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function [mu, parameters] = mode_and_variance_to_mean(m,s2,distribution,lower_bound,upper_bound)
% This function computes the mean of a distribution given the mode and variance of this distribution.
%
% INPUTS
% m [double] scalar, mode of the distribution.
% s2 [double] scalar, variance of the distribution.
% distribution [integer] scalar for the distribution shape
% 1 gamma
% 2 inv-gamma-2
% 3 inv-gamma-1
% 4 beta
% lower_bound [double] scalar, lower bound of the random variable support (optional).
% upper_bound [double] scalar, upper bound of the random variable support (optional).
%
% OUTPUT
% mu [double] scalar, mean of the distribution.
% parameters [double] 2*1 vector, parameters of the distribution.
%
% Copyright (C) 2009 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 <http://www.gnu.org/licenses/>.
% Check input aruments.
if ~(nargin==3 || nargin==5 || nargin==4 )
error('mode_and_variance_to mean:: 3 or 5 input arguments are needed!')
end
% Set defaults bounds.
if nargin==3
switch distribution
case 1
lower_bound = 0;
upper_bound = Inf;
case 3
lower_bound = 0;
upper_bound = Inf;
case 2
lower_bound = 0;
upper_bound = Inf;
case 4
lower_bound = 0;
upper_bound = 1;
otherwise
error('Unknown distribution!')
end
end
if nargin==4
switch distribution
case 1
upper_bound = Inf;
case 3
upper_bound = Inf;
case 2
upper_bound = Inf;
case 4
upper_bound = 1;
otherwise
error('Unknown distribution!')
end
end
if (distribution==1)% Gamma distribution
if m<lower_bound
error('The mode has to be greater than the lower bound!')
end
if (m-lower_bound)<1e-12
error('The gamma distribution should be specified with the mean and variance.')
end
m = m - lower_bound ;
beta = -.5*m*(1-sqrt(1+4*s2/(m*m))) ;
alpha = (m+beta)/beta ;
parameters(1) = alpha;
parameters(2) = beta;
mu = alpha*beta + lower_bound ;
return
end
if (distribution==2)% Inverse Gamma - 2 distribution
if m<lower_bound+2*eps
error('The mode has to be greater than the lower bound!')
end
m = m - lower_bound ;
if isinf(s2)
nu = 4;
s = 2*m;
else
delta = 2*(m*m/s2);
poly = [ 1 , -(8+delta) , 20-4*delta , -(16+4*delta) ];
all_roots = roots(poly);
real_roots = all_roots(find(abs(imag(all_roots))<2*eps));
nu = real_roots(find(real_roots>2));
s = m*(nu+2);
end
parameters(1) = nu;
parameters(2) = s;
mu = s/(nu-2) + lower_bound;
return
end
if (distribution==3)% Inverted Gamma 1 distribution
if m<lower_bound+2*eps
error('The mode has to be greater than the lower bound!')
end
m = m - lower_bound ;
if isinf(s2)
nu = 2;
s = 3*(m*m);
else
[mu, parameters] = mode_and_variance_to_mean(m,s2,2);
nu = sqrt(parameters(1));
nu2 = 2*nu;
nu1 = 2;
err = s2/(m*m) - (nu+1)/(nu-2) + .5*(nu+1)*(gamma((nu-1)/2)/gamma(nu/2))^2;
while abs(nu2-nu1) > 1e-12
if err < 0
nu1 = nu;
if nu < nu2
nu = nu2;
else
nu = 2*nu;
nu2 = nu;
end
else
nu2 = nu;
end
nu = (nu1+nu2)/2;
err = s2/(m*m) - (nu+1)/(nu-2) + .5*(nu+1)*(gamma((nu-1)/2)/gamma(nu/2))^2;
end
s = (nu+1)*(m*m) ;
end
parameters(1) = nu;
parameters(2) = s;
mu = sqrt(.5*s)*gamma(.5*(nu-1))/gamma(.5*nu) + lower_bound ;
return
end
if (distribution==4)% Beta distribution
if m<lower_bound
error('The mode has to be greater than the lower bound!')
end
if m>upper_bound
error('The mode has to be less than the upper bound!')
end
if (m-lower_bound)<1e-12
error('The beta distribution should be specified with the mean and variance.')
end
if (upper_bound-m)<1e-12
error('The beta distribution should be specified with the mean and variance.')
end
ll = upper_bound-lower_bound;
m = (m-lower_bound)/ll;
s2 = s2/(ll*ll);
delta = m^2/s2;
poly = NaN(1,4);
poly(1) = 1;
poly(2) = 7*m-(1-m)*delta-3;
poly(3) = 16*m^2-14*m+3-2*m*delta+delta;
poly(4) = 12*m^3-16*m^2+7*m-1;
all_roots = roots(poly);
real_roots = all_roots(find(abs(imag(all_roots))<2*eps));
idx = find(real_roots>1);
if length(idx)>1
error('Multiplicity of solutions for the beta distribution specification.')
elseif isempty(idx)
disp('No solution for the beta distribution specification. You should reduce the variance.')
error();
end
alpha = real_roots(idx);
beta = ((1-m)*alpha+2*m-1)/m;
parameters(1) = alpha;
parameters(2) = beta;
mu = alpha/(alpha+beta)*(upper_bound-lower_bound)+lower_bound;
return
end
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