Significance test of rank cross-correlations

Description

Significance test of Spearman's Rho or Kendall's Tau between time series of short-range dependent random variables. The test is based on the asymptotic normal distributions of the estimators.

Usage

corTESTsrd(x, y,
           iid=TRUE, method="spearman",
           alternative="two.sided",
           kernelf=function(z){return(ifelse(abs(z) <= 1, (1 - z^2)^2, 0))},
           bwf=function(n){3*n^(1/4)})

Arguments

Details

Calculates an estimate of the rank correlation coefficient between the inputs x and y, which are assumed to be evenly spaced time series with equal time-increments, and performs a significance test for the rank correlation coefficient with \mathcal{H}_0: \rho_S/\tau=0 against an alternative specified by the user. The function returns the estimate of the rank correlation coefficient and a p-value. Missing observations (NA) are allowed, but will prompt a warning. Ties are not allowed.

The test statistic and the corresponding p-value are based on the distribution of the respective estimator under the assumption of independence between the inputs x and y, and an additional assumption regarding the dependence structure of the inputs on their own past. The distribution of the test statistic is modelled as a normal distribution.

If the option iid is TRUE, the inputs are assumed to be realizations of independent and identically distributed random variables. In this case the asymptotic variance of the test statistic is given by \frac{1}{n-1} for Spearman's Rho and as \frac{2(2n+5)}{9n(n-1)} for Kendall's Tau, see Gibbons and Chakraborti (2003), equations 3.13 and 2.29 in chapter 11, respectively.

If the option iid is FALSE, the inputs are assumed to be realizations of short-range dependent random variables (see Corollary 1 in Lun et al., 2022). The asymptotic variance of the test statistic is modelled as \frac{1}{n}(1+2\sum_{h=1}^{\infty} \rho_S^X(h) \rho_S^Y(h)) for Spearman's Rho and as \frac{4}{9n}(1+2\sum_{h=1}^{\infty} \rho_S^X(h) \rho_S^Y(h)) for Kendall's Tau. Here \rho_S^X(h) refers to the Spearman autocorrelation of the first input x for lag h, and the analogue applies to \rho_S^X(h). In this case the asymptotic variance of the test statistic is estimated (see Corollary 2 in Lun et al., 2022). For this estimation procedure a kernel-function together with a bandwidth is used, which can be specified by the user.

Value

Estimate of rank correlation coefficient and p-value of corresponding hypothesis test.

References

J. D. Gibbons, and S. Chakraborti, Nonparametric statistical inference (4th Edition). CRC press, 2003.

D. Lun, S. Fischer, A. Viglione, and G. Blöschl, Significance testing of rank cross-correlations between autocorrelated time series with short-range dependence, Journal of Applied Statistics, 2022, 1-17. doi: 10.1080/02664763.2022.2137115.

Examples

#Demonstration
sam_size = 50
nsim = 1000

pval_iid <- rep(NA, nsim)
pval_srd <- rep(NA, nsim)
#iid-simulation: if we have iid observations the modified test
#is able to maintain the desired significance level
for(j in c(1:nsim)) {
  x <- rnorm(n=sam_size)
  y <- rnorm(n=sam_size)
  pval_iid[j] <- corTESTsrd(x, y, iid=TRUE, method="spearman")[2]
  pval_srd[j] <- corTESTsrd(x, y, iid=FALSE, method="spearman")[2]
}
sum(pval_iid <= 0.05)/nsim
sum(pval_srd <= 0.05)/nsim

#ar(1)-simulation: if we have srd-observations the modified test
#counteracts the inflation of type-I-errors
for(j in c(1:nsim)) {
  x <- as.numeric(arima.sim(model=list(ar=c(0.8)), n=sam_size))
  y <- as.numeric(arima.sim(model=list(ar=c(0.8)), n=sam_size))
  pval_iid[j] <- corTESTsrd(x, y, iid=TRUE, method="spearman")[2]
  pval_srd[j] <- corTESTsrd(x, y, iid=FALSE, method="spearman")[2]
}
sum(pval_iid <= 0.05)/nsim
sum(pval_srd <= 0.05)/nsim

#the test can be made more conservative be choosing a bigger bandwidth,
#but this decreases the power
bwfbig <- function(n) {10*(n^(1/4))}
#ar(1)-simulation: if we have srd-observations the modified test
#counteracts the inflation of type-I-errors
for(j in c(1:nsim)) {
  x <- as.numeric(arima.sim(model=list(ar=c(0.8)), n=sam_size))
  y <- as.numeric(arima.sim(model=list(ar=c(0.8)), n=sam_size))
  pval_iid[j] <- corTESTsrd(x, y, iid=TRUE, method="spearman")[2]
  pval_srd[j] <- corTESTsrd(x, y, iid=FALSE, method="spearman", bwf=bwfbig)[2]
}
sum(pval_iid<=0.05)/nsim
sum(pval_srd<=0.05)/nsim

Asymptotic variance of estimator of Kendall's Tau between independent iid random variables

Description

Calculates the asymptotic variance of the estimator of Kendall's Tau between iid observations of independent random variables. The expression comes from the asymptotic normal distribution of the estimator, see e.g. equation 2.30 in chapter 11 of Gibbons and Chakraborti (2003).

Usage

kendall_var_independent(n)

Arguments

Value

Asymptotic variance of estimator.

References

J. D. Gibbons, and S. Chakraborti, Nonparametric statistical inference (4th Edition). CRC press, 2003.

Examples

kendall_var_independent(n=50)

Estimator of long-run variance for Kendall's Tau

Description

Estimates the long-run variance of the estimator of Kendall's Tau between short-range dependent observations of independent random variables. The expression comes from the asymptotic normal distribution of estimator, see Corollary 2 in Lun et al. (2022).

Usage

long_run_var_est_kendall_H0(x, y,
           kernelf=function(z) {return(ifelse(abs(z) <= 1,(1 - z^2)^2, 0))},
           bwf=function(nnn){3*nnn^(1/4)})

Arguments

Value

Estimate of long-run variance of estimator.

References

D. Lun, S. Fischer, A. Viglione, and G. Blöschl, Significance testing of rank cross-correlations between autocorrelated time series with short-range dependence, Journal of Applied Statistics, 2022, 1-17. doi: 10.1080/02664763.2022.2137115.

Examples

long_run_var_est_kendall_H0(x=rnorm(50),y=rnorm(50))

Estimator of long-run variance for Spearman's Rho

Description

Estimates the long-run variance of the estimator of Spearman's Rho between short-range dependent observations of independent random variables. The expression comes from the asymptotic normal distribution of the estimator, see Corollary 2 in Lun et al. (2022).

Usage

long_run_var_est_spearman_H0(x, y,
           kernelf=function(z) {return(ifelse(abs(z) <= 1,(1 - z^2)^2, 0))},
           bwf=function(nnn){3*nnn^(1/4)})

Arguments

Value

Estimate of long-run variance of estimator.

References

D. Lun, S. Fischer, A. Viglione, and G. Blöschl, Significance testing of rank cross-correlations between autocorrelated time series with short-range dependence, Journal of Applied Statistics, 2022, 1-17. doi: 10.1080/02664763.2022.2137115.

Examples

long_run_var_est_spearman_H0(x=rnorm(50),y=rnorm(50))

Estimator of Spearman autocorrelations

Description

Computes Spearman-autocorrelations when the observations contain NAs. If the observations do not contain NAs, the function is consistent with the function acf applied to ranks.

Usage

rankacf(x,lag.max=length(x)-2)

Arguments

Value

Estimated spearman autocorrelation up to lag.max.

Examples

x = rnorm(10)
rankacf(x)
acf(rank(x),plot=FALSE,lag.max = length(x)-2)$acf[-1,,1]

Asymptotic variance of estimator of Spearman's Rho between independent iid random variables

Description

Calculates the asymptotic variance of the estimator of Spearman's Rho between iid observations of independent random variables. The expression comes from the asymptotic normal distribution of the estimator, see e.g. equation 3.13 in chapter 11 of Gibbons and Chakraborti (2003).

Usage

spearman_var_independent(n)

Arguments

Value

Asymptotic variance of estimator.

References

J. D. Gibbons, and S. Chakraborti, Nonparametric statistical inference (4th Edition). CRC press, 2003.

Examples

spearman_var_independent(n=50)