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Compare Maximum-Likelihood and Cramér–von Mises Estimates

Visually compare parameter estimates for a normal distribution using maximum-likelihood estimation and optimizing the Cramér–von Mises test statistic.

    
Wolfram Language code: data = BlockRandom[SeedRandom[2]; RandomVariate[NormalDistribution[], 100]];
Wolfram Language code: cvmFit[μ_ ? NumericQ, σ_ ? NumericQ, dist_] := CramerVonMisesTest[data, dist[μ, σ], "TestStatistic"]; cvmPar = Quiet[FindMinimum[cvmFit[μ, σ, NormalDistribution], {{μ}, {σ}}]];mlePar = FindDistributionParameters[data, NormalDistribution[μ, σ]];
Wolfram Language code: Show[Plot3D[cvmFit[μ, σ, NormalDistribution], {μ, -3, 3}, {σ, 0, 6}, ColorFunction -> ColorData["SouthwestColors"], PlotRange -> {{-2, 2}, {0, 3}, {-10, 35}}, MeshFunctions -> {#3&}, MeshStyle -> Gray, Mesh -> 30, BoxRatios -> 1], Graphics3D[{Darker@Blue, Thickness[.01], Line[{{μ, σ, 40}, {μ, σ, -20}} /. cvmPar[[2]]]}], Graphics3D[{Darker@Red, Thickness[.01], Line[{{μ, σ, 40}, {μ, σ, -20}} /. mlePar]}]]