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function [ldens,Dldens,D2ldens] = lpdfig1(x,s,nu)
% Evaluates the logged INVERSE-GAMMA-1 PDF at x.
%
% X ~ IG1(s,nu) if X = sqrt(Y) where Y ~ IG2(s,nu) and Y = inv(Z) with Z ~ G(nu/2,2/s) (Gamma distribution)
%
% See L. Bauwens, M. Lubrano and J-F. Richard [1999, appendix A] for more details.
%
%
% INPUTS
% x [double] m*n matrix of locations,
% s [double] m*n matrix or scalar, First INVERSE-GAMMA-1 distribution parameters,
% nu [double] m*n matrix or scalar, Second INVERSE-GAMMA-1 distribution parameters.
%
% OUTPUTS
% ldens [double] m*n matrix of logged INVERSE-GAMMA-1 densities evaluated at x.
% Dldens [double] m*n matrix of first derivatives of logged INVERSE-GAMMA-1 densities.
% D2ldens [double] m*n matrix of second derivatives of logged matrix of logged INVERSE-GAMMA-1 densities.
%
% SPECIAL REQUIREMENTS
% none
% Copyright © 2004-2021 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/>.
ldens = -Inf( size(x) ) ;
idx = find( x>0 ) ;
if length(s)==1
ldens(idx) = log(2) - gammaln(.5*nu) - .5*nu*(log(2)-log(s)) - (nu+1)*log(x(idx)) - .5*s./(x(idx).*x(idx)) ;
else
ldens(idx) = log(2) - gammaln(.5*nu(idx)) - .5*nu(idx).*(log(2)-log(s(idx))) - (nu(idx)+1).*log(x(idx)) - .5*s(idx)./(x(idx).*x(idx)) ;
end
if nargout >1
Dldens = ldens ;
if length(s)==1
Dldens(idx) = - (nu+1)./(x(idx)) + s./(x(idx).^3) ;
else
Dldens(idx) = - (nu(idx)+1)./(x(idx)) + s(idx)./(x(idx).^3) ;
end
end
if nargout == 3
D2ldens = ldens ;
if length(s)==1
D2ldens(idx) = (nu+1)./(x(idx).^2) - 3*s(idx)./(x(idx).^4) ;
else
D2ldens(idx) = (nu(idx)+1)./(x(idx).^2) - 3*s(idx)./(x(idx).^4) ;
end
end
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