function [shape, scale] = weibull_specification(mu, sigma2, lb, name)

% Returns the hyperparameters of the Weibull distribution given the expectation and variance.
%
% INPUTS
%
%
% OUTPUTS
%
%

% Copyright © 2015-2023 Dynare Team
%
% This file is part of Dynare.
%
% Dynare is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% Dynare is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with Dynare.  If not, see <https://www.gnu.org/licenses/>.

if nargin<3
    lb = 0;
end

if nargin>3 && ~isempty(name)
    name1 = sprintf('for %s ', name);
    name2 = sprintf(' (for %s)', name);
else
    name1 = '';
    name2 = '';
end

if mu<lb
    error('The prior expectation (%f) %scannot be smaller than the lower bound of the Weibull distribution (%f)!', mu, name1, lb)
end

if isinf(sigma2)
    error('The variance of the Gamma distribution has to be finite%s!', name2)
end

scale = NaN;
shape = NaN;

mu = mu-lb;
mu2 = mu*mu;

eqn = @(k) gammaln(1+2./k) - 2*gammaln(1+1./k) - log(1+sigma2/mu2);
eqn2 = @(k) eqn(k).*eqn(k);

kstar = fminbnd(eqn2, 1e-9, 100);
[shape, fval, exitflag]  = fzero(eqn, kstar);

if exitflag<1
    shape = NaN;
    return
end

scale = mu/gamma(1+1/shape);

return % --*-- Unit tests --*--

%@test:1
debug = false;
scales = 1:.01:5;
shapes = .5:.01:2;
n_scales = length(scales);
n_shapes = length(shapes);
mu = NaN(n_scales, n_shapes);
s2 = NaN(n_scales, n_shapes);
for i=1:n_shapes
    g1 = gamma(1+1/shapes(i));
    g2 = gamma(1+2/shapes(i));
    g3 = g1*g1;
    for j=1:n_scales
        mu(j, i) = scales(j)*g1;
        s2(j, i) = scales(j)*scales(j)*(g2-g3);
    end
end
if debug
   success = [];
   failed1 = [];
   failed1_ = [];
   failed2 = [];
end
try
   for i=1:n_shapes
       for j=1:n_scales
          if debug
              disp(sprintf('... mu=%s and s2=%s', num2str(mu(j,i)),num2str(s2(j,i))))
          end
          if ~isnan(mu(j,i)) && ~isnan(s2(j,i)) && ~isinf(mu(j,i)) && ~isinf(s2(j,i))
              [shape, scale] = weibull_specification(mu(j,i), s2(j,i));
              if isnan(scale)
                 t = false;
              else
                 if abs(scales(j)-scale)<1e-9 && abs(shapes(i)-shape)<1e-9
                     t = true;
                 else
                     t = false;
                 end
              end
              if ~t && debug
                 failed1 = [failed1; mu(j,i) s2(j,i)];
                 failed1_ = [failed1_; shapes(i) scales(j)];
                 error('UnitTest','Cannot compute scale and shape hyperparameters for mu=%s and s2=%s', num2str(mu(j,i)), num2str(s2(j,i)))
              end
              if debug
                  success = [success; mu(j,i) s2(j,i)];
              end
          else
              failed2 = [failed2; shapes(i) scales(j)];
              continue % Pass this test
          end
      end
  end
catch
   t = false;
end

if debug
    figure(1)
    plot(success(:,1),success(:,2),'ok');
    if ~isempty(failed1)
        hold on
        plot(failed1(:,1),failed1(:,2),'or');
        hold off
        figure(2)
        plot(failed1_(:,1),failed1_(:,2),'or')
    end
    if ~isempty(failed2)
        figure(2)
        plot(failed2(:,1),failed2(:,2),'or');
    end
end
T = all(t);
%@eof:1