modeHunting {modehunt}

Multiscale analysis of a density on all possible intervals

Description

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.

Usage

modeHunting(X.raw, lower = -Inf, upper = Inf, crit.vals, min.int = FALSE)

Arguments

X.raw

Vector of observations.

lower

Lower support point of f, if known.

upper

Upper support point of f, if known.

crit.vals

2-dimensional vector giving the critical values for the desired level.

min.int

If min.int = TRUE, the set of minimal intervals is output, otherwise all intervals with a test statistic above the critical value are given.

Details

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.

Value

Dp

The set \mathcal{D}^+(\alpha) (or \bf{D}^+(\alpha)), based on the test statistic with additive correction \Gamma.

Dm

The set \mathcal{D}^-(\alpha) (or \bf{D}^-(\alpha)), based on the test statistic with \Gamma.

Dp.noadd

The set \mathcal{D}^+(\alpha) (or \bf{D}^+(\alpha)), based on the test statistic without \Gamma.

Dm.noadd

The set \mathcal{D}^+(\alpha) (or \bf{D}^-(\alpha)), based on the test statistic without \Gamma.

Note

Critical values for modeHunting and some combinations of n and \alpha are provided in the data set cvModeAll. Critical values for other values of n and \alpha can be generated using criticalValuesAll.

Parts of this function were derived from MatLab code provided on Lutz Duembgen's webpage,
http://www.staff.unibe.ch/duembgen.

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Guenther Walther, [e-mail],
www-stat.stanford.edu/~gwalther

References

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.

See Also

modeHuntingApprox, modeHuntingBlock, and cvModeAll.

Examples

## for examples type
help("mode hunting")
## and check the examples there