Simultanous confidence statements for the existence and location of local increases and decreases of a density f, computed via the block procedure.
modeHuntingBlock(X.raw, lower = -Inf, upper = Inf, d0 = 2,
m0 = 10, fm = 2, crit.vals, min.int = FALSE)
The set \(\mathcal{D}^+(\alpha)\) (or \(\bf{D}^+(\alpha)\)).
The set \(\mathcal{D}^-(\alpha)\) (or \(\bf{D}^-(\alpha)\)).
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 (in their respective block) 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 modeHuntingBlock uses the test statistic \(T^+_n({\bf X}, \mathcal{B}_r)\),
where \(\mathcal{B}_r\) contains all intervals of Block \(r\), \(r=1,\ldots,\#blocks\).
Critical values for each block individually are received via finding an \(\tilde \alpha\) such that
$$P(B_n({\bf{X}}) > q_{r,\tilde \alpha / (r+tail)^\gamma} \ for \ at \ least \ one \ r) \le \alpha,$$
where \(q_{r,\alpha}\) is the \((1-\alpha)\)--quantile of the distribution of \(T^+_n({\bf X}, \mathcal{B}_r).\) We then define the sets \(\mathcal{D}^\pm(\alpha)\) as
$$\mathcal{D}^\pm(\alpha) := \Bigl\{\mathcal{I}_{jk} \ : \ \pm T_{jk}({\bf{X}}) > q_{r,\tilde \alpha / (r+tail)^\gamma} \, , \ r = 1,\ldots \#blocks\Bigr\}.$$
Note that \(\gamma\) and \(tail\) are automatically determined by \(crit.vals\).
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, modeHuntingApprox, and cvModeBlock.
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