ADCFplot {dCovTS}

Auto-distance correlation plot

Description

The function plots the estimated auto-distance correlation function obtained by ADCF and provides confindence intervals by employing three bootstrap based methods.

Usage

ADCFplot(x, MaxLag = 15, alpha = 0.05, b = 499, bootMethod =
c("Wild Bootstrap", "Subsampling", "Independent Bootstrap"), ylim = NULL, main = NULL)

Arguments

x

A numeric vector or univariate time series.

MaxLag

The maximum lag order at which to plot ADCF. Default is 15.

alpha

The significance level used to construct the (1-\alpha)% empirical critical values.

b

The number of bootstrap replications for constructing the (1-\alpha)% empirical critical values. Default is 499.

bootMethod

A character string indicating the method to use for obtaining the (1-\alpha)% critical values. Possible choices are "Wild Bootstrap" (the default), "Independent Bootstrap" and "Subsampling".

ylim

A numeric vector of length 2 indicating the y limits of the plot. The default value, NULL, indicates that the range (0,v), where v is the maximum number between 1 and the empirical critical values, should be used.

main

The title of the plot.

Details

Fokianos and Pitsillou (2018) showed that the sample auto-distance covariance function ADCV (and thus ADCF) can be expressed as a V-statistic of order two, which under the null hypothesis of independence is degenerate. Thus, constructing a plot analogous to the traditional autocorrelation plot where the confidence intervals are obtained simultaneously, turns to be a complicated task. To overcome this issue, the (1-\alpha)% confidence intervals shown in the plot (dotted blue horizontal line) are computed simultaneously via Monte Carlo simulation, and in particular via the independent wild bootstrap approach (Dehling and Mikosch, 1994; Shao, 2010; Leucht and Neumann, 2013). The reader is referred to Fokianos and Pitsillou (2018) for the steps followed. mADCFplot returns an analogous plot of the estimated auto-distance correlation function for a multivariate time series.

One can also compute the pairwise (1-\alpha)% critical values via the subsampling approach suggested by Zhou (2012, Section 5.1).That is, the critical values are obtained at each lag separately. The block size of the procedure is based on the minimum volatility method proposed by Politis et al. (1999, Section 9.4.2). In addition, the function provides the ordinary independent bootstrap methodology to derive simultaneous (1-\alpha)% critical values.

Value

A plot of the estimated ADCF values. It also returns a list including:

ADCF

The sample auto-distance correlation function for all lags specified by MaxLag.

bootMethod

The method followed for computing the (1-\alpha)% confidence intervals of the plot.

critical.value

The critical value shown in the plot.

Note

When the critical values are obtained via the Subsampling methodology, the function returns a plot that starts from lag 1.

The function plots only the biased estimator of ADCF.

Author(s)

Maria Pitsillou, Michail Tsagris and Konstantinos Fokianos.

References

Dehling, H. and T. Mikosch (1994). Random quadratic forms and the bootstrap for U-statistics. Journal of Multivariate Analysis, 51, 392-413.

Dominic, E, K. Fokianos and M. Pitsillou Maria (2019). An Updated Literature Review of Distance Correlation and Its Applications to Time Series. International Statistical Review, 87, 237-262.

Fokianos K. and Pitsillou M. (2018). Testing independence for multivariate time series via the auto-distance correlation matrix. Biometrika, 105, 337-352.

Leucht, A. and M. H. Neumann (2013). Dependent wild bootstrap for degenerate U- and V- statistics. Journal of Multivariate Analysis, 117, 257-280.

Pitsillou M. and Fokianos K. (2016). dCovTS: Distance Covariance/Correlation for Time Series. R Journal, 8, 324-340.

Politis, N. P., J. P. Romano and M. Wolf (1999). Subsampling. New York: Springer.

Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105, 218-235.

Zhou, Z. (2012). Measuring nonlinear dependence in time series, a distance correlation approach. Journal of Time Series Analysis, 33, 438-457.

See Also

ADCF, ADCV, mADCFplot

Examples

### x <- rnorm(200)
### ADCFplot(x, bootMethod = "Subs")