quad function

Quadratic distribution

Computes the pdf, cdf, value at risk and expected shortfall for the quadratic distribution given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef(x) = \alpha (x - \beta)^2,\\&\displaystyleF(x) = \frac {\alpha}{3} \left[ (x - \beta)^3 + (\beta - a)^3 \right],\\&\displaystyle{\rm VaR}_p (X) = \beta + \left[ \frac {3 p}{\alpha} - (\beta - a)^3 \right]^{1/3},\\&\displaystyle{\rm ES}_p (X) = \beta + \frac {\alpha}{4 p} \left\{ \left[ \frac {3 p}{\alpha} - (\beta - a)^3 \right]^{4/3} - (\beta - a)^4 \right\}\end{array} [REMOVE_ME_2]$

for $a \leq x \leq b$, $0 < p < 1$, $-\infty < a < \infty$ , the first location parameter, and $-\infty < a < b < \infty$, the second location parameter, where $\alpha = \frac {12}{(b - a)^3}$ and $\beta = \frac {a + b}{2}$.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the quadratic distribution given by

for $a \leq x \leq b$, $0 < p < 1$, $-\infty < a < \infty$ , the first location parameter, and $-\infty < a < b < \infty$, the second location parameter, where $\alpha = \frac {12}{(b - a)^3}$ and $\beta = \frac {a + b}{2}$.

dquad(x, a=0, b=1, log=FALSE)
pquad(x, a=0, b=1, log.p=FALSE, lower.tail=TRUE)
varquad(p, a=0, b=1, log.p=FALSE, lower.tail=TRUE)
esquad(p, a=0, b=1)

Arguments

• x: scaler or vector of values at which the pdf or cdf needs to be computed
• p: scaler or vector of values at which the value at risk or expected shortfall needs to be computed
• a: the value of the first location parameter, can take any real value, the default is zero
• b: the value of the second location parameter, can take any real value but must be greater than a, the default is 1
• log: if TRUE then log(pdf) are returned
• log.p: if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
• lower.tail: if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Returns

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

x=runif(10,min=0,max=1)
dquad(x)
pquad(x)
varquad(x)
esquad(x)