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function t = wblinv(proba, scale, shape)
% Inverse cumulative distribution function.
%
% INPUTS
% - proba [double] Probability, scalar or vector with values in [0,1].
% - scale [double] Positive hyperparameter (scalar).
% - shape [double] Positive hyperparameter (scalar).
%
% OUTPUTS
% - t [double] Quantile(s), same size as proba.
% 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/>.
% Check input arguments.
if nargin<3
error('Three input arguments required!')
end
if ~isnumeric(proba) || ~isreal(proba) || any(proba(:)<0) || any(proba(:)>1)
error('First input argument must be a real scalar or vector with values in [0,1] (probability)!')
end
if ~isnumeric(scale) || ~isscalar(scale) || ~isreal(scale) || scale<=0
error('Second input argument must be a real positive scalar (scale parameter of the Weibull distribution)!')
end
if ~isnumeric(shape) || ~isscalar(shape) || ~isreal(shape) || shape<=0
error('Third input argument must be a real positive scalar (shape parameter of the Weibull distribution)!')
end
% Vectorized closed-form inverse CDF.
t = zeros(size(proba));
t(proba > 1-2*eps()) = Inf;
k = proba >= 2*eps() & proba <= 1-2*eps();
if any(k(:))
t(k) = exp(log(scale) + log(-log(1-proba(k)))./shape);
end
return % --*-- Unit tests --*--
%@test:1
try
x = wblinv(0, 1, 2);
t(1) = true;
catch
t(1) = false;
end
if t(1)
t(2) = isequal(x, 0);
end
T = all(t);
%@eof:1
%@test:2
try
x = wblinv(1, 1, 2);
t(1) = true;
catch
t(1) = false;
end
if t(1)
t(2) = isinf(x);
end
T = all(t);
%@eof:2
%@test:3
scales = [.5, 1, 5];
shapes = [.1, 1, 2];
x = NaN(9,1);
try
k = 0;
for i=1:3
for j=1:3
k = k+1;
x(k) = wblinv(.5, scales(i), shapes(j));
end
end
t(1) = true;
catch
t(1) = false;
end
if t(1)
k = 1;
for i=1:3
for j=1:3
k = k+1;
t(k) = abs(x(k-1)-scales(i)*log(2)^(1/shapes(j)))<1e-12;
end
end
end
T = all(t);
%@eof:3
%@test:4
debug = false;
scales = [ .5, 1, 5];
shapes = [ 1, 2, 3];
x = NaN(9,1);
p = 1e-1;
try
k = 0;
for i=1:3
for j=1:3
k = k+1;
x(k) = wblinv(p, scales(i), shapes(j));
end
end
t(1) = true;
catch
t(1) = false;
end
if t(1)
k = 1;
for i=1:3
for j=1:3
k = k+1;
shape = shapes(j);
scale = scales(i);
density = @(z) exp(lpdfgweibull(z,shape,scale));
if debug
[shape, scale, x(k-1)]
end
if isoctave
s = quadv(density, 0, x(k-1),1e-10);
else
s = integral(density, 0, x(k-1));
end
if debug
[s, abs(p-s)]
end
if isoctave
t(k) = abs(p-s)<1e-9;
else
t(k) = abs(p-s)<1e-12;
end
end
end
end
T = all(t);
%@eof:4
%@test:5
% Test with vector-valued proba input.
% Set the hyperparameters of a Weibull distribution.
scale = .5;
shape = 1.5;
% Build a probability vector including boundary cases.
q = [0, 0.25, 0.5, 0.75, 1];
try
x = wblinv(q, scale, shape);
t(1) = true;
catch
t(1) = false;
end
% Check the results.
if t(1)
t(2) = isequal(size(x), size(q));
t(3) = isequal(x(1), 0); % p = 0 → x = 0
t(4) = isinf(x(end)); % p = 1 → x = Inf
% Check round-trip consistency with wblcdf for interior points.
for i = 2:length(q)-1
t(i+3) = abs(wblcdf(x(i), scale, shape) - q(i)) < 1e-12;
end
end
T = all(t);
%@eof:5
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