Truncated Pareto quantile plot
Extension of the Pareto QQ-plot as described in Beirlant et al. (2016).
trParetoQQ(data, r = 1, DT, kstar = NULL, plot = TRUE, main = "TPa QQ-plot", ...)
data |
Vector of n observations. |
r |
Trimming parameter, default is |
DT |
Vector of n-1 estimates for the truncation odds D_T obtained from |
kstar |
Value for k used to construct the plot. When |
plot |
Logical indicating if the quantiles should be plotted in a Pareto QQ-plot, default is |
main |
Title for the plot, default is |
... |
Additional arguments for the |
The Pareto QQ-plot for truncated data plots
(-\log(\hat{D}_{T,r,k^*,n}+j/(n+1)), \log(X_{n-j+1,n}) )
for j=1,…,n.
The value for k^* can be be given by the user or can be determined automatically. In the latter case, we use the k^* that maximises the absolute value of the correlation between -\log(\hat{D}_{T,r,k^*,n}+j/(n+1)) and \log(X_{n-j+1,n}) for j=1,…,k and k^*>10.
When taking D_T=0, one obtains the ordinary Pareto QQ-plot.
Note that the definition here differs slightly from the one in Beirlant et al. (2016). We plot the empirical and theoretical quantiles the other way around and therefore have to add a minus (before the log).
See Beirlant et al. (2016) for more details.
A list with following components:
pqq.the |
Vector of theoretical quantiles -\log(\hat{D}_{T,r,k^*,n}+j/(n+1)), see Details. |
pqq.emp |
Vector of the empirical quantiles from the log-transformed data. |
kstar |
Optimal value for k or input argument |
DTstar |
Estimate of D_T corresponding to |
Tom Reynkens.
Beirlant, J., Fraga Alves, M.I. and Gomes, M.I. (2016). "Tail fitting for Truncated and Non-truncated Pareto-type Distributions." Extremes, 19, 429–462.
# Endpoint of truncated Pareto distribution endpoint <- qpareto(0.99, shape=2) # Generate sample from truncated Pareto distribution X <- rtpareto(1000, shape=2, endpoint=endpoint) # Ordinary Pareto QQ-plot ParetoQQ(X) # Truncated Hill estimates gamma <- trHill(X)$gamma # Estimates for truncation odds dt <- trDT(X, gamma=gamma)$DT # Truncated Pareto QQ-plot trParetoQQ(X, DT=dt)
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