LogisticBetaDistribution

LogisticBeta(α, β)

The Logistic-Beta distribution, also known Generalized Logistic distribution of Type IV.

\[f(x) = \frac{1}{\mathrm{B}(\alpha,\beta)} \frac{e^{-\beta x}}{(1 + e^{-x})^{(\alpha+\beta)}}, \quad -\infty < x < \infty\]

The distribution is defined by:

If Y ∼ Beta(α, β), then X = log(Y/(1-Y)) ∼ LogisticBeta(α, β).

The general LogisticBeta(α, β, μ, σ) is obtained by the location-scale construction:

LogisticBeta(α, β, μ, σ) = μ + σ*LogisticBeta(α, β)

d = LogisticBeta(1/2, 1/2)   # LogisticBeta distribution with shape parameters α and β

params(d)          # Get the parameters, i.e. (α and β)
pdf(d, 0.5)        # Probability density function at x = 0.5
mean(d)            # Mean 

External links

• Logistic-Beta distribution on Wikipedia

Examples

julia> d = LogisticBeta(1/2, 1/2)
julia> rand(d, 4)'
1×4 adjoint(::Vector{Float64}) with eltype Float64:
 1.00851  0.640297  0.566234  2.16941
julia> pdf(d, 1)
julia> cdf(d, 1)
julia> dist_general = 3 + 2*LogisticBeta(1/2, 1/2)
julia> pdf(dist_general, 1)

rand(d::LogisticBeta[, n::Integer])

Draw n random numbers from the logistic-beta distribution d.

cdf(d::LogisticBeta, x::Real)

Compute the cdf of the logistic-beta distribution d at x.

logpdf(d::LogisticBeta, x::Real)

Compute the logpdf of the logistic-beta distribution d at x.

pdf(d::LogisticBeta, x::Real)

Compute the pdf of the logistic-beta distribution d at x.

mean(d::LogisticBeta)

Compute the mean of the logistic-beta distribution d.

quantile(d::LogisticBeta, p::Real)

Compute the p-quantile of the logistic-beta distribution d.

std(d::LogisticBeta)

Compute the standard deviation of the logistic-beta distribution d.

var(d::LogisticBeta)

Compute the variance of the logistic-beta distribution d.

mode(d::LogisticBeta)

Compute the mode of the logistic-beta distribution d.

skewness(d::LogisticBeta)

Compute the skewness of the logistic-beta distribution d.