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References
Banerjee, M. and Wellner, J. (2001). Likelihood ratio tests for monotone functions, Ann. Stat., 29(6), 1699–1731 .
Bickel, P.J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates, Ann. Stat., 1,1071–1095.
Birge, L. (1989). The Grenander estimator: a nonasymptotic approach, Ann. Stat., 17(4), 1532–1549 .
Bowman, A.W. (1984). An alternative method of cross-validation for the smoothing of density estimates, Biometrika, 71(2), 353–360 .
Breiman, L., Meisel, W., and Purcell, E. (1977). Variable kernel estimates of multivariate densities, Technometrics, 19,135–144.
Brown, L. (1992). Minimaxity, more or less, in Statistical Decision Theory and Related Topics, Vol. V, J. Berger and S.S. Gupta (eds.), Springer-Verlag, New York, 1–18.
Brown, L. (2000). An essay on statistical decision theory, J. Am. Stat. Assoc., 95,1277–1281.
Chernoff, H. (1964). Estimation of the mode, Ann. Inst. Stat. Math., 16,31–41.
Chow, Y.S., Geman, S. and Wu, L.D. (1983). Consistent cross-validated density estimation, Ann. Stat., 11(1), 25–38 .
DasGupta, A. (2000). Some results on the curse of dimensionality and sample size recommendations, Cal. Stat. Assoc. Bull., 50(199/200), 157–177.
Devroye, L. (1983). On arbitrarily slow rates of convergence in density estimation, Z. Wahr. Verw. Geb., 62(4), 475–483 .
Devroye, L. (1987). A Course in Density Estimation, Birkhäuser, Boston.
Devroye, L. (1992). A note on the usefulness of superkernels in density estimation, Ann. Stat., 20,2037–2056.
Devroye, L. and Penrod, C.S. (1984). The consistency of automatic kernel density estimates, Ann. Stat., 12(4), 1231–1249 .
Devroye, L. and Lugosi, G. (1999). Variable kernel estimates: on the impossibility of tuning the parameters, in High Dimensional Probability, Ginè, E. and D. Mason (eds.) Vol. II, Birkhäuser, Boston, 405–424.
Devroye, L. and Lugosi, G. (2000). Combinatorial Methods in Density Estimation, Springer, New York.
Epanechnikov, V.A. (1969). Nonparametric estimates of a multivariate probability density, Theory Prob. Appl., 14,153–158.
Farrell, R.H. (1972). On the best attainable asymptotic rates of convergence in estimation of a density at a point, Ann. Math. Stat., 43,170–180.
Gasser, T., Müller, H., and Mammitzsch, V. (1985). Kernels for nonparametric curve estimations, J.R. Stat. Soc. B, 47,238–252.
Grenander, U. (1956). On the theory of mortality measurement, Skand. Aktuarietidskr, 39,70–96.
Groeneboom, P. (1985). Estimating a monotone density, in Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, L. Le cam and R. A. O eslen (eds.) Wadsworth, Belmont, CA, 539–555.
Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions, Prob. Theory Related Fields, 81,79–109.
Groeneboom, P. and Wellner, J. (2001). Computing Chernoff’s distribution, J. Comput. Graph. Stat., 10,388–400.
Hall, P. (1982). Cross-validation in density estimation, Biometrika, 69(2), 383–390 .
Hall, P. (1983). Large sample optimality of least squares cross-validation in density estimation, Ann. Stat., 11(4), 1156–1174 .
Hall, P. and Marron, J.S. (1988). Choice of kernel order in density estimation, Ann. Stat., 16,161–173.
Hall, P., Minnotte, M.C., and Zhang, C., (2004). Bump hunting with non-Gaussian kernels, Ann. Stat., 32(5), 2124–2141 .
Hall, P., Sheather, S.J., Jones, M.C., and Marron, J.S. (1991). On optimal data-based bandwidth selection in kernel density estimation, Biometrika, 78(2), 263–269 .
Hall, P. and Wand, M.P. (1988). Minimizing mathcal L_1 distance in nonparametric density estimation, J. Multivar. Anal., 26(1), 59–88 .
Hardle, W., Muller, M., Sperlich, S., and Werwatz, A. (2004). Nonparametric and Semiparametric Models, Springer, New York.
Huber, P.J. (1985). Projection pursuit, with discussion, Ann. Stat., 13(2), 435–525 .
Jones, M.C., Marron, J.S., and Sheather, S.J. (1996). A brief survey of bandwidth selection for density estimation, J. Am. Stat. Assoc., 91(433), 401–407 .
Jones, M.C. and Sheather, S.J. (1991). Using nonstochastic terms to advantage in kernel-based estimation of integrated squared density derivatives, Stat. Prob. Lett., 11(6), 511–514 .
Müller, H. (1984). Smooth optimum kernel estimators of densities, regression curves, and modes, Ann. Stat., 12,766–774.
Parzen, E. (1962). On estimation of a probability density function and mode, Ann. Math. Stat., 33,1065–1076.
Rao, B.L.S.P. (1969). Estimation of a unimodal density, Sankhya, Ser. A, 31,23–36.
Revesz, P. (1984). Density estimation, in Handbook of Statistics, Vol. 4, P.R. Krishnaiah and P.K. Sen (eds.), North-Holland, Amsterdam, 531–549.
Rice, J. (1986). Bandwidth choice for differentiation, J. Multivar. Anal., 19(2), 251–264.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function, Ann. Math. Stat., 27,832–835.
Sager, T. (1975). Consistency in nonparametric estimation of the mode, Ann. Stat., 3,698–706.
Schuster, E.F. and Gregory, G.G. (1981). On the inconsistency of maximum likelihood density estimators, in Computer Science and Statistics, Proceedings of the 13th Symposium on Interface, W.F. Eddy Springer, New York, 295–298.
Scott, D.W. (1992). Multivariate Density Estimation: Theory, Practice and Visualization, John Wiley, New York.
Scott, D.W. and Wand, M.P. (1991). Feasibility of multivariate density estimates, Biometrika, 78(1), 197–205 .
Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall, London.
Stone, C.J. (1984). An asymptotically optimal window selection rule for kernel density estimates, Ann. Stat., 12(4), 1285–1297 .
Stone, M. (1974). Cross-validatory choice and assessment of statistical predictions, J.R. Stat. Soc. Ser. B, 36,111–147.
Stuetzle, W. and Mittal, Y. (1979). Some comments on the asymptotic behavior of robust smoothers, in Proceedings of the Heidelberg Workshop, T. Gasser and M. Rosenblatt (eds.), Lecture Notes in Mathematics, Springer, Berlin, 191–195.
Thompson, J. and Tapia, R. (1990). Nonparametric Function Estimation, Modelling and Simulation, SIAM, Philadelphia.
Venter, J. (1967). On estimation of the mode, Ann. Math. Stat., 38,1446–1455.
Wang, X. and Zidek, J.V. (2005). Selecting likelihood weights by cross-validation, Ann. Stat., 33(2), 463–500 .
Wegman, E. (1971). A note on the estimation of the mode, Ann. Math. Stat., 42,1909–1915.
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DasGupta, A. (2008). Density Estimation. In: Asymptotic Theory of Statistics and Probability. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-75971-5_32
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