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)vector of quantiles.
shape parameter \(\alpha\), must be positive.
rate parameter \(\lambda\), must be positive.
alternative parameterization to the rate parameter, scale = 1 / rate.
logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).
kth-moment.
cut-off value.
logical; if TRUE (default) truncated mean for values <= d, otherwise, for values > d.
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