modeHunting {modehunt} | R Documentation |
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.
modeHunting(X.raw, lower = -Inf, upper = Inf, crit.vals, min.int = FALSE)
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. |
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.
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. |
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.
Kaspar Rufibach, kaspar.rufibach@gmail.com
Guenther Walther, gwalther@stanford.edu,
www-stat.stanford.edu/~gwalther
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.
modeHuntingApprox
, modeHuntingBlock
, and cvModeAll
.
## for examples type help("mode hunting") ## and check the examples there