Perform an Edgeworth Expansion to Approximate a Distribution
Use central moments of data to approximate a population distribution via an Edgeworth expansion.
pdf = With[{len = 4}, PDF[NormalDistribution[], x] (1 + Sum[1 / (2 ^ (s / 2 + r) s!) HermiteH[s + 2 r, x / Sqrt[2]] BellY[s, r, Table[Cumulant[k] / (k (k - 1)), {k, 3, s - r + 3}]], {s, 1, len - 2}, {r, 1, s}])];data = Standardize[Total[RandomReal[1, {3, 10 ^ 5}]]];
dist = ProbabilityDistribution[MomentEvaluate[pdf, data], {x, -∞, ∞}];Show[Histogram[data, Automatic, "ProbabilityDensity", ChartStyle -> Directive[EdgeForm[Gray], Hue[.15, .3, .9]]], Plot[{PDF[dist, z], PDF[NormalDistribution[], z]}, {z, -3, 3}, PlotStyle -> {Directive[Thick, Hue[.6, 1, .6]], Directive[Thick, Dashed, Red]}], ImageSize -> 500]