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| """ | ||
| LocationScale(μ,σ,ρ) | ||
| A location-scale transformed distribution with location parameter `μ`, | ||
| scale parameter `σ`, and given univariate distribution `ρ`. | ||
| If ``Z`` is a random variable with distribution `ρ`, then the distribution of the random | ||
| variable | ||
| ```math | ||
| X = μ + σ Z | ||
| ``` | ||
| is the location-scale transformed distribution with location parameter `μ` and scale | ||
| parameter `σ`. | ||
| If `ρ` is a discrete distribution, the probability mass function of | ||
| the transformed distribution is given by | ||
| ```math | ||
| P(X = x) = P\\left(Z = \\frac{x-μ}{σ} \\right). | ||
| ``` | ||
| If `ρ` is a continuous distribution, the probability density function of | ||
| the transformed distribution is given by | ||
| ```math | ||
| f(x) = \\frac{1}{σ} ρ \\! \\left( \\frac{x-μ}{σ} \\right). | ||
| ``` | ||
| ```julia | ||
| LocationScale(μ,σ,ρ) # location-scale transformed distribution | ||
| params(d) # Get the parameters, i.e. (μ, σ, and the base distribution) | ||
| location(d) # Get the location parameter | ||
| scale(d) # Get the scale parameter | ||
| ``` | ||
| External links | ||
| [Location-Scale family on Wikipedia](https://en.wikipedia.org/wiki/Location%E2%80%93scale_family) | ||
| """ | ||
| struct LocationScale{T<:Real, S<:ValueSupport, D<:UnivariateDistribution{S}} <: UnivariateDistribution{S} | ||
| μ::T | ||
| σ::T | ||
| ρ::D | ||
| function LocationScale{T,S,D}(μ::T, σ::T, ρ::D; check_args=true) where {T<:Real, S<:ValueSupport, D<:UnivariateDistribution{S}} | ||
| check_args && @check_args(LocationScale, σ > zero(σ)) | ||
| new{T, S, D}(μ, σ, ρ) | ||
| end | ||
| end | ||
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| function LocationScale(μ::T, σ::T, ρ::UnivariateDistribution; check_args=true) where {T<:Real} | ||
| _T = promote_type(eltype(ρ), T) | ||
| D = typeof(ρ) | ||
| S = value_support(D) | ||
| return LocationScale{_T,S,D}(_T(μ), _T(σ), ρ; check_args=check_args) | ||
| end | ||
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| LocationScale(μ::Real, σ::Real, ρ::UnivariateDistribution) = LocationScale(promote(μ, σ)..., ρ) | ||
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| # aliases | ||
| const ContinuousLocationScale{T<:Real,D<:ContinuousUnivariateDistribution} = LocationScale{T,Continuous,D} | ||
| const DiscreteLocationScale{T<:Real,D<:DiscreteUnivariateDistribution} = LocationScale{T,Discrete,D} | ||
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| Base.eltype(::Type{<:LocationScale{T}}) where T = T | ||
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| minimum(d::LocationScale) = d.μ + d.σ * minimum(d.ρ) | ||
| maximum(d::LocationScale) = d.μ + d.σ * maximum(d.ρ) | ||
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| LocationScale(μ::Real, σ::Real, d::LocationScale) = LocationScale(μ + d.μ * σ, σ * d.σ, d.ρ) | ||
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| #### Conversions | ||
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| convert(::Type{LocationScale{T}}, μ::Real, σ::Real, ρ::D) where {T<:Real, D<:UnivariateDistribution} = LocationScale(T(μ),T(σ),ρ) | ||
| convert(::Type{LocationScale{T}}, d::LocationScale{S}) where {T<:Real, S<:Real} = LocationScale(T(d.μ),T(d.σ),d.ρ, check_args=false) | ||
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| #### Parameters | ||
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| location(d::LocationScale) = d.μ | ||
| scale(d::LocationScale) = d.σ | ||
| params(d::LocationScale) = (d.μ,d.σ,d.ρ) | ||
| partype(::LocationScale{T}) where {T} = T | ||
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| #### Statistics | ||
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| mean(d::LocationScale) = d.μ + d.σ * mean(d.ρ) | ||
| median(d::LocationScale) = d.μ + d.σ * median(d.ρ) | ||
| mode(d::LocationScale) = d.μ + d.σ * mode(d.ρ) | ||
| modes(d::LocationScale) = d.μ .+ d.σ .* modes(d.ρ) | ||
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| var(d::LocationScale) = d.σ^2 * var(d.ρ) | ||
| std(d::LocationScale) = d.σ * std(d.ρ) | ||
| skewness(d::LocationScale) = skewness(d.ρ) | ||
| kurtosis(d::LocationScale) = kurtosis(d.ρ) | ||
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| isplatykurtic(d::LocationScale) = isplatykurtic(d.ρ) | ||
| isleptokurtic(d::LocationScale) = isleptokurtic(d.ρ) | ||
| ismesokurtic(d::LocationScale) = ismesokurtic(d.ρ) | ||
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| entropy(d::ContinuousLocationScale) = entropy(d.ρ) + log(d.σ) | ||
| entropy(d::DiscreteLocationScale) = entropy(d.ρ) | ||
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| mgf(d::LocationScale,t::Real) = exp(d.μ*t) * mgf(d.ρ,d.σ*t) | ||
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| #### Evaluation & Sampling | ||
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| pdf(d::ContinuousLocationScale,x::Real) = pdf(d.ρ,(x-d.μ)/d.σ) / d.σ | ||
| pdf(d::DiscreteLocationScale, x::Real) = pdf(d.ρ,(x-d.μ)/d.σ) | ||
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| logpdf(d::ContinuousLocationScale,x::Real) = logpdf(d.ρ,(x-d.μ)/d.σ) - log(d.σ) | ||
| logpdf(d::DiscreteLocationScale, x::Real) = logpdf(d.ρ,(x-d.μ)/d.σ) | ||
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| # additional definitions are required to fix ambiguity errors and incorrect defaults | ||
| for f in (:cdf, :ccdf, :logcdf, :logccdf) | ||
| _f = Symbol(:_, f) | ||
| @eval begin | ||
| $f(d::LocationScale, x::Real) = $_f(d, x) | ||
| $f(d::DiscreteLocationScale, x::Real) = $_f(d, x) | ||
| $f(d::DiscreteLocationScale, x::Integer) = $_f(d, x) | ||
| $_f(d::LocationScale, x::Real) = $f(d.ρ, (x - d.μ) / d.σ) | ||
| end | ||
| end | ||
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| quantile(d::LocationScale,q::Real) = d.μ + d.σ * quantile(d.ρ,q) | ||
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| rand(rng::AbstractRNG, d::LocationScale) = d.μ + d.σ * rand(rng, d.ρ) | ||
| cf(d::LocationScale, t::Real) = cf(d.ρ,t*d.σ) * exp(1im*t*d.μ) | ||
| gradlogpdf(d::ContinuousLocationScale, x::Real) = gradlogpdf(d.ρ,(x-d.μ)/d.σ) / d.σ | ||
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| #### Syntactic sugar for simple transforms of distributions, e.g., d + x, d - x, and so on | ||
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| Base.:+(d::UnivariateDistribution, x::Real) = LocationScale(x, one(x), d) | ||
| Base.:+(x::Real, d::UnivariateDistribution) = d + x | ||
| Base.:*(x::Real, d::UnivariateDistribution) = LocationScale(zero(x), x, d) | ||
| Base.:*(d::UnivariateDistribution, x::Real) = x * d | ||
| Base.:-(d::UnivariateDistribution, x::Real) = d + -x | ||
| Base.:/(d::UnivariateDistribution, x::Real) = inv(x) * d | ||
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