Loglogistic Distribution

Loglogistic distribution with shape parameter \(\tau\) and scale parameter \(\lambda\).

dLlogis(x, shape, rate = 1/scale, scale = 1/rate)

pLlogis(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE)

expValLlogis(shape, rate = 1/scale, scale = 1/rate)

varLlogis(shape, rate = 1/scale, scale = 1/rate)

kthMomentLlogis(k, shape, rate = 1/scale, scale = 1/rate)

expValLimLlogis(d, shape, rate = 1/scale, scale = 1/rate)

expValTruncLlogis(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE)

stopLossLlogis(d, shape, rate = 1/scale, scale = 1/rate)

meanExcessLlogis(d, shape, rate = 1/scale, scale = 1/rate)

VatRLlogis(kap, shape, rate = 1/scale, scale = 1/rate)

TVatRLlogis(kap, shape, rate = 1/scale, scale = 1/rate)

Arguments

x

vector of quantiles.
shape

shape parameter \(\tau\), must be positive
rate

rate parameter \(\beta\), 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 :

  • dLlogis gives the probability density function (PDF).

  • pLlogis gives the cumulative density function (CDF).

  • expValLlogis gives the expected value.

  • varLlogis gives the variance.

  • kthMomentLlogis gives the kth moment.

  • expValLimLlogis gives the limited mean.

  • expValTruncLlogis gives the truncated mean.

  • stopLossLlogis gives the stop-loss.

  • meanExcessLlogis gives the mean excess loss.

  • VatRLlogis gives the Value-at-Risk.

  • TVatRLlogis gives the Tail Value-at-Risk.

Invalid parameter values will return an error detailing which parameter is problematic.

Details

The loglogistic distribution with shape parameter \(\tau\) and scale parameter \(\lambda\) has density: $$\frac{\tau \lambda^\tau x^{\tau -1}}{(\lambda^{\tau }+x^{\tau })^{2}}$$ for \(x \in \mathcal{R}^+\), \(\lambda, \tau > 0\).

Examples

dLlogis(x = 2, shape = 2, scale = 4)
#> [1] 0.01514793

# With scale parameter
pLlogis(q = 3, shape = 3, scale = 5)
#> [1] 0.9997038

# With rate parameter
pLlogis(q = 3, shape = 3, rate = 0.2)
#> [1] 0.9997038

# Survival function
pLlogis(q = 3, shape = 3, rate = 0.2, lower.tail = FALSE)
#> [1] 0.0002962085

expValLlogis(shape = 2, scale = 4)
#> [1] 6.283185

varLlogis(shape = 3, scale = 4)
#> [1] 15.29977

kthMomentLlogis(k = 3, shape = 5, scale = 4)
#> [1] 126.8454

expValLimLlogis(d = 2, shape = 2, scale = 4)
#> [1] 1.85459

# With rate parameter
expValTruncLlogis(d = 2, shape = 2, scale = 4)
#> [1] 0.2545904

# Values greater than d
expValTruncLlogis(d = 2, shape = 2, scale = 4, less.than.d = FALSE)
#> [1] 6.028595

stopLossLlogis(d = 2, shape = 2, scale = 4)
#> [1] 4.428595

meanExcessLlogis(d = 3, shape = 2, scale = 4)
#> [1] 5.795595

# With scale parameter
VatRLlogis(kap = .2, shape = 2, scale = 4)
#> [1] 2

# With rate parameter
VatRLlogis(kap = .2, shape = 2, rate = 0.25)
#> [1] 2

# With scale parameter
TVatRLlogis(kap = .2, shape = 3, scale = 4)
#> [1] 5.588911

# With rate parameter
TVatRLlogis(kap = .2, shape = 3, rate = 0.25)
#> [1] 5.588911