Pareto 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)

Arguments

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 TRUE (default) truncated mean for values <= d, otherwise, for values > d.

kap

probability.

Value

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.

Details

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 \(x \in \mathcal{R}^+\), \(\alpha, \lambda > 0\).

Examples

# 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