# bOptEmpProc: Bandwidth Parameter Estimation In npcp: Some Nonparametric CUSUM Tests for Change-Point Detection in Possibly Multivariate Observations

## Description

In the context of the standard CUSUM test based on the sample mean or in a particular empirical process setting, the following functions estimate the bandwidth parameter controlling the serial dependence when generating dependent multiplier sequences using the 'moving average approach'; see Section 5 of the third reference. The function function `bOpt()` is called in the functions `cpMean()`, `cpVar()`, `cpGini()`, `cpAutocov()`, `cpCov()` and `cpTau()` when `b` is set to `NULL`. The function function `bOptEmpProc()` is called in the functions `cpDist()`, `cpCopula()`, `cpAutocop()` and `stDistAutocop()` when `b` is set to `NULL`.

## Usage

 ```1 2 3 4``` ```bOpt(influ, weights = c("parzen", "bartlett")) bOptEmpProc(x, m=5, weights = c("parzen", "bartlett"), L.method=c("max","median","mean","min")) ```

## Arguments

 `influ` a numeric containing the relevant influence coefficients, which, in the case of the standard CUSUM test based on the sample mean, are simply the available observations; see also the last reference. `x` a data matrix whose rows are continuous observations. `weights` a string specifying the kernel for creating the weights used in the generation of dependent multiplier sequences within the 'moving average approach'; see Section 5 of the third reference. `m` a strictly positive integer specifying the number of points of the uniform grid on (0,1)^d (where d is `ncol(x)`) involved in the estimation of the bandwidth parameter; see Section 5 of the third reference. The number of points of the grid is given by `m^ncol(x)` so that `m` needs to be decreased as d increases. `L.method` a string specifying how the parameter `L` involved in the estimation of the bandwidth parameter is computed; see Section 5 of the third reference.

## Details

The implemented approach results from an adaptation of the procedure described in the first two references (see also the references therein). The use of theses functions in a context different from that considered in the third or fourth reference may not be meaningful.

Acknowledgment: Part of the code of the function results from an adaptation of R code of C. Parmeter and J. Racine, itself an adaptation of Matlab code by A. Patton.

## Value

A strictly positive integer.

## References

D.N. Politis and H. White (2004), Automatic block-length selection for the dependent bootstrap, Econometric Reviews 23(1), pages 53-70.

D.N. Politis, H. White and A.J. Patton (2004), Correction: Automatic block-length selection for the dependent bootstrap, Econometric Reviews 28(4), pages 372-375.

A. B<c3><bc>cher and I. Kojadinovic (2016), A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing, Bernoulli 22:2, pages 927-968, http://arxiv.org/abs/1306.3930.

A. B<c3><bc>cher and I. Kojadinovic (2016), Dependent multiplier bootstraps for non-degenerate U-statistics under mixing conditions with applications, Journal of Statistical Planning and Inference 170 pages 83-105, http://arxiv.org/abs/1412.5875.

`cpDist()`, `cpCopula()`, `cpAutocop()`, `stDistAutocop()`, `cpMean()`, `cpVar()`, `cpGini()`, `cpAutocov()`, `cpCov()` and `cpTau()`.