Simultanous confidence statements for the existence and location of local increases and decreases of a density f, computed on the approximating set of intervals.
modeHuntingApprox(X.raw, lower = -Inf, upper = Inf,
d0 = 2, m0 = 10, fm = 2, crit.vals, min.int = FALSE)
The set \(\mathcal{D}^+(\alpha)\) (or \(\bf{D}^+(\alpha)\)), based on the test statistic with additive correction \(\Gamma\).
The set \(\mathcal{D}^-(\alpha)\) (or \(\bf{D}^-(\alpha)\)), based on the test statistic with \(\Gamma\).
The set \(\mathcal{D}^+(\alpha)\) (or \(\bf{D}^+(\alpha)\)), based on the test statistic without \(\Gamma\).
The set \(\mathcal{D}^+(\alpha)\) (or \(\bf{D}^-(\alpha)\)), based on the test statistic without \(\Gamma\).
Vector of observations.
Lower support point of \(f\), if known.
Upper support point of \(f\), if known.
Initial parameter for the grid resolution.
Initial parameter for the number of observations in one block.
Factor by which \(m\) is increased from block to block.
2-dimensional vector giving the critical values for the desired level.
If min.int = TRUE, the set of minimal intervals is output, otherwise all intervals with a test
statistic above the critical value are given.
Kaspar Rufibach, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch
Guenther Walther, gwalther@stanford.edu,
https://gwalther.su.domains/
See blocks for details how \(\mathcal{I}_{app}\) is generated and modeHunting for
a proper introduction to the notation used here.
The function modeHuntingApprox computes \(\mathcal{D}^\pm(\alpha)\) based on the two
test statistics \(T_n^+({\bf{X}}, \mathcal{I}_{app})\) and \(T_n({\bf{X}}, \mathcal{I}_{app})\).
If min.int = TRUE, the set \(\mathcal{D}^\pm(\alpha)\) is replaced by the set \({\bf{D}}^\pm(\alpha)\)
of its minimal elements. An interval \(J \in \mathcal{D}^\pm(\alpha)\) is called minimal if
\(\mathcal{D}^\pm(\alpha)\) contains no proper subset of \(J\). This minimization post-processing
step typically massively reduces the number of intervals. If we are mainly interested in locating the ranges
of increases and decreases of \(f\) as precisely as possible, the intervals in
\(\mathcal{D}^\pm(\alpha) \setminus \bf{D}^\pm(\alpha)\) do not contain relevant information.
Duembgen, L. and Walther, G. (2008). Multiscale Inference about a density. Ann. Statist., 36, 1758--1785.
Rufibach, K. and Walther, G. (2010). A general criterion for multiscale inference. J. Comput. Graph. Statist., 19, 175--190.
modeHunting, modeHuntingBlock, and cvModeApprox.
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help("mode hunting")
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