modeHuntingBlock {modehunt}R Documentation

Multiscale analysis of a density via block procedure

Description

Simultanous confidence statements for the existence and location of local increases and decreases of a density f, computed via the block procedure.

Usage

modeHuntingBlock(X.raw, lower = -Inf, upper = Inf, d0 = 2, 
    m0 = 10, fm = 2, crit.vals, min.int = FALSE)

Arguments

X.raw

Vector of observations.

lower

Lower support point of ff, if known.

upper

Upper support point of ff, if known.

d0

Initial parameter for the grid resolution.

m0

Initial parameter for the number of observations in one block.

fm

Factor by which mm is increased from block to block.

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 (in their respective block) are given.

Details

See blocks for details how Iapp\mathcal{I}_{app} is generated and modeHunting for a proper introduction to the notation used here. The function modeHuntingBlock uses the test statistic Tn+(X,Br)T^+_n({\bf X}, \mathcal{B}_r), where Br\mathcal{B}_r contains all intervals of Block rr, r=1,,#blocksr=1,\ldots,\#blocks. Critical values for each block individually are received via finding an α~\tilde \alpha such that

P(Bn(X)>qr,α~/(r+tail)γ for at least one r)α,P(B_n({\bf{X}}) > q_{r,\tilde \alpha / (r+tail)^\gamma} \ for \ at \ least \ one \ r) \le \alpha,

where qr,αq_{r,\alpha} is the (1α)(1-\alpha)–quantile of the distribution of Tn+(X,Br).T^+_n({\bf X}, \mathcal{B}_r). We then define the sets D±(α)\mathcal{D}^\pm(\alpha) as

D±(α):={Ijk : ±Tjk(X)>qr,α~/(r+tail)γ, r=1,#blocks}.\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 γ\gamma and tailtail are automatically determined by crit.valscrit.vals.

If min.int = TRUE, the set D±(α)\mathcal{D}^\pm(\alpha) is replaced by the set D±(α){\bf{D}}^\pm(\alpha) of its minimal elements. An interval JD±(α)J \in \mathcal{D}^\pm(\alpha) is called minimal if D±(α)\mathcal{D}^\pm(\alpha) contains no proper subset of JJ. 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 ff as precisely as possible, the intervals in D±(α)D±(α)\mathcal{D}^\pm(\alpha) \setminus \bf{D}^\pm(\alpha) do not contain relevant information.

Value

Dp

The set D+(α)\mathcal{D}^+(\alpha) (or D+(α)\bf{D}^+(\alpha)).

Dm

The set D(α)\mathcal{D}^-(\alpha) (or D(α)\bf{D}^-(\alpha)).

Note

Critical values for some combinations of nn and α\alpha are provided in the data sets cvModeBlock. Critical values for other values of nn and α\alpha can be generated using criticalValuesApprox.

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.

See Also

modeHunting, modeHuntingApprox, and cvModeBlock.

Examples

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

[Package modehunt version 1.0.7 Index]