Limited mean of the Gamma distribution

Limited mean of the Gamma distribution with shape parameter \(\alpha\) and rate parameter \(\beta\).

Elim_gamma(d, shape, rate = 1/scale, scale = 1/rate)

Arguments

d	cut-off value.
shape	shape parameter \(\alpha\), must be positive integer.
rate	\(\beta\) is the rate parameter, must be positive.
scale	alternative parameterization to rate parameter, scale = 1 / rate.

Value

Function :

• MGF_gamma gives the moment generating function (MGF).

• E_gamma gives the expected value.

• V_gamma gives the variance.

• kthmoment_gamma gives the kth moment.

• Etrunc_gamma gives the truncated mean.

• SL_gamma gives the stop-loss.

• Elim_gamma gives the limited mean.

• Mexcess_gamma gives the mean excess loss.

• TVaR_gamma gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Details

The Gamma distribution with shape parameter \(\alpha\) and rate parameter \(\beta\) has density: $$f\left(x\right) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1}% \textrm{e}^{-\beta x}$$ for \(x \in \mathcal{R}^+\), \(\beta, \alpha > 0\).

See also

Other Gamma Distribution: E_gamma(), Etrunc_gamma(), MGF_gamma(), Mexcess_gamma(), SL_gamma(), TVaR_gamma(), V_gamma(), VaR_gamma(), kthmoment_gamma()

Examples

# With scale parameter
Elim_gamma(d = 2, shape = 3, scale = 4)
#> [1] 1.992244

# With rate parameter
Elim_gamma(d = 2, shape = 3, rate = 0.25)
#> [1] 1.992244