ReIns (version 1.0.7)

LognormalQQ_der: Derivative plot of the log-normal QQ-plot

Description

Computes the derivative plot of the log-normal QQ-plot. These values can be plotted as a function of the data or as a function of the tail parameter \(k\).

Usage

LognormalQQ_der(data, k = FALSE, plot = TRUE, 
                main = "Derivative plot of log-normal QQ-plot", ...)

Arguments

data

Vector of \(n\) observations.

plot

Logical indicating if the derivative values should be plotted, default is TRUE.

k

Logical indicating if the derivative values are plotted as a function of the tail parameter \(k\) (k=TRUE) or as a function of the logarithm of the data (k=FALSE). Default is FALSE.

main

Title for the plot, default is "Derivative plot of log-normal QQ-plot".

…

Additional arguments for the plot function, see plot for more details.

Value

A list with following components:

xval

Vector of the x-values of the plot (\(k\) or \(\log X_{n-k,n}\)).

yval

Vector of the derivative values.

Details

The derivative plot of a log-normal QQ-plot is $$(k, H_{k,n}/N_{k,n})$$ or $$(\log X_{n-k,n}, H_{k,n}/N_{k,n})$$ with \(H_{k,n}\) the Hill estimates and $$N_{k,n} = (n+1)/(k+1) \phi(\Phi^{-1}(a)) - \Phi^{-1}(a).$$ Here is \(a=1-(k+1)/(n+1)\), \(\phi\) the standard normal PDF and \(\Phi\) the standard normal CDF.

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

References

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

See Also

LognormalQQ, Hill, MeanExcess, ParetoQQ_der, WeibullQQ_der

Examples

# NOT RUN {
data(norwegianfire)

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

# Derivate plot
LognormalQQ_der(norwegianfire$size[norwegianfire$year==76])
# }