# Test on the effect of concomitant covariate on the extremes of the response variable Given observed data, perform a Kolmogorov-Smirnov type test comparing the cumulative distribution function of the concomitant covariate, defined as $X \mid Y > t$, with $t$ being the threshold, against the cumulative distribution function of the random vector of covariate. ```r scedastic.test(data, k, M = 1000L, xg, ng, bayes = TRUE, C = 5L, alpha = 0.05) ``` ## Arguments - `data`: design matrix of dimension `n` by `2` containing the complete data for the dependent variable (first column) and covariate (second column) in [0,1] - `k`: integer, number of exceedances for the generalized Pareto - `M`: integer, number of samples to draw from the posterior distrinution of the law of the concomitant covariate. Default: 1000 - `xg`: vector of covariate grid of dimension `ng` by `1` containing a sequence between zero and the last value of the corresponding covariate - `ng`: length of covariate grid - `bayes`: logical indicating the bootstrap method. If `FALSE`, a frequentist bootstrap on the empirical cumulative distribution function of the concomitant covariate is performed. Default to `TRUE` - `C`: integer, hypermparameter entering the posterior distributyion of the law of the concomitant covariate. Default: 5 - `alpha`: double, significance level for the critical value of the test, computed as the $(1-alpha)$ level empirical quantile of the sample of distances between the empirical cumulative distribution function of the concomitant and complete covariate. Default: 0.05 ## Returns a list with components * `Delta` maximum observed distance between the empirical distribution functions of the concomitant and complete covariate * `DeltaM` vector of length M containing the sample of maximum distances between the empirical distribution function of the concomitant complete covariate * `critical` double, critical value for the test statistic, computed as the $(1-alpha)$ level empirical quantile of DeltaM * `pval` double, p-value ## Examples ```r ## Not run: # generate data set.seed(1234) n <- 500 samp <- evd::rfrechet(n,0,1:n,4) # set effective sample size and threshold k <- 50 threshold <- sort(samp,decreasing = TRUE)[k+1] # preliminary mle estimates of scale and shape parameters mlest <- evd::fpot(samp, threshold, control=list(maxit = 500)) # empirical bayes procedure proc <- estPOT( samp, k = k, pn = c(0.01, 0.005), type = "continuous", method = "bayesian", prior = "empirical", start = as.list(mlest$estimate), sig0 = 0.1) # conditional predictive density estimation yg <- seq(0, 50, by = 2) nyg <- length(yg) # estimation of scedasis function # setting M <- 1e3 C <- 5 alpha <- 0.05 bw <- .5 nsim <- 5000 burn <- 1000 # create covariate # in sample obs n_in = n # number of years ahead nY = 1 n_out = 365 * nY # total obs n_tot = n_in + n_out # total covariate (in+out sample period) x <- seq(0, 1, length = n_tot) # in sample grid dimension for covariate ng_in <- 150 xg <- seq(0, x[n_in], length = ng_in) # in+out of sample grid xg <- c(xg, seq(x[n_in + 1], x[(n_tot)], length = ng_in)) # in+out sample grid dimension nxg <- length(xg) xg <- array(xg, c(nxg, 1)) # in sample observations samp_in <- samp[1:n_in] ssamp_in <- sort(samp_in, decreasing = TRUE, index = TRUE) x_in <- x[1:n_in] # in sample covariate xs <- x_in[ssamp_in$ix[1:k]] # in sample concomitant covariate # test on covariate effect test <- scedastic.test( cbind(samp, x[1:n]), k, M, array(xg[1:ng_in], c(ng_in, 1)), ng_in, TRUE, C, 0.05 ) ## End(Not run) ``` ## References Dombry, C., S. Padoan and S. Rizzelli (2025). Asymptotic theory for Bayesian inference and prediction: from the ordinary to a conditional Peaks-Over-Threshold method, arXiv:2310.06720v2.

scedastic.test function