# modeHunting: Multiscale analysis of a density on all possible intervals In modehunt: Multiscale Analysis for Density Functions

## 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

 1 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 α \in (0, 1) and the corresponding critical value c_{jk}(α) two sets of intervals

\mathcal{D}^\pm(α) = \Bigl\{ \mathcal{I}_{jk} \ : \ \pm T_{jk}({\bf{X}} ) > c_{jk}(α) \Bigr\}

where \mathcal{I}_{jk}:=(X_{(j)},X_{(k)}) for 0≤ j < k ≤ n+1, k-j> 1 and c_{jk} are appropriate critical values.

Specifically, the function modeHunting computes \mathcal{D}^\pm(α) based on the two test statistics

T_n^+({\bf{X}}, \mathcal{I}) = \max_{(j,k) \in \mathcal{I}} \Bigl( |T_{jk}({\bf{X}})| / σ_{jk} - Γ \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}})| / σ_{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 ≤ j < k ≤ n+1, \ k - j > 1\Bigr\}.

We introduced the local test statistics

T_{jk}({\bf{X}}) := ∑_{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 σ_{jk} := √{(k-j-1)/3} and the additive correction term Γ(δ) := √{2 \log(e / δ)} for δ > 0.

If min.int = TRUE, the set \mathcal{D}^\pm(α) is replaced by the set {\bf{D}}^\pm(α) of its minimal elements. An interval J \in \mathcal{D}^\pm(α) is called minimal if \mathcal{D}^\pm(α) 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(α) \setminus \bf{D}^\pm(α) do not contain relevant information.

## Value

 Dp The set \mathcal{D}^+(α) (or \bf{D}^+(α)), based on the test statistic with additive correction Γ. Dm The set \mathcal{D}^-(α) (or \bf{D}^-(α)), based on the test statistic with Γ. Dp.noadd The set \mathcal{D}^+(α) (or \bf{D}^+(α)), based on the test statistic without Γ. Dm.noadd The set \mathcal{D}^+(α) (or \bf{D}^-(α)), based on the test statistic without Γ.

## Note

Critical values for modeHunting and some combinations of n and α are provided in the data set cvModeAll. Critical values for other values of n and α 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, kaspar.rufibach@gmail.com,
http://www.kasparrufibach.ch

Guenther Walther, gwalther@stanford.edu,
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.

modeHuntingApprox, modeHuntingBlock, and cvModeAll.
 1 2 3 ## for examples type help("mode hunting") ## and check the examples there