# Fitting a distribution not supported by Distributions.jl

**URL:** <https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818>\
**Category:** Statistics\
**Tags:** distributions\
**Created:** [July 17, 2025, 4:57pm UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818 "2025-07-17T16:57:15Z")\
**Posts on this page:** 7\
**Page:** 1

<div class="post-metadata">

**Author:** ![henry2004y](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/henry2004y/32/9284_2.png) [@henry2004y](https://discourse.julialang.org/u/henry2004y)\
**Post date:** [July 17, 2025, 4:57pm UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818/1 "2025-07-17T16:57:15Z")

</div>

Hi,

I want to fit an analytical distribution in Julia from discrete points. For instance, this specific [Levy distribution](https://juliastats.org/Distributions.jl/stable/univariate/#Distributions.Levy) is listed in Distributions.jl, but the `fit` method is not supported for this type: [Distribution Fitting · Distributions.jl](https://juliastats.org/Distributions.jl/stable/fit/#Distribution-Fitting)

```julia
using Distributions

D = Levy()
x = rand(D, 1000)
d = fit(Levy, x)

```

```julia
ERROR: suffstats is not implemented for (Levy, Vector{Float64}).
Stacktrace:
 [1] error(s::String)
   @ Base .\error.jl:35
 [2] suffstats(dt::Type{Levy}, xs::Vector{Float64})
   @ Distributions C:\Users\hyzho\.julia\packages\Distributions\YQSrn\src\genericfit.jl:5
 [3] fit_mle(dt::Type{Levy}, x::Vector{Float64})
   @ Distributions C:\Users\hyzho\.julia\packages\Distributions\YQSrn\src\genericfit.jl:27
 [4] fit(dt::Type{Levy}, x::Vector{Float64})
   @ Distributions C:\Users\hyzho\.julia\packages\Distributions\YQSrn\src\genericfit.jl:46
 [5] top-level scope
   @ REPL[20]:1

```

In general, how should I fit these kinds of analytic distributions?

---

<div class="post-metadata">

**Author:** ![bertschi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bertschi/32/33462_2.png) [@bertschi](https://discourse.julialang.org/u/bertschi)\
**Post date:** [July 17, 2025, 6:14pm UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818/2 "2025-07-17T18:14:41Z")

</div>

From what I understand, `fit` is only implemented for some distributions especially when an efficient ML estimator is available.  
Depending on your needs, you can always reach for an optimization package

```julia
using Optim, ComponentArrays

p₀ = ComponentVector(; μ = 0.0, logσ = 0.0)
mle = Optim.optimize(p -> - sum(logpdf.(Levy(p.μ, exp(p.logσ)), x)), p₀, LBFGS())

```

or go for a Bayesian approach

```julia
using Turing

@model function fitme(x)
    μ ~ Normal(0, 10)
    σ ~ truncated(TDist(3), lower = 0)
    x .~ Levy(μ, σ)
end

chains = sample(fitme(x), NUTS(), 1_000)

```

---

<div class="post-metadata">

**Author:** ![henry2004y](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/henry2004y/32/9284_2.png) [@henry2004y](https://discourse.julialang.org/u/henry2004y)\
**Post date:** [July 17, 2025, 6:27pm UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818/3 "2025-07-17T18:27:00Z")

</div>

> [@bertschi](#):
>
> `using Optim, ComponentArrays`

Thanks for the tips! I need some time to learn about the usage of these optimization packages. For instance when using `Optim`, how can I check the fitted parameters (in this Levy case μ and σ)?

---

<div class="post-metadata">

**Author:** ![bertschi](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/bertschi/32/33462_2.png) [@bertschi](https://discourse.julialang.org/u/bertschi)\
**Post date:** [July 17, 2025, 6:49pm UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818/4 "2025-07-17T18:49:35Z")

</div>

`Optim.minimizer(mle)` will give you the optimized parameters. Note that I have used `logσ` as I like using unconstrained optimizers, thus you will need to take the exponential of the optimized value.

---

<div class="post-metadata">

**Author:** ![henry2004y](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/henry2004y/32/9284_2.png) [@henry2004y](https://discourse.julialang.org/u/henry2004y)\
**Post date:** [July 17, 2025, 8:06pm UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818/5 "2025-07-17T20:06:35Z")

</div>

I have another related question to the usage of `Optim`. Now I wrap the calls into the following function:

```julia
function fit_levy(x)
   p₀ = ComponentVector(; μ = 0.0, logσ = 0.0)
   mle = Optim.optimize(p -> - sum(logpdf.(Levy(p.μ, exp(p.logσ)), x)), p₀, LBFGS())
   param = Optim.minimizer(mle)
   μ = param.μ
   σ = exp(param.logσ)

   μ, σ
end

```

I found that the optimization process is sensitive to the scales of the numbers:

```julia
julia> a = [ 25.77444756189654
        1941.9959559504289
           30.90865782846204
         789.5316974005192
          67.17259874993165
             29.16640194223711
          28.6202910812033
         933.401245894768
          86.69688566840875
             39.04475622775034]
10-element Vector{Float64}:
   25.77444756189654
 1941.9959559504289
   30.90865782846204
  789.5316974005192
   67.17259874993165
   29.16640194223711
   28.6202910812033
  933.401245894768
   86.69688566840875
   39.04475622775034

julia> μ1, σ1 = fit_levy(a)
(24.207532552609223, 7.545442547618322)

julia> μ1, σ1 = fit_levy(a / 100)
(0.2420753224182992, 0.07545443480661769)

julia> μ1, σ1 = fit_levy(a / 1000)
(0.024207499550423738, 0.007545541553505919)

julia> μ1, σ1 = fit_levy(a / 10000)
ERROR: DomainError with 0.0:
Levy: the condition σ > zero(σ) is not satisfied.
Stacktrace:
  [1] #331
    @ C:\Users\hyzho\.julia\packages\Distributions\YQSrn\src\univariate\continuous\levy.jl:31 [inlined]
  [2] check_args
    @ C:\Users\hyzho\.julia\packages\Distributions\YQSrn\src\utils.jl:89 [inlined]
  [3] #Levy#330
    @ C:\Users\hyzho\.julia\packages\Distributions\YQSrn\src\univariate\continuous\levy.jl:31 [inlined]
  [4] Levy
    @ C:\Users\hyzho\.julia\packages\Distributions\YQSrn\src\univariate\continuous\levy.jl:30 [inlined]
  [5] (::var"#3#4"{Vector{Float64}})(p::ComponentVector{Float64, Vector{Float64}, Tuple{Axis{(μ = 1, logσ = 2)}}})
    @ Main D:\Research\Turbulence\analysis\fit_distributions.jl:20
  [6] value_and_gradient!
    @ C:\Users\hyzho\.julia\packages\DifferentiationInterface\alBlj\ext\DifferentiationInterfaceFiniteDiffExt\onearg.jl:329 [inlined]
  [7] (::NLSolversBase.var"#fg!#14"{…})(_g::ComponentVector{…}, _x::ComponentVector{…})
    @ NLSolversBase C:\Users\hyzho\.julia\packages\NLSolversBase\n7XXO\src\objective_types\oncedifferentiable.jl:53
  [8] value_gradient!!(obj::OnceDifferentiable{…}, x::ComponentVector{…})
    @ NLSolversBase C:\Users\hyzho\.julia\packages\NLSolversBase\n7XXO\src\interface.jl:82
  [9] value_gradient!(obj::OnceDifferentiable{…}, x::ComponentVector{…})
    @ NLSolversBase C:\Users\hyzho\.julia\packages\NLSolversBase\n7XXO\src\interface.jl:69
 [10] value_gradient!(obj::Optim.ManifoldObjective{OnceDifferentiable{…}}, x::ComponentVector{Float64, Vector{…}, Tuple{…}})
    @ Optim C:\Users\hyzho\.julia\packages\Optim\7krni\src\Manifolds.jl:50
 [11] (::LineSearches.var"#ϕdϕ#6"{…})(α::Float64)
    @ LineSearches C:\Users\hyzho\.julia\packages\LineSearches\b4CwT\src\LineSearches.jl:83
 [12] (::LineSearches.HagerZhang{…})(ϕ::Function, ϕdϕ::LineSearches.var"#ϕdϕ#6"{…}, c::Float64, phi_0::Float64, dphi_0::Float64)
    @ LineSearches C:\Users\hyzho\.julia\packages\LineSearches\b4CwT\src\hagerzhang.jl:147
 [13] HagerZhang
    @ C:\Users\hyzho\.julia\packages\LineSearches\b4CwT\src\hagerzhang.jl:103 [inlined]
 [14] perform_linesearch!(state::Optim.LBFGSState{…}, method::LBFGS{…}, d::Optim.ManifoldObjective{…})
    @ Optim C:\Users\hyzho\.julia\packages\Optim\7krni\src\utilities\perform_linesearch.jl:58
 [15] update_state!(d::OnceDifferentiable{…}, state::Optim.LBFGSState{…}, method::LBFGS{…})
    @ Optim C:\Users\hyzho\.julia\packages\Optim\7krni\src\multivariate\solvers\first_order\l_bfgs.jl:220
 [16] optimize(d::OnceDifferentiable{…}, initial_x::ComponentVector{…}, method::LBFGS{…}, options::Optim.Options{…}, state::Optim.LBFGSState{…})
    @ Optim C:\Users\hyzho\.julia\packages\Optim\7krni\src\multivariate\optimize\optimize.jl:65
 [17] optimize
    @ C:\Users\hyzho\.julia\packages\Optim\7krni\src\multivariate\optimize\optimize.jl:43 [inlined]
 [18] #optimize#77
    @ C:\Users\hyzho\.julia\packages\Optim\7krni\src\multivariate\optimize\interface.jl:274 [inlined]
 [19] optimize
    @ C:\Users\hyzho\.julia\packages\Optim\7krni\src\multivariate\optimize\interface.jl:264 [inlined]
 [20] optimize(f::Function, initial_x::ComponentVector{…}, method::LBFGS{…})
    @ Optim C:\Users\hyzho\.julia\packages\Optim\7krni\src\multivariate\optimize\interface.jl:264
 [21] fit_levy(x::Vector{Float64})
    @ Main D:\Research\Turbulence\analysis\fit_distributions.jl:20
 [22] top-level scope
    @ REPL[32]:1
Some type information was truncated. Use `show(err)` to see complete types.

```

Is there a way to avoid the above condition unsatisfied error?

---

<div class="post-metadata">

**Author:** ![henry2004y](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/henry2004y/32/9284_2.png) [@henry2004y](https://discourse.julialang.org/u/henry2004y)\
**Post date:** [July 17, 2025, 8:32pm UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818/6 "2025-07-17T20:32:10Z")

</div>

And yet another small question is, how can I impose constraints to the parameters when performing optimizations? For instance, in the Levy distribution case, I may wish to have a nonnegative location parameter mu.

---

<div class="post-metadata">

**Author:** ![huang\_min](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/huang_min/32/34337_2.png) [@huang\_min](https://discourse.julialang.org/u/huang_min)\
**Post date:** [July 18, 2025, 1:14am UTC](https://discourse.julialang.org/t/fitting-a-distribution-not-supported-by-distributions-jl/130818/7 "2025-07-18T01:14:42Z")

</div>

logmu also works