ClearAll;Clear[f1];Clear[f2];
$DefaultFont = {"Times-Roman",10};
Needs["Statistics`DataManipulation`"];
Needs["Graphics`Graphics`"];
cumulative = {312/545,413/545,487/545,522/545,537/545,1};
datapoints = {{0,0},{1,312/545},{2,413/545},
			  {3,487/545},{4,522/545},{5,537/545},{6,1}};
density = {312/545,101/545,74/545,35/545,15/545,8/545};
bound[z_,a_,b_]:= Which[z<a,a,z>b,b,a<=z<=b,z];
F1[A_,c_,x_] = 4 A c x/(1 + A c^2) - A x^2;
F2[A_,c_,x_] = 1 - ((1 - A c^2)^2 (c/x)^(2A c^2/(1 - A c^2)))/(1 + A c^2);

f1[A_,c_,x_] = 2 A(c-x) + 2 A c(1 - A c^2)/(1 + A c^2);
f2[A_,c_,x_] = (2A c(1 - A c^2) (c/x)^((1 + A c^2)/(1 - A c^2)))/(1 + A c^2);

FF[a_,c_,x_] = Which[x<=1.5,F1[a,c,x],x>1.5,F2[a,c,x]];

error[a_,c_] = Evaluate[Apply[Plus,Table[(FF[a,c,n]-cumulative[[n]])^2,{n,1,6}]]];

unconstrainedminimizer[a_]:= c/.FindMinimum[error[a,c],{c,1.85,1.86,0,2}][[2]];
constrainedminimizer[a_]:= bound[unconstrainedminimizer[a],1.7,2];

Sstar[a_]:= error[a,constrainedminimizer[a]];
plot = Plot[Sstar[a],{a,0.12,0.15},PlotPoints->100];
ahat = a/.FindMinimum[Sstar[a],{a,0.13,0.131,0.12,0.15}][[2]]
Sstar[ahat]
				
Show[plot,PlotRange->{{0.12,0.15},{0.00655,0.00704}},
			   AxesOrigin->{0.119,0.00653},AxesLabel->{"A","z"},
			   PlotLabel->"Figure 24.8\n  z = S*(A)"]