modeHunting(X.raw, lower = -Inf, upper = Inf, crit.vals, min.int = FALSE)min.int = TRUE, the set of minimal intervals is output, otherwise all intervals with a test
statistic above the critical value are given.modeHunting, modeHuntingApprox, and
modeHuntingBlock compute for a given level $\alpha \in (0, 1)$ and the corresponding
critical value $c_{jk}(\alpha)$ two sets of intervals
$$\mathcal{D}^\pm(\alpha) = \Bigl{ \mathcal{I}_{jk} \ : \ \pm T_{jk}({\bf{X}} ) > c_{jk}(\alpha) \Bigr}$$
where $\mathcal{I}_{jk}:=(X_{(j)},X_{(k)})$ for $0\le j < k \le n+1, k-j> 1$ and $c_{jk}$ are
appropriate critical values.
Specifically, the function modeHunting computes $\mathcal{D}^\pm(\alpha)$ based on the two
test statistics
$$T_n^+({\bf{X}}, \mathcal{I}) = \max_{(j,k) \in \mathcal{I}} \Bigl( |T_{jk}({\bf{X}})| / \sigma_{jk} - \Gamma \Bigl(\frac{k-j}{n+2}\Bigr)\Bigr)$$
and
$$T_n({\bf{X}}, \mathcal{I}) = \max_{(j,k) \in \mathcal{I}} ( |T_{jk}({\bf{X}})| / \sigma_{jk} ),$$
using the set $\mathcal{I} := \mathcal{I}_{all}$ of all intervals spanned by two observations
$(X_{(j)}, X_{(k)})$:
$$\mathcal{I}_{all} = \Bigl{(j, \ k ) \ : \ 0 \le j < k \le n+1, \ k - j > 1\Bigr}.$$
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 $\sigma_{jk} := \sqrt{(k-j-1)/3}$ and the additive correction term
$\Gamma(\delta) := \sqrt{2 \log(e / \delta)}$ for $\delta > 0$.
If min.int = TRUE, the set $\mathcal{D}^\pm(\alpha)$ is replaced by the set ${\bf{D}}^\pm(\alpha)$
of its minimal elements. An interval $J \in \mathcal{D}^\pm(\alpha)$ is called minimal if
$\mathcal{D}^\pm(\alpha)$ 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
$\mathcal{D}^\pm(\alpha) \setminus \bf{D}^\pm(\alpha)$ do not contain relevant information.modeHuntingApprox, modeHuntingBlock, and cvModeAll.## for examples type
help("mode hunting")
## and check the examples there}
<keyword>htest</keyword>
<keyword>nonparametric</keyword>Run the code above in your browser using DataLab