Some Movies illustrating Exact MISE in Kernel Density Estimation Frederic Udina, March 1995. J.S. Marron and M.P. Wand published in 1992 a paper in The Annals of Statistics about exact computation of the MISE. This is from his abstract: An exact and easily computable expression for the mean integrated squared error (MISE) for the kernel density estimator of a general normal mixture density, is given for Gaussian kernels of arbitrary order. This provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis. The family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes. [...] In joint work with J. S. Marron, animated versions of some of the figures in the paper were built, using the sample size as the evolving quantity over time. We present it here, along with directions for accessing it from the WWWeb (or by FTP). Following you will find: 1.- Some relevant facts from the Marron-Wand paper. 2.- How to get the animation files. 3.- How to be able to display animations. 4.- References ************ 1.- SOME RELEVANT FACTS FROM THE MARRON-WAND PAPER. In theorem 2.1. for a given normal mixture density and a Gaussian kernel (of a given order), the MISE(h,n) is given for each bandwidth h and sample size n. It is given as a sum of a squared bias and a variance part. In section 3, a collection of particular normal mixture densities is described. The corresponding density function plots are shown in the paper's figure 1. In section 4 a detailed study about the exact MISE for these densities is conducted. We take figure 2 or 3 as the starting point for our movies. In these figures, MISE(h) is plotted against Log10(h) for density number 11 and sample size 100. The MISE(h) is shown together with its components IV(h) and ISB(h) (integrated variance and squared bias). Our movies are made by building graphs similar to figure 2, taking Log(MISE(h)) in the vertical axis and making the sample size vary between one and one million. 2.- HOW TO GET THE ANIMATION FILES. Our files can be accessed by ftp or by WWW-http. Detailed instructions follow at the end of this section. There are files, with suffix .mov, containing animations in QuickTime format. In the subdirectory density-plots there are postscript files containing the graphs of the 15 density functions. All the files are compressed using the GNU compressor gzip. So, the files end with '.gz'. Only a few of them are also in uncompressed form. By ftp: ftp halley.upf.es cd stat/movies/Marron-Wand-92-Densities get dens#.mov.gz or prompt mget dens*.mov.gz or even cd density-plots get Marron-Wand-Density-1.ps.gz Using Mosaic or any WWW client: http://halley.upf.es/stat/movies/Marron-Wand-92-Densities You can get Mosaic configured so that when clicking our files, Mosaic automatically download it, uncompress it and call the appropriate viewer for showing you the movie. Help from a sysadmin could be useful here. 3.- HOW TO DISPLAY ANIMATIONS. This depends on the machine and the operating system you are using. We recommend the Cross-platform WWW page: http://emb121.rh.psu.edu/xplat/xplat.html where a lot of information about doing multi-media on different systems is kept. Unix/X-Windows: You can get 'xanim', a movie player by Mark Podlipec. To get xanim, try: ftp.shell.portal.com:/pub/podlipec/xanim26977.tar.Z Unix compressed ftp.shell.portal.com:/pub/podlipec/xanim26977.tar.gz GNU zipped or http://www.portal.com/~podlipec/home.html MS-Windows: Apple Computer has a QuickTime for Windows kit. You can find it on several Mac servers. Macintosh: Simple Player is a standard and free software for displaying Quick Time animations. Popcorn(tm) is a good player too. 4.- REFERENCES J. S. Marron and M. P. Wand, ``Exact Mean Integrated Squared Error'' The annals of statistics 1992, Vol. 20, No. 2, 712--736. Frederic Udina udina@upf.es http://libiya.upf.es/