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function [a, b] = beta_specification ( mu, sigma2, lb, ub, name) % --*-- Unitary tests --*--
% Returns the hyperparameters of the beta distribution given the expectation and variance.
%
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% INPUTS
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% - mu [double] Expectation of the Gamma random variable.
% - sigma2 [double] Variance of the Gamma random variable.
% - lb [double] Lower bound of the domain (default is zero).
% - ub [double] Upper bound of the domain (default is one).
%
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% OUTPUTS
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% - a [double] First hyperparameter of the Beta density.
% - b [double] Second hyperparameter of the Beta density.
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% Copyright © 2015-2017 Dynare Team
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%
% 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
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% along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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if nargin < 3
lb = 0 ;
ub = 1 ;
end
if nargin < 4
ub = 1 ;
end
if nargin > 4 && ~ isempty ( name )
name1 = sprintf ( ' of %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 Beta distribution (%f)!' ] , mu , name1 , lb )
end
if mu > ub
error ( ' The prior expectation (%f) %scannot be greater than the upper bound of the Beta distribution (%f)!' , mu , name1 , ub )
end
len = ub - lb ;
mu = ( mu - lb ) / len ;
sigma2 = sigma2 / ( len * len ) ;
if sigma2 > ( 1 - mu ) * mu
error ( ' Beta prior%s. Given the declared prior expectation, prior lower and upper bounds, the prior std. has to be smaller than %f.' , name2 , sqrt ( ( 1 - mu ) * mu ) )
end
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a = ( 1 - mu ) * mu * mu / sigma2 - mu ;
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b = a * ( 1 / mu - 1 ) ;
%@test:1
%$ try
%$ [a, b] = beta_specification(.5, .05);
%$ t(1) = true;
%$ catch
%$ t(1) = false;
%$ end
%$
%$ if t(1)
%$ t(2) = abs(0.5-a/(a+b))<1e-12;
%$ t(3) = abs(0.05-a*b/((a+b)^2*(a+b+1)))<1e-12;
%$ end
%$ T = all(t);
%@eof:1
%@test:2
%$ try
%$ [a, b] = beta_specification(0.5, .05, 1, 2);
%$ t(1) = false;
%$ catch
%$ t(1) = true;
%$ end
%$
%$ T = all(t);
%@eof:2
%@test:3
%$ try
%$ [a, b] = beta_specification(2.5, .05, 1, 2);
%$ t(1) = false;
%$ catch
%$ t(1) = true;
%$ end
%$
%$ T = all(t);
%@eof:3
%@test:4
%$ try
%$ [a, b] = beta_specification(.4, .6*.4+1e-4);
%$ t(1) = false;
%$ catch
%$ t(1) = true;
%$ end
%$
%$ T = all(t);
%@eof:4
%@test:5
%$ try
%$ [a, b] = beta_specification(.4, .6*.4+1e-6);
%$ t(1) = false;
%$ catch
%$ t(1) = true;
%$ end
%$ try
%$ [a, b] = beta_specification(.4, .6*.4-1e-6);
%$ t(2) = true;
%$ catch
%$ t(2) = false;
%$ end
%$
%$ T = all(t);
%@eof:5