Compound Negative Binomial Distribution

Computes various risk measures (mean, variance, Value-at-Risk (VatR), and Tail Value-at-Risk (TVatR)) for the compound Negative Binomial distribution.

pCompNBinom(
  x,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

expValCompNBinom(
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

varCompNBinom(
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  distr_severity = "Gamma"
)

VatRCompNBinom(
  kap,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

TVatRCompNBinom(
  kap,
  vark,
  size,
  prob,
  shape,
  rate = 1/scale,
  scale = 1/rate,
  k0,
  distr_severity = "Gamma"
)

Arguments

x	vector of quantiles.
size	Number of successful trials.
prob	Probability of success in each trial.
shape	shape parameter \(\alpha\), must be positive.
rate	rate parameter \(\beta\), must be positive.
scale	alternative parameterization to the rate parameter, scale = 1 / rate.
k0	point up to which to sum the distribution for the approximation.
distr_severity	Choice of severity distribution. "gamma" (default) "lognormal" only for the expected value and variance.
kap	probability.
vark	Value-at-Risk (VaR) calculated at the given probability kap.

Value

Function :

• pCompNBinom gives the cumulative density function.

• expValCompNBinom gives the expected value.

• varCompNBinom gives the variance.

• TVatRCompNBinom gives the Tail Value-at-Risk.

• VatRCompNBinom gives the Value-at-Risk.

Returned values are approximations for the cumulative density function, TVatR, and VatR.

Details

The compound negative binomial distribution has density ....

Examples

pCompNBinom(x = 2, size = 1, prob = 0.2, shape = log(1000) - 0.405,
          rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
#> [1] 0.2004906


expValCompNBinom(size = 4, prob = 0.2, shape = 0, scale = 1,
         distr_severity = "Lognormal")
#> [1] 26.37954

varCompNBinom(size = 1, prob = 0.2, shape = log(1000) - 0.405, rate = 0.9^2,
          distr_severity = "Lognormale")
#> [1] 5011632

VatRCompNBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
#> [1] 7.833102

vark_calc <- VatRCompNBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59,
            rate = 0.9^2, k0 = 1E2, distr_severity = "Gamma")
TVatRCompNBinom(kap = 0.9, size = 1, prob = 0.2, shape = 0.59, rate = 0.9^2,
            vark = vark_calc, k0 = 1E2, distr_severity = "Gamma")
#> [1] 11.75336