Description Usage Arguments Details Value Note Author(s) References See Also Examples
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.
1 | 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 |
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.
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,
http://www.kasparrufibach.ch
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. (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
|
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