fixing error in xval function in gumbel.jl (#1859)

* fixing error in xval function in gumbel.jl

* implement Gumbel quantile function using xval

* adding Gumbel functions: ccdf, logccdf, cquantile, invlogcdf invlogccdf, mgf, cgf, cf

* Update src/univariate/continuous/gumbel.jl

Co-authored-by: David Widmann <devmotion@users.noreply.github.com>

* Update src/univariate/continuous/gumbel.jl

Co-authored-by: David Widmann <devmotion@users.noreply.github.com>

* Update src/univariate/continuous/gumbel.jl

Co-authored-by: David Widmann <devmotion@users.noreply.github.com>

---------

Co-authored-by: David Widmann <devmotion@users.noreply.github.com>

src/univariate/continuous/gumbel.jl

@@ -85,7 +85,7 @@ entropy(d::Gumbel) = log(d.θ) + 1 + MathConstants.γ
 #### Evaluation
 
 zval(d::Gumbel, x::Real) = (x - d.μ) / d.θ
-xval(d::Gumbel, z::Real) = x * d.θ + d.μ
+xval(d::Gumbel, z::Real) = z * d.θ + d.μ
 
 function pdf(d::Gumbel, x::Real)
     z = zval(d, x)
@@ -98,8 +98,17 @@ function logpdf(d::Gumbel, x::Real)
 end
 
 cdf(d::Gumbel, x::Real) = exp(-exp(-zval(d, x)))
+ccdf(d::Gumbel, x::Real) = -expm1(-exp(-zval(d, x)))
 logcdf(d::Gumbel, x::Real) = -exp(-zval(d, x))
+logccdf(d::Gumbel, x::Real) = log1mexp(-exp(-zval(d, x)))
 
-quantile(d::Gumbel, p::Real) = d.μ - d.θ * log(-log(p))
+quantile(d::Gumbel, p::Real) = xval(d, -log(-log(p)))
+cquantile(d::Gumbel, p::Real) = xval(d, -log(-log1p(-p)))
+invlogcdf(d::Gumbel, lp::Real) = xval(d, -log(-lp))
+invlogccdf(d::Gumbel, lp::Real) = xval(d, -log(-log1mexp(lp)))
 
 gradlogpdf(d::Gumbel, x::Real) = expm1(-zval(d, x)) / d.θ
+
+mgf(d::Gumbel, t::Real) = gamma(1 - d.θ * t) * exp(d.μ * t)
+cgf(d::Gumbel, t::Real) = loggamma(1 - d.θ * t) + d.μ * t
+cf(d::Gumbel, t::Real) = gamma(1 - im * d.θ * t) * cis(d.μ * t)