Now let’s plot the dual bound \(D(s)\) for various \(\theta\) (this checks the outer root-finding). We also see that \(s\) has to be sufficiently large in order to find a root \(h(s,t) = 0\) for \(t < s/d\). D s/d) res }) f <- sapply( seq_along(s), function(i) sapply(t, function(t.) qrmtools ::: dual_bound_2(s, t = t., d = d, pF = pF2) - qrmtools ::: dual_bound_2_deriv_term(s, t = t., d = d, pF = pF2))) palette <- colorRampPalette( c( "maroon3", "darkorange2", "royalblue3"), space = "Lab") cols <- palette( 6) if(doPDF) pdf( file = (file <- paste0( "fig_worst_VaR_hom_dual_h_Par=2_d=",d, ".pdf")), width = 6, height = 6) plot(t, f, type = "l", log = "x", xlim = range(t), ylim = range(f), col = cols, xlab = "t", ylab = expression( "h(s,t) for d = 8 and F being Par(2)")) lines(t, f, col = cols) lines(t, f, col = cols) lines(t, f, col = cols) lines(t, f, col = cols) lines(t, f, col = cols) abline( h = 0, lty = 2) legend( "topright", lty = rep( 1, 6), col = cols, bty = "n", legend = as.expression( lapply( 1 : 6, function(i) substitute(s =s., list( s.
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