# Serial Independence Test for Continuous Time Series Via Empirical Copula

### Description

Computes the serial independence test based on the empirical copula
process as proposed in Ghoudi et al.(2001) and Genest and Rémillard (2004).
The test, which is the serial analog of `indepTest`

, can be
seen as composed of three steps:

- (i)
a simulation step, which consists in simulating the distribution of the test statistics under serial independence for the sample size under consideration;

- (ii)
the test itself, which consists in computing the approximate p-values of the test statistics with respect to the empirical distributions obtained in step (i);

- (iii)
the display of a graphic, called a

*dependogram*, enabling to understand the type of departure from serial independence, if any.

More details can be found in the articles cited in the reference section.

### Usage

1 2 | ```
serialIndepTestSim(n, lag.max, m=lag.max+1, N=1000, verbose = interactive())
serialIndepTest(x, d, alpha=0.05)
``` |

### Arguments

`n` |
length of the time series when simulating the distribution of the test statistics under serial independence. |

`lag.max` |
maximum lag. |

`m` |
maximum cardinality of the subsets of 'lags' for which a test
statistic is to be computed. It makes sense to consider |

`N` |
number of repetitions when simulating under serial independence. |

`verbose` |
a logical specifying if progress
should be displayed via |

`x` |
numeric vector containing the time series whose serial independence is to be tested. |

`d` |
object of class |

`alpha` |
significance level used in the computation of the critical values for the test statistics. |

### Details

See the references below for more details, especially the third and fourth ones.

### Value

The function `serialIndepTestSim()`

returns an object of S3 class
`"serialIndepTestDist"`

with list components `sample.size`

,
`lag.max`

, `max.card.subsets`

, `number.repetitons`

,
`subsets`

(list of the subsets for which test statistics have
been computed), `subsets.binary`

(subsets in binary 'integer'
notation), `dist.statistics.independence`

(a `N`

line matrix
containing the values of the test statistics for each subset and each
repetition) and `dist.global.statistic.independence`

(a vector a
length `N`

containing the values of the serial version of the
global Cramér-von Mises test statistic for each repetition — see
last reference p.175).

The function `serialIndepTest()`

returns an object of S3 class
`"indepTest"`

with list components `subsets`

,
`statistics`

, `critical.values`

, `pvalues`

,
`fisher.pvalue`

(a p-value resulting from a combination *à la*
Fisher of the subset statistic p-values), `tippett.pvalue`

(a
p-value resulting from a combination *à la* Tippett of the subset
statistic p-values), `alpha`

(global significance level of the
test), `beta`

(1 - `beta`

is the significance level per
statistic), `global.statistic`

(value of the global Cramér-von
Mises statistic derived directly from the serial independence
empirical copula process — see last reference p 175) and
`global.statistic.pvalue`

(corresponding p-value).

The former argument `print.every`

is deprecated and not
supported anymore; use `verbose`

instead.

### References

Deheuvels, P. (1979). La fonction de dépendance empirique et ses
propriétés: un test non paramétrique d'indépendance, *Acad. Roy.
Belg. Bull. Cl. Sci.*, 5th Ser. 65:274–292.

Deheuvels, P. (1981), A non parametric test for independence,
*Publ. Inst. Statist. Univ. Paris*. 26:29–50.

Genest, C. and Rémillard, B. (2004)
Tests of independence and randomness based on the empirical copula
process.
*Test* **13**, 335–369.

Genest, C., Quessy, J.-F., and Rémillard, B. (2006)
Local efficiency of a Cramer-von Mises test of independence.
*Journal of Multivariate Analysis* **97**, 274–294.

Genest, C., Quessy, J.-F., and Rémillard, B. (2007)
Asymptotic local efficiency of Cramér-von Mises tests for multivariate
independence. *The Annals of Statistics* **35**, 166–191.

### See Also

`indepTest`

,
`multIndepTest`

,
`multSerialIndepTest`

,
`dependogram`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | ```
## AR 1 process
ar <- numeric(200)
ar[1] <- rnorm(1)
for (i in 2:200)
ar[i] <- 0.5 * ar[i-1] + rnorm(1)
x <- ar[101:200]
## In order to test for serial independence, the first step consists
## in simulating the distribution of the test statistics under
## serial independence for the same sample size, i.e. n=100.
## As we are going to consider lags up to 3, i.e., subsets of
## {1,...,4} whose cardinality is between 2 and 4 containing {1},
## we set lag.max=3. This may take a while...
d <- serialIndepTestSim(100,3)
## The next step consists in performing the test itself:
test <- serialIndepTest(x,d)
## Let us see the results:
test
## Display the dependogram:
dependogram(test,print=TRUE)
## NB: In order to save d for future use, the saveRDS() function can be used.
``` |