Kernel density estimation via diffusion

[1] Abramson, I. S. (1982). On bandwidth variation in kernel estimates—a square root law. <i>Ann. Statist.</i> <b>10</b> 1217–1223.

[2] Azencott, R. (1984). Density of diffusions in small time: Asymptotic expansions. In <i>Seminar on Probability, XVIII. Lecture Notes in Math.</i> <b>1059</b> 402–498. Springer, Berlin.

[4] Botev, Z. I. (2007). Kernel density estimation using Matlab. Available at <a href="http://www.mathworks.us/matlabcentral/fileexchange/authors/27236">http://www.mathworks.us/matlabcentral/fileexchange/authors/27236</a>.

[5] Botev, Z. I. (2007). Nonparametric density estimation via diffusion mixing. Technical report, Dept. Mathematics, Univ. Queensland. Available at <a href="http://espace.library.uq.edu.au">http://espace.library.uq.edu.au</a>.

[6] Chaudhuri, P. and Marron, J. S. (2000). Scale space view of of curve estimation. <i>Ann. Statist.</i> <b>28</b> 408–428.

[7] Choi, E. and Hall, P. (1999). Data sharpening as a prelude to density estimation. <i>Biometrika</i> <b>86</b> 941–947.

[8] Cohen, J. K., Hagin, F. G. and Keller, J. B. (1972). Short time asymptotic expansions of solutions of parabolic equations. <i>J. Math. Anal. Appl.</i> <b>38</b> 82–91.

[9] Csiszár, I. (1972). A class of measures of informativity of observation channels. <i>Period. Math. Hungar.</i> <b>2</b> 191–213.

[10] Devrôye, L. (1997). Universal smoothing factor selection in density estimation: Theory and practice. <i>Test</i> <b>6</b> 223–320.

[13] Feller, W. (1952). The parabolic differential equations and the associated semi-groups of transformations. <i>Ann. of Math. (2)</i> <b>55</b> 468–519.

[15] Hall, P. (1990). On the bias of variable bandwidth curve estimators. <i>Biometrika</i> <b>77</b> 523–535.

[16] Hall, P., Hu, T. C. and Marron, J. S. (1995). Improved variable window kernel estimates of probability densities. <i>Ann. Ststist.</i> <b>23</b> 1–10.

[17] Hall, P. and Marron, J. S. (1987). Estimation of integrated squared density derivatives. <i>Statist. Probab. Lett.</i> <b>6</b> 109–115.

[18] Hall, P. and Minnotte, M. C. (2002). High order data sharpening for density estimation. <i>J. R. Stat. Soc. Ser. B Stat. Methodol.</i> <b>64</b> 141–157.

[19] Hall, P. and Park, B. U. (2002). New methods for bias correction at endpoints and boundaries. <i>Ann. Statist.</i> <b>30</b> 1460–1479.

[20] Hall, P. and Park, B. U. (2002). New methods for bias correction at endpoints and boundaries. <i>Ann. Statist.</i> <b>30</b> 1460–1479.

[21] Havrda, J. H. and Charvat, F. (1967). Quantification methods of classification processes: Concepts of structural <i>α</i> entropy. <i>Kybernetika (Prague)</i> <b>3</b> 30–35.

[22] Jones, M. C. and Foster, P. J. (1996). A simple nonnegative boundary correction method for kernel density estimation. <i>Statist. Sinica</i> <b>6</b> 1005–1013.

[23] Jones, M. C., Marron, J. S. and Park, B. U. (1991). A simple root n bandwidth selector. <i>Ann. Statist.</i> <b>19</b> 1919–1932.

[24] Jones, M. C., Marron, J. S. and Sheather, S. J. (1993). Simple boundary correction for kernel density estimation. <i>Statist. Comput.</i> <b>3</b> 135–146.

[25] Jones, M. C., Marron, J. S. and Sheather, S. J. (1996). A brief survey of bandwidth selection for density estimation. <i>J. Amer. Statist. Assoc.</i> <b>91</b> 401–407.

[26] Jones, M. C., Marron, J. S. and Sheather, S. J. (1996). Progress in data-based bandwidth selection for kernel density estimation. <i>Comput. Statist.</i> <b>11</b> 337–381.

[27] Jones, M. C., McKay, I. J. and Hu, T. C. (1994). Variable location and scale kernel density estimation. <i>Ann. Inst. Statist. Math.</i> <b>46</b> 521–535.

[28] Jones, M. C. and Signorini, D. F. (1997). A comparison of higher-order bias kernel density estimators. <i>J. Amer. Statist. Assoc.</i> <b>92</b> 1063–1073.

[29] Kannai, Y. (1977). Off diagonal short time asymptotics for fundamental solutions of diffusion equations. <i>Comm. Partial Differential Equations</i> <b>2</b> 781–830.

[31] Karunamuni, R. J. and Alberts, T. (2005). A generalized reflection method of boundary correction in kernel density estimation. <i>Canad. J. Statist.</i> <b>33</b> 497–509.

[32] Karunamuni, R. J. and Zhang, S. (2008). Some improvements on a boundary corrected kernel density estimator. <i>Statist. Probab. Lett.</i> <b>78</b> 499–507.

[33] Kerm, P. V. (2003). Adaptive kernel density estimation. <i>Statist. J.</i> <b>3</b> 148–156.

[35] Ladyženskaja, O. A., Solonnikov, V. A. and Ural’ceva, N. N. (1967). <i>Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs</i> <b>23</b> xi+648. Amer. Math. Soc., Providence, RI.

[37] Lehmann, E. L. (1990). Model specification: The views of fisher and neyman, and later developments. <i>Statist. Sci.</i> <b>5</b> 160–168.

[38] Loader, C. R. (1999). Bandwidth selection: Classical or plug-in. <i>Ann. Statist.</i> <b>27</b> 415–438.

[39] Loftsgaarden, D. O. and Quesenberry, C. P. (1965). A nonparametric estimate of a multivariate density function. <i>Ann. Math. Statist.</i> <b>36</b> 1049–1051.

[40] Marron, J. S. (1985). An asymptotically efficient solution to the bandwidth problem of kernel density estimation. <i>Ann. Statist.</i> <b>13</b> 1011–1023.

[41] Marron, J. S. and Ruppert, D. (1996). Transformations to reduce boundary bias in kernel density-estimation. <i>J. Roy. Statist. Soc. Ser. B</i> <b>56</b> 653–671.

[42] Marron, J. S. and Wand, M. P. (1992). Exact mean integrated error. <i>Ann. Statist.</i> <b>20</b> 712–736.

[43] Molchanov, S. A. (1975). Diffusion process and Riemannian geometry. <i>Russian Math. Surveys</i> <b>30</b> 1–63.

[44] Park, B. U., Jeong, S. O. and Jones, M. C. (2003). Adaptive variable location kernel density estimators with good performance at boundaries. <i>J. Nonparametr. Stat.</i> <b>15</b> 61–75.

[45] Park, B. U. and Marron, J. S. (1990). Comparison of data-driven bandwidith selectors. <i>J. Amer. Statist. Assoc.</i> <b>85</b> 66–72.

[46] Samiuddin, M. and El-Sayyad, G. M. (1990). On nonparametric kernel density estimates. <i>Biometrika</i> <b>77</b> 865.

[48] Sheather, S. J. and Jones, M. C. (1991). A reliable data-based bandwidth selection method for kernel density estimation. <i>J. Roy. Statist. Soc. Ser. B</i> <b>53</b> 683–690.

[51] Terrell, G. R. and Scott, D. W. (1992). Variable kernel density estimation. <i>Ann. Statist.</i> <b>20</b> 1236–1265.

[52] Wand, M. P. and Jones, M. C. (1994). Multivariate plug-in bandwidth selection. <i>Comput. Statist.</i> <b>9</b> 97–117.

[3] Bellman, R. (1961). <i>A Brief Introduction to Theta Functions</i>. Holt, Rinehart and Winston, New York.

[11] Doucet, A., de Freitas, N. and Gordon, N. (2001). <i>Sequential Monte Carlo Methods in Practice</i>. Springer, New York.

[12] Ethier, S. N. and Kurtz, T. G. (2009). <i>Markov Processes. Characterization and Convergence</i>. Wiley, New York.

[14] Friedman, A. (1964). <i>Partial Differential Equations of Parabolic Type</i>. Prentice Hall, Englewood Cliffs, NJ.

[30] Kapur, J. N. and Kesavan, H. K. (1987). <i>Generalized Maximum Entropy Principle (With Applications)</i>. Standford Educational Press, Waterloo, ON.

[34] Kloeden, P. E. and Platen, E. (1999). <i>Numerical Solution of Stochastic Differential Equations</i>. Springer, Berlin.

[36] Larsson, S. and Thomee, V. (2003). <i>Partial Differential Equations with Numerical Methods</i>. Springer, Berlin.

[47] Scott, D. W. (1992). <i>Multivariate Density Estimation. Theory, Practice and Visualization</i>. Wiley, New York.

[49] Silverman, B. W. (1986). <i>Density Estimation for Statistics and Data Analysis</i>. Chapman and Hall, London.

[50] Simonoff, J. S. (1996). <i>Smoothing Methods in Statistics</i>. Springer, New York.

[53] Wand, M. P. and Jones, M. C. (1995). <i>Kernel Smoothing</i>. Chapman and Hall, London.