# cpAutocop: Test for Change-Point Detection in Univariate Observations... In npcp: Some Nonparametric CUSUM Tests for Change-Point Detection in Possibly Multivariate Observations

## Description

Nonparametric test for change-point detection particularly sensitive to changes in the autocopula of univariate continuous observations. Approximate p-values for the test statistic are obtained by means of a multiplier approach. Details can be found in the first reference.

## Usage

 ```1 2 3``` ```cpAutocop(x, lag = 1, b = NULL, bivariate = FALSE, weights = c("parzen", "bartlett"), m = 5, N = 1000, init.seq = NULL, include.replicates = FALSE) ```

## Arguments

 `x` a one-column matrix containing continuous observations. `lag` an integer specifying at which lag to consider the autocopula; the autocopula is a (`lag+1`)-dimensional copula. `b` strictly positive integer specifying the value of the bandwidth parameter determining the serial dependence when generating dependent multiplier sequences using the 'moving average approach'; see Section 5 of the second reference. If set to `NULL`, `b` will be estimated using the function `bOptEmpProc()`; see the first reference. `bivariate` a logical specifying whether the test should focus only on the bivariate margin of the (`lag+1`)-dimensional autocopula obtained from the first and the last dimension. `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 second reference. `m` a strictly positive integer specifying the number of points of the uniform grid on (0,1) involved in the estimation of the bandwidth parameter; see Section 5 of the second reference. `N` number of multiplier replications. `init.seq` a sequence of independent standard normal variates of length `N * (nrow(x) - lag + 2 * (b - 1))` used to generate dependent multiplier sequences. `include.replicates` a logical specifying whether the object of `class` `htest` returned by the function (see below) will include the multiplier replicates.

## Details

The approximate p-value is computed as

(0.5 + sum(S[i] >= S, i=1, .., N)) / (N+1),

where S and S[i] denote the test statistic and a multiplier replication, respectively. This ensures that the approximate p-value is a number strictly between 0 and 1, which is sometimes necessary for further treatments.

## Value

An object of `class` `htest` which is a list, some of the components of which are

 `statistic` value of the test statistic. `p.value` corresponding approximate p-value. `cvm` the values of the `length(x)-lag-1` intermediate Cramér-von Mises change-point statistics; the test statistic is defined as the maximum of those. `b` the value of parameter `b`.

## Note

This is a tests for a continuous univariate time series.

## References

A. Bücher, J.-D. Fermanian and I. Kojadinovic (2019), Combining cumulative sum change-point detection tests for assessing the stationarity of univariate time series, Journal of Time Series Analysis 40, pages 124-150, http://arxiv.org/abs/1709.02673.

A. Bücher 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.

`cpAutocov()` for a related test based on the autocovariance.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```## AR1 example n <- 200 k <- n/2 ## the true change-point x <- matrix(c(arima.sim(list(ar = -0.5), n = k), arima.sim(list(ar = 0.5), n = n - k))) cp <- cpAutocop(x) cp ## Estimated change-point which(cp\$cvm == max(cp\$cvm)) ## AR2 example n <- 200 k <- n/2 ## the true change-point x <- matrix(c(arima.sim(list(ar = c(0,-0.5)), n = k), arima.sim(list(ar = c(0,0.5)), n = n - k))) cpAutocop(x) cpAutocop(x, lag = 2) cpAutocop(x, lag = 2, bivariate = TRUE) ```

### Example output

```	Test of change-point detection sensitive to changes in the
2-dimensional autocopula

data:  x
cvmmax = 13.264, p-value = 0.001499

cvm95
95

Test of change-point detection sensitive to changes in the
2-dimensional autocopula

data:  x
cvmmax = 3.11, p-value = 0.2792

Test of change-point detection sensitive to changes in the
3-dimensional autocopula

data:  x
cvmmax = 13.641, p-value = 0.02448

Test of change-point detection sensitive to changes in the bivariate
serial copula at lag 2

data:  x
cvmmax = 23.072, p-value = 0.005495
```

npcp documentation built on July 16, 2020, 5:07 p.m.