Simultanous confidence statements for the existence and location of local increases and decreases of a density f, computed on all intervals spanned by two observations.
modeHunting(X.raw, lower = -Inf, upper = Inf, 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.
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/
In general, the methods modeHunting, modeHuntingApprox, and
modeHuntingBlock compute for a given level \(\alpha \in (0, 1)\) and the corresponding
critical value \(c_{jk}(\alpha)\) two sets of intervals
$$\mathcal{D}^\pm(\alpha) = \Bigl\{ \mathcal{I}_{jk} \ : \ \pm T_{jk}({\bf{X}} ) > c_{jk}(\alpha) \Bigr\}$$
where \(\mathcal{I}_{jk}:=(X_{(j)},X_{(k)})\) for \(0\le j < k \le n+1, k-j> 1\) and \(c_{jk}\) are appropriate critical values.
Specifically, the function modeHunting computes \(\mathcal{D}^\pm(\alpha)\) based on the two
test statistics
$$T_n^+({\bf{X}}, \mathcal{I}) = \max_{(j,k) \in \mathcal{I}} \Bigl( |T_{jk}({\bf{X}})| / \sigma_{jk} - \Gamma \Bigl(\frac{k-j}{n+2}\Bigr)\Bigr)$$
and
$$T_n({\bf{X}}, \mathcal{I}) = \max_{(j,k) \in \mathcal{I}} ( |T_{jk}({\bf{X}})| / \sigma_{jk} ),$$
using the set \(\mathcal{I} := \mathcal{I}_{all}\) of all intervals spanned by two observations \((X_{(j)}, X_{(k)})\):
$$\mathcal{I}_{all} = \Bigl\{(j, \ k ) \ : \ 0 \le j < k \le n+1, \ k - j > 1\Bigr\}.$$
We introduced the local test statistics
$$T_{jk}({\bf{X}}) := \sum_{i=j+1}^{k-1} ( 2 X_{(i; j, k)} - 1) 1\{X_{(i; j, k)} \in (0,1)\},$$
for local order statistics
$$X_{(i; j, k)} := \frac{X_{(i)}-X_{(j)}}{X_{(k)} - X_{(j)}},$$
the standard deviation \(\sigma_{jk} := \sqrt{(k-j-1)/3}\) and the additive correction term \(\Gamma(\delta) := \sqrt{2 \log(e / \delta)}\) for \(\delta > 0\).
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.
modeHuntingApprox, modeHuntingBlock, and cvModeAll.
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