Solve Optimization Problems in Density Estimation

Leverage the symbolic capabilities of KernelMixtureDistribution to solve for the least-squares cross-validation bandwidth. This method uses leave-one-out cross-validation to select a bandwidth that minimizes the integrated squared error of the resulting estimate.
 In[1]:= Xd = BlockRandom[SeedRandom[12]; RandomVariate[NormalDistribution[], 25]]; Rk[h_, data_] := With[{n = Length[data]}, 1/(h Sqrt[\[Pi]]) (Exp[-((Subtract @@@ Subsets[data, {2}])^2/(4 h^2))].ConstantArray[1/n^2, Total[Range[1, n - 1]]] + 1/(2 n))] Ro[h_, data_] := Total[1/((Length[data] - 1) h Sqrt[2 \[Pi]]) Table[ Plus @@ Exp[-(data[[i]] - Delete[data, {i}])^2/(2 h^2)], {i, Length[data]}]] LSCV[h_, data_] := With[{n = Length[data]}, Rk[h, data] - 2/n Ro[h, data]] bw = h /. FindMinimum[LSCV[h, d], {h}][[2]];
 In[2]:= X\[ScriptCapitalD] = KernelMixtureDistribution[d, bw];
 In[3]:= XShow[Plot[LSCV[h, d], {h, 0.03, 2}, PlotLabel -> Text[Style[Row[{"h \[Rule] ", bw}], Bold, FontFamily -> "Verdana", FontSize -> 14]], Frame -> True, Axes -> None, Filling -> None, PlotStyle -> Thick, PlotRange -> {{0, 1.99}, {-.39, 0}}, ImageSize -> {570, 374}], Graphics[{Lighter[Blend[{Red, Orange}], 0.3], Dashed, Thick, Line[{{bw, -.385}, {bw, .005}}]}], Epilog -> Inset[Plot[PDF[\[ScriptCapitalD], x], {x, -3.5, 3}, Filling -> Axis, FillingStyle -> Lighter[Blend[{Red, Orange}], 0.4], Axes -> {True, False}, ImageSize -> 360], {1.3, -.24}]]
 Out[3]=