Pareto.RdPareto distribution with shape parameter \(\alpha\) and rate parameter \(\lambda\).
dPareto(x, shape, rate = 1/scale, scale = 1/rate) pPareto(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE) expValPareto(shape, rate = 1/scale, scale = 1/rate) varPareto(shape, rate = 1/scale, scale = 1/rate) kthMomentPareto(k, shape, rate = 1/scale, scale = 1/rate) expValLimPareto(d, shape, rate = 1/scale, scale = 1/rate) expValTruncPareto(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE) stopLossPareto(d, shape, rate = 1/scale, scale = 1/rate) meanExcessPareto(d, shape, rate = 1/scale, scale = 1/rate) VatRPareto(kap, shape, rate = 1/scale, scale = 1/rate) TVatRPareto(kap, shape, rate = 1/scale, scale = 1/rate)
| x | vector of quantiles. |
|---|---|
| shape | shape parameter \(\alpha\), must be positive. |
| rate | rate parameter \(\lambda\), must be positive. |
| scale | alternative parameterization to the rate parameter, scale = 1 / rate. |
| q | vector of quantiles. |
| lower.tail | logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\). |
| k | kth-moment. |
| d | cut-off value. |
| less.than.d | logical; if |
| kap | probability. |
Function :
dPareto gives the probability density function (PDF).
pPareto gives the cumulative density function (CDF).
expValPareto gives the expected value.
varPareto gives the variance.
kthMomentPareto gives the kth moment.
expValLimPareto gives the limited mean.
expValTruncPareto gives the truncated mean.
stopLossPareto gives the stop-loss.
meanExcessPareto gives the mean excess loss.
VatRPareto gives the Value-at-Risk.
TVatRPareto gives the Tail Value-at-Risk.
Invalid parameter values will return an error detailing which parameter is problematic.
The Pareto distribution with rate parameter \(\lambda\) as well as shape parameter \(\alpha\) has density: $$f\left(x\right) = \frac{\alpha\lambda^{\alpha}}% {(\lambda + x)^{\alpha + 1}}$$ for \(x \in \mathcal{R}^+\), \(\alpha, \lambda > 0\).
# With scale parameter dPareto(x = 2, shape = 2, scale = 5)#> [1] 0.007513148# With rate parameter dPareto(x = 2, shape = 2, rate = .20)#> [1] 0.007513148# With scale parameter pPareto(q = 2, shape = 2, scale = 5)#> [1] 0.9917355# With rate parameter pPareto(q = 2, shape = 2, rate = 0.20)#> [1] 0.9917355# Survival function pPareto(q = 2, shape = 2, rate = 0.20, lower.tail = FALSE)#> [1] 0.008264463# With scale parameter expValPareto(shape = 5, scale = 0.5)#> [1] 0.5# With rate parameter expValPareto(shape = 5, rate = 2)#> [1] 0.5# With scale parameter varPareto(shape = 5, scale = 0.5)#> [1] 0.4166667# With rate parameter varPareto(shape = 5, rate = 2)#> [1] 0.4166667# With scale parameter kthMomentPareto(k = 4, shape = 5, scale = 0.5)#> [1] 16# With rate parameter kthMomentPareto(k = 4, shape = 5, rate = 2)#> [1] 16# With scale parameter expValLimPareto(d = 4, shape = 5, scale = 0.5)#> [1] 0.4938272# With rate parameter expValLimPareto(d = 4, shape = 5, rate = 2)#> [1] 0.4938272# With scale parameter expValTruncPareto(d = 4, shape = 5, scale = 0.5)#> [1] 0.4773663# With rate parameter expValTruncPareto(d = 4, shape = 5, rate = 2)#> [1] 0.4773663# With scale parameter stopLossPareto(d = 2, shape = 5, scale = 0.5)#> [1] 0.03125# With rate parameter stopLossPareto(d = 2, shape = 5, rate = 2)#> [1] 0.03125# With scale parameter meanExcessPareto(d = 6, shape = 5, scale = 0.5)#> [1] 2# With rate parameter meanExcessPareto(d = 6, shape = 5, rate = 2)#> [1] 2# With scale parameter VatRPareto(kap = .99, shape = 5, scale = 0.5)#> [1] 3.023773# With rate parameter VatRPareto(kap = .99, shape = 5, rate = 2)#> [1] 3.023773# With scale parameter TVatRPareto(kap = .99, shape = 5, scale = 0.5)#> [1] 4.279716# With rate parameter TVatRPareto(kap = .99, shape = 5, rate = 2)#> [1] 4.279716