modeHunting {modehunt}R Documentation

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 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 > 1Bigr}.

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 σ_{jk} := sqrt{(k-j-1)/3} and the additive correction term Γ(delta) := sqrt{2 log(e / delta)} for delta > 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

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

modeHuntingApprox, modeHuntingBlock, and cvModeAll.

Examples

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

[Package modehunt version 1.0.4 Index]