# Computes number of observations for each block

### Description

In Rufibach and Walther (2010) a new multiscale mode hunting procedure is presented that compares the local test statistics with critical values given by blocks. Blocks are collection of intervals on a given grid that contain roughly the same number of original observations.

### Usage

1 | ```
blocks(n, m0 = 10, fm = 2)
``` |

### Arguments

`n` |
Number of observations. |

`m0` |
Initial parameter that determines the number of observations in one block. |

`fm` |
Factor by which |

### Details

In our block procedure, we only consider a subset *\mathcal{I}_{app}* of all possible intervals
*\mathcal{I}_{all}* where

*\mathcal{I}_{all} = \Bigl\{(j, \ k ) \ : \ 0 ≤ j < k ≤ n+1, \ k - j > 1\Bigr\}.*

This subset *\mathcal{I}_{app}* is computed as follows:

Set *d_1, m_1, f_m > 1*. Then:

*for \ \ r = 1,…,\#blocks*

*d_r := round(d_1 f_m^{(r-1)/2}), \ m_r := m_1 f_m^{r-1}.*

Include *(j,k)* in *\mathcal{I}_{app}* if

(a) *j, k \in \{1+i d_r, \ i = 0, 1, … \}* \ \ (we only consider every *d*–th observation) and

(b) *m_r ≤ k-j-1 ≤ 2m_r-1* \ \ (*\mathcal{I}_{jk}* contains between *m_r* and *2m_r - 1* observations)

*end \ \ for*

### Value

*b \times 2*–matrix, where *b* is the number of blocks and the columns contain the lower
and the upper number of observations that form each block.

### Note

The asymptotic results in Rufibach and Walther (2010) are only derived for *f_m = 2*.

### Author(s)

Kaspar Rufibach, kaspar.rufibach@gmail.com,

http://www.kasparrufibach.ch

Guenther Walther, gwalther@stanford.edu,

www-stat.stanford.edu/~gwalther

### References

Rufibach, K. and Walther, G. (2010).
A general criterion for multiscale inference.
*J. Comput. Graph. Statist.*, **19**, 175–190.

### See Also

This function is called by `modeHuntingBlock`

.