Mode testing, critical bandwidth and excess mass

The identification of peaks or maxima in probability densities, by mode testing or bump hunting, has become an important problem in applied fields. For real random variables, this task has been approached in the statistical literature from different perspectives, with the proposal of testing procedures which are based on kernel density estimators or on the quantification of excess mass. However, none of the existing proposals for testing the number of modes provides a satisfactory performance in practice. In this work, a new procedure which combines the previous approaches (smoothing and excess mass) is presented together with a revision on the previous proposals. The new method is compared with the existing ones in an extensive simulation study, showing a superior behaviour, with good calibration and power results. Theoretical justification for its performance is also obtained. A real data example on philatelic data is also included for illustration purposes, revising previous approaches and discussing the results with the new procedure.

[1]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[2]  R. Tibshirani,et al.  An introduction to the bootstrap , 1993 .

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

[4]  James Stephen Marron,et al.  Mode testing via the excess mass estimate , 2001 .

[5]  Ming-Yen Cheng,et al.  Calibrating the excess mass and dip tests of modality , 1998 .

[6]  K. Roeder.  Density estimation with confidence sets exemplified by superclusters and voids in the galaxies , 1990 .

[7]  James Stephen Marron,et al.  Some asymptotics for multimodality tests based on kernel density estimates , 1992 .

[8]  Matthew P. Wand,et al.  Kernel Smoothing , 1995 .

[9]  David J. Marchette,et al.  The Bumpy Road to the Mode Forest , 1998 .

[10]  A. Izenman,et al.  Philatelic Mixtures and Multimodal Densities , 1988 .

[11]  D. W. Scott,et al.  The Mode Tree: A Tool for Visualization of Nonparametric Density Features , 1993 .

[12]  P. Deb.  Finite Mixture Models , 2008 .

[13]  G. Sawitzki,et al.  Excess Mass Estimates and Tests for Multimodality , 1991 .

[14]  G. McLachlan,et al.  Modelling the distribution of stamp paper thickness via finite normal mixtures: The 1872 Hidalgo stamp issue of Mexico revisited , 1997 .

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

[16]  Jude F. Mitchell,et al.  Differential Attention-Dependent Response Modulation across Cell Classes in Macaque Visual Area V4 , 2007, Neuron.

[17]  I. Salgado-Ugarte,et al.  Nonparametric assessment of multimodality for univariate data , 1998 .

[18]  J. Marron,et al.  Simultaneous Density Estimation of Several Income Distributions , 1992, Econometric Theory.

[19]  J. Hartigan,et al.  The Dip Test of Unimodality , 1985 .

[20]  J. Marron,et al.  SiZer for Exploration of Structures in Curves , 1999 .

[21]  Joseph P. Romano.  On weak convergence and optimality of kernel density estimates of the mode , 1988 .

[22]  Uwe Einmahl,et al.  Uniform in bandwidth consistency of kernel-type function estimators , 2005 .

[23]  Julian D. Olden,et al.  Small fish, big fish, red fish, blue fish: size-biased extinction risk of the world's freshwater and marine fishes. , 2007 .

[24]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[25]  M. York,et al.  ON THE CALIBRATION OF SILVERMAN'S TEST FOR MULTIMODALITY , 2001 .