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, q

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.

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