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 f, if known.
upper Upper support point of f, if known.
d0 Initial parameter for the grid resolution.
m0 Initial parameter for the number of observations in one block.
fm Factor by which m 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 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,...,#blocks. Critical values for each block individually are received via finding an tilde α such that

P(B_n({bf{X}}) > q_{r,tilde α / (r+tail)^gamma} for at least one r) <= α,

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

mathcal{D}^pm(α) := Bigl{mathcal{I}_{jk} : pm T_{jk}({bf{X}}) > q_{r,tilde α / (r+tail)^gamma} , , r = 1,... #blocksBigr}.

Note that gamma and tail are automatically determined by crit.vals.

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

Note

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

Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com

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. (2007). A general criterion for multiscale inference. Preprint, Department of Statistics, Stanford University.

See Also

modeHunting, modeHuntingApprox, and cvModeBlock.

Examples

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

[Package modehunt version 1.0.4 Index]