Distributacalcul (version 0.4.0)

Pareto: Pareto Distribution

Description

Pareto distribution with shape parameter \(\alpha\) and rate parameter \(\lambda\).

Usage

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)

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.

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.

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)

# With rate parameter
dPareto(x = 2, shape = 2, rate = .20)

# With scale parameter
pPareto(q = 2, shape = 2, scale = 5)

# With rate parameter
pPareto(q = 2, shape = 2, rate = 0.20)

# Survival function
pPareto(q = 2, shape = 2, rate = 0.20, lower.tail = FALSE)

# With scale parameter
expValPareto(shape = 5, scale = 0.5)

# With rate parameter
expValPareto(shape = 5, rate = 2)

# With scale parameter
varPareto(shape = 5, scale = 0.5)

# With rate parameter
varPareto(shape = 5, rate = 2)

# With scale parameter
kthMomentPareto(k = 4, shape = 5, scale = 0.5)

# With rate parameter
kthMomentPareto(k = 4, shape = 5, rate = 2)

# With scale parameter
expValLimPareto(d = 4, shape = 5, scale = 0.5)

# With rate parameter
expValLimPareto(d = 4, shape = 5, rate = 2)

# With scale parameter
expValTruncPareto(d = 4, shape = 5, scale = 0.5)

# With rate parameter
expValTruncPareto(d = 4, shape = 5, rate = 2)

# With scale parameter
stopLossPareto(d = 2, shape = 5, scale = 0.5)

# With rate parameter
stopLossPareto(d = 2, shape = 5, rate = 2)

# With scale parameter
meanExcessPareto(d = 6, shape = 5, scale = 0.5)

# With rate parameter
meanExcessPareto(d = 6, shape = 5, rate = 2)

# With scale parameter
VatRPareto(kap = .99, shape = 5, scale = 0.5)

# With rate parameter
VatRPareto(kap = .99, shape = 5, rate = 2)

# With scale parameter
TVatRPareto(kap = .99, shape = 5, scale = 0.5)

# With rate parameter
TVatRPareto(kap = .99, shape = 5, rate = 2)