Journal ArticleDOI
Some asymptotics for multimodality tests based on kernel density estimates
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An asymptotic formula is given for the expected number of modes of a kernel density estimator, and this establishes the rate of convergence of the critically smoothed bandwidth.Abstract:
A test due to B.W. Silverman for modality of a probability density is based on counting modes of a kernel density estimator, and the idea of critical smoothing. An asymptotic formula is given for the expected number of modes. This, together with other methods, establishes the rate of convergence of the critically smoothed bandwidth. These ideas are extended to provide insight concerning the behaviour of the test based on bootstrap critical values.read more
Citations
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SiZer for Exploration of Structures in Curves
TL;DR: Assessment of Significant ZERo crossings of derivatives results in the SiZer map, a graphical device for display of significance of features with respect to both location and scale.
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Scale space view of curve estimation
TL;DR: In this paper, the authors study the scale space surface from a statistical viewpoint and provide new insights into nonparametric smoothing procedures and yields useful techniques for statistical exploration of features in the data.
Journal ArticleDOI
Multiscale testing of qualitative hypotheses
Lutz Dümbgen,Vladimir Spokoiny +1 more
TL;DR: In this paper, the authors propose two classes of tests of qualitative nonparametric hypotheses about f such as monotonicity or concavity, constructed via a new class of multiscale statistics and an extension of Levy's modulus of continuity of Brownian motion.
Journal ArticleDOI
A comparative study of several smoothing methods in density estimation
TL;DR: A critical up-to-date review of the main methods currently available can be found in this article, where the authors provide some new insights on the important problem of estimating the minimization criteria and on the choice of pilot bandwidths in bootstrap-based methods.
Journal ArticleDOI
Testing Monotonicity of Regression
TL;DR: In this paper, a test of monotonicity of a regression function is proposed based on the size of a critical bandwidth, the amount of smoothing necessary to force a nonparametric regression estimate to be monotone.
References
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BookDOI
Density estimation for statistics and data analysis
TL;DR: The Kernel Method for Multivariate Data: Three Important Methods and Density Estimation in Action.
Journal ArticleDOI
The Dip Test of Unimodality
J. A. Hartigan,P. M. Hartigan +1 more
TL;DR: The dip test as mentioned in this paper measures multimodality in a sample by the maximum difference, over all sample points, between the empirical distribution function, and the unimodal distribution function that minimizes that maximum difference.
Journal ArticleDOI
Weak and Strong Uniform Consistency of the Kernel Estimate of a Density and its Derivatives
TL;DR: In this article, the estimation of a density and its derivatives by the kernel method is considered, and uniform consistency properties over the whole real line are studied under certain conditions on the density and on the behavior of the window width which are necessary and sufficient for weak and strong uniform consistency of the estimate of the density derivatives.
Journal ArticleDOI
Estimating a Smooth Monotone Regression Function
TL;DR: In this paper, the problem of estimating a smooth monotone regression function $m$ is studied, where the estimator is composed of a smoothing step and an isotonisation step.