteiga, W., Sanchez-Sellero, C., and Wand, M.P. (1996) Accuracy of binned kernel functional approximations. Computational Statistics and Data Analysis, 22, 1-16 [given in book as Gonzalez-Manteiga, Sanchez-Sellero, and Wand (1995)].
Hall, P. and Turlach, B.A. (1996) Contribution to discussion of papers by Seifert and Gasser, Marron, and Cleveland and Loader. In Statistical Theory and Computational Aspects of Smoothing, eds. W. Hardle and M.G. Schimek, Physica-Verlag, Heidelberg, 80-84 [given in book as Hall and Turlach (1995)].
Hall, P. and Wand, M.P. (1996) On the accuracy of binned kernel density estimators. Journal of Multivariate Analysis, 56, 165-184.
Hardle, W. and Marron, J.S. (1995) Fast and simple scatterplot smoothing. Computational Statistics and Data Analysis, 20, 1-17.
Herrmann, E. (1997) Local bandwidth choice in kernel regression estimation. Journal of Computational and Graphical Statistics, 6, 35-54 [given in book as Herrmann (1996)].
Hjort, N.L. and Jones, M.C. (1996) Locally parametric density estimation. Annals of Statistics, 24, 1619-1647.
Huang, L.-S. (1997) Testing goodness-of-fit based on a roughness measure. Journal of the American Statistical Association, 92, 1399-1402 [given in book as Huang (1995)].
Janssen, P., Marron, J.S., Veraverbeke, N., and Sarle, W. (1995) Scale measures for bandwidth selection. Journal of Nonparametric Statistics, 5, 359-380.
Jones, M.C. (1996) On close relations of local likelihood density estimation. Test, 5, 345-356 [given in book as Jones (1995c)].
Jones, M.C. and Foster, P.J. (1996) A simple nonnegative boundary correction method for kernel density estimation. Statistica Sinica, 6, 1005-1013 [given in book as Jones and Foster (1995)].
Jones, M.C., Marron, J.S., and Sheather, S.J. (1996) A brief survey of bandwidth selection for density estimation. Journal of the American Statistical Association, 91, 401-407 [given in book as Jones, Marron, and Sheather (1995)].
Jones, M.C., Samiuddin, M., Al-Harbey, A.H., and Maatouk, T.A.H. (1998) The edge frequency polygon. Biometrika, 85, 235-239 [given in book as Jones, Samiuddin, Al-Harbey, and Maatouk (1995)].
Jones, M.C. and Signorini, D.F. (1997) A comparison of higher-order bias kernel density estimators. Journal of the American Statistical Association, 92, 1063-1073 [given in book as Jones and Signorini (1996)].
Koehler, K.J. and Gan, F.F. (1990) Chi-squared goodness-of-fit tests: cell selection and power. Communications in Statistics - Simulation and Computation, 19, 1265-1278 [given in book as Koehler and Gan (1987)].
Loader, C.R. (1996) Local likelihood density estimation. Annals of Statistics, 24, 1602-1618.
Lugosi, G. and Nobel, A. (1996) Consistency of data-driven histogram methods for density estimation and classification. Annals of Statistics, 24, 687-706.
Manchester, L. (1996) Empirical influence for robust smoothing. Australian Journal of Statistics, 38, 275-290 [given in book as Manchester (1995)].
Marchette, D.J., Priebe, C.E., Rogers, G.W., and Solka, J.L. (1996) Filtered kernel density estimation. Computational Statistics, 11, 95-112.
Marron, J.S. (1996) A personal view of smoothing and statistics (with discussion). In Statistical Theory and Computational Aspects of Smoothing, eds. W. Hardle and M.G. Schimek, Physica-Verlag, Heidelberg, 1-9; 80-112 [given in book as Marron (1995b)].
Marron, J.S. (1998) Assessing bandwidth selectors with visual error criteria. Computational Statistics, 13, 511-528 [given in book as Marron (1996)].
Minnotte, M.C. (1996) The bias-optimized frequency polygon. Computational Statistics, 11, 35-48.
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Ray, B.K. and Tsay, R.S. (1996) Iterative bandwidth selection for nonparametric regression with long-range dependent errors. In Proceedings of the International Conference on Applied Probability and Time Series, Volume II (1995), ed. M. Rosenblatt, Springer-Verlag, New York, 339-351 [given in book as Ray and Tsay (1995)].
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Russell, J.M., Simonoff, J.S., and Nightingale, J. (1997) Nursing behaviors of beluga calves (delphinapterus leucas) born in captivity. Zoo Biology, 16, 247-262 [given in book as Russell, Simonoff, and Nightingale (1995)].
Seifert, B. and Gasser, T. (1996) Variance properties of local polynomials and ensuing modifications (with discussion). In Statistical Theory and Computational Aspects of Smoothing, eds. W. Hardle and M.G. Schimek, Physica-Verlag, Heidelberg, 50-102; 121-127 [given in book as Seifert and Gasser (1995)].
Seifert, B. and Gasser, T. (1996) Finite sample variance of local polynomials: analysis and solutions. Journal of the American Statistical Association, 91, 267-275.
Simonoff, J.S. and Udina, F. (1997) Measuring the stability of histogram appearance when the anchor position is changed. Computational Statistics and Data Analysis, 23, 335-353 [given in book as Simonoff and Udina (1996)].
Smith, M. and Kohn, R. (1996) Nonparametric regression using Bayesian variable selection. Journal of Econometrics, 75, 317-343.
Stone, C.J., Hansen, M.H., Kooperberg, C., and Truong, Y.K. (1997) Polynomial splines and their tensor products in extended linear modeling (with discussion). Annals of Statistics, 25, 1371-1470 [given in book as Stone, Hansen, Kooperberg, and Truong (1995)].
Turlach, B.A. and Wand, M.P. (1996) Fast computation of auxiliary quantities in local polynomial regression. Journal of Computational and Graphical Statistics, 5, 337-350.
Van Es, A.J. and Hoogstrate, A.J. (1998) How much do plug-in bandwidth selectors adapt to non-smoothness? Journal of Nonparametric Statistics, 8, 185-197 [given in book as Van Es and Hoogstrate (1995)].
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Wand, M.P. (1997) Data-based choice of histogram bin width. American Statistician, 51, 59-64 [given in book as Wand (1994a)].
Wang, K. and Gasser, T. (1996) Optimal rate for estimating local bandwidth in kernel estimators of regression functions. Scandinavian Journal of Statistics, 23, 303-312.