ParetoQQ function

Pareto quantile plot

Computes the empirical quantiles of the log-transform of a data vector and the theoretical quantiles of the standard exponential distribution. These quantiles are then plotted in a Pareto QQ-plot with the theoretical quantiles on the $x$-axis and the empirical quantiles on the $y$-axis.

ParetoQQ(data, plot = TRUE, main = "Pareto QQ-plot", ...)

Arguments

• data: Vector of $n$ observations.
• plot: Logical indicating if the quantiles should be plotted in a Pareto QQ-plot, default is TRUE.
• main: Title for the plot, default is "Pareto QQ-plot".
• ...: Additional arguments for the plot function, see plot for more details.

Details

It can be easily seen that a log-transformed Pareto random variable is exponentially distributed. We can hence obtain a Pareto QQ-plot from an exponential QQ-plot by replacing the empirical quantiles from the data vector by the empirical quantiles from the log-transformed data. We hence plot

for $i=1,...,n,$

with $X_{i,n}$ the $i$-th order statistic of the data.

See Section 4.1 of Albrecher et al. (2017) for more details.

Returns

A list with following components: - pqq.the: Vector of the theoretical quantiles from a standard exponential distribution.

• pqq.emp: Vector of the empirical quantiles from the log-transformed data.

References

Albrecher, H., Beirlant, J. and Teugels, J. (2017). Reinsurance: Actuarial and Statistical Aspects, Wiley, Chichester.

Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.

Author(s)

Tom Reynkens based on S-Plus code from Yuri Goegebeur.

See Also

ParetoQQ_der, ExpQQ, genQQ, LognormalQQ, WeibullQQ

Examples

data(norwegianfire)

# Exponential QQ-plot for Norwegian Fire Insurance data for claims in 1976.
ExpQQ(norwegianfire$size[norwegianfire$year==76])

# Pareto QQ-plot for Norwegian Fire Insurance data for claims in 1976.
ParetoQQ(norwegianfire$size[norwegianfire$year==76])