function [H, pValue, W] = swtest(x, alpha)
%SWTEST Shapiro-Wilk parametric hypothesis test of composite normality.
% [H, pValue, SWstatistic] = SWTEST(X, ALPHA) performs the
% Shapiro-Wilk test to determine if the null hypothesis of
% composite normality is a reasonable assumption regarding the
% population distribution of a random sample X. The desired significance
% level, ALPHA, is an optional scalar input (default = 0.05).
%
% The Shapiro-Wilk and Shapiro-Francia null hypothesis is:
% "X is normal with unspecified mean and variance."
%
% This is an omnibus test, and is generally considered relatively
% powerful against a variety of alternatives.
% Shapiro-Wilk test is better than the Shapiro-Francia test for
% Platykurtic sample. Conversely, Shapiro-Francia test is better than the
% Shapiro-Wilk test for Leptokurtic samples.
%
% When the series 'X' is Leptokurtic, SWTEST performs the Shapiro-Francia
% test, else (series 'X' is Platykurtic) SWTEST performs the
% Shapiro-Wilk test.
%
% [H, pValue, SWstatistic] = SWTEST(X, ALPHA)
%
% Inputs:
% X - a vector of deviates from an unknown distribution. The observation
% number must exceed 3 and less than 5000.
%
% Optional inputs:
% ALPHA - The significance level for the test (default = 0.05).
%
% Outputs:
% SWstatistic - The test statistic (non normalized).
%
% pValue - is the p-value, or the probability of observing the given
% result by chance given that the null hypothesis is true. Small values
% of pValue cast doubt on the validity of the null hypothesis.
%
% H = 0 => Do not reject the null hypothesis at significance level ALPHA.
% H = 1 => Reject the null hypothesis at significance level ALPHA.
%
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% Copyright (c) 17 March 2009 by Ahmed Ben Sada %
% Department of Finance, IHEC Sousse - Tunisia %
% Email: ahmedbensaida@yahoo.com %
% $ Revision 3.0 $ Date: 18 Juin 2014 $ %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% References:
%
% - Royston P. "Remark AS R94", Applied Statistics (1995), Vol. 44,
% No. 4, pp. 547-551.
% AS R94 -- calculates Shapiro-Wilk normality test and P-value
% for sample sizes 3 <= n <= 5000. Handles censored or uncensored data.
% Corrects AS 181, which was found to be inaccurate for n > 50.
% Subroutine can be found at: http://lib.stat.cmu.edu/apstat/R94
%
% - Royston P. "A pocket-calculator algorithm for the Shapiro-Francia test
% for non-normality: An application to medicine", Statistics in Medecine
% (1993a), Vol. 12, pp. 181-184.
%
% - Royston P. "A Toolkit for Testing Non-Normality in Complete and
% Censored Samples", Journal of the Royal Statistical Society Series D
% (1993b), Vol. 42, No. 1, pp. 37-43.
%
% - Royston P. "Approximating the Shapiro-Wilk W-test for non-normality",
% Statistics and Computing (1992), Vol. 2, pp. 117-119.
%
% - Royston P. "An Extension of Shapiro and Wilk's W Test for Normality
% to Large Samples", Journal of the Royal Statistical Society Series C
% (1982a), Vol. 31, No. 2, pp. 115-124.
%
%
% Ensure the sample data is a VECTOR.
%
if numel(x) == length(x)
x = x(:); % Ensure a column vector.
else
error(' Input sample ''X'' must be a vector.');
end
%
% Remove missing observations indicated by NaN's and check sample size.
%
x = x(~isnan(x));
if length(x) < 3
error(' Sample vector ''X'' must have at least 3 valid observations.');
end
if length(x) > 5000
warning('Shapiro-Wilk test might be inaccurate due to large sample size ( > 5000).');
end
%
% Ensure the significance level, ALPHA, is a
% scalar, and set default if necessary.
%
if (nargin >= 2) && ~isempty(alpha)
if ~isscalar(alpha)
error(' Significance level ''Alpha'' must be a scalar.');
end
if (alpha <= 0 || alpha >= 1)
error(' Significance level ''Alpha'' must be between 0 and 1.');
end
else
alpha = 0.05;
end
% First, calculate the a's for weights as a function of the m's
% See Royston (1992, p. 117) and Royston (1993b, p. 38) for details
% in the approximation.
x = sort(x); % Sort the vector X in ascending order.
n = length(x);
mtilde = norminv(((1:n)' - 3/8) / (n + 1/4));
weights = zeros(n,1); % Preallocate the weights.
if kurtosis(x) > 3
% The Shapiro-Francia test is better for leptokurtic samples.
weights = 1/sqrt(mtilde'*mtilde) * mtilde;
%
% The Shapiro-Francia statistic W' is calculated to avoid excessive
% rounding errors for W' close to 1 (a potential problem in very
% large samples).
%
W = (weights' * x)^2 / ((x - mean(x))' * (x - mean(x)));
% Royston (1993a, p. 183):
nu = log(n);
u1 = log(nu) - nu;
u2 = log(nu) + 2/nu;
mu = -1.2725 + (1.0521 * u1);
sigma = 1.0308 - (0.26758 * u2);
newSFstatistic = log(1 - W);
%
% Compute the normalized Shapiro-Francia statistic and its p-value.
%
NormalSFstatistic = (newSFstatistic - mu) / sigma;
% Computes the p-value, Royston (1993a, p. 183).
pValue = 1 - normcdf(NormalSFstatistic, 0, 1);
else
% The Shapiro-Wilk test is better for platykurtic samples.
c = 1/sqrt(mtilde'*mtilde) * mtilde;
u = 1/sqrt(n);
% Royston (1992, p. 117) and Royston (1993b, p. 38):
PolyCoef_1 = [-2.706056 , 4.434685 , -2.071190 , -0.147981 , 0.221157 , c(n)];
PolyCoef_2 = [-3.582633 , 5.682633 , -1.752461 , -0.293762 , 0.042981 , c(n-1)];
% Royston (1992, p. 118) and Royston (1993b, p. 40, Table 1)
PolyCoef_3 = [-0.0006714 , 0.0250540 , -0.39978 , 0.54400];
PolyCoef_4 = [-0.0020322 , 0.0627670 , -0.77857 , 1.38220];
PolyCoef_5 = [0.00389150 , -0.083751 , -0.31082 , -1.5861];
PolyCoef_6 = [0.00303020 , -0.082676 , -0.48030];
PolyCoef_7 = [0.459 , -2.273];
weights(n) = polyval(PolyCoef_1 , u);
weights(1) = -weights(n);
if n > 5
weights(n-1) = polyval(PolyCoef_2 , u);
weights(2) = -weights(n-1);
count = 3;
phi = (mtilde'*mtilde - 2 * mtilde(n)^2 - 2 * mtilde(n-1)^2) / ...
(1 - 2 * weights(n)^2 - 2 * weights(n-1)^2);
else
count = 2;
phi = (mtilde'*mtilde - 2 * mtilde(n)^2) / ...
(1 - 2 * weights(n)^2);
end
% Special attention when n = 3 (this is a special case).
if n == 3
% Royston (1992, p. 117)
weights(1) = 1/sqrt(2);
weights(n) = -weights(1);
phi = 1;
end
%
% The vector 'WEIGHTS' obtained next corresponds to the same coefficients
% listed by Shapiro-Wilk in their original test for small samples.
%
weights(count : n-count+1) = mtilde(count : n-count+1) / sqrt(phi);
%
% The Shapiro-Wilk statistic W is calculated to avoid excessive rounding
% errors for W close to 1 (a potential problem in very large samples).
%
W = (weights' * x) ^2 / ((x - mean(x))' * (x - mean(x)));
%
% Calculate the normalized W and its significance level (exact for
% n = 3). Royston (1992, p. 118) and Royston (1993b, p. 40, Table 1).
%
newn = log(n);
if (n >= 4) && (n <= 11)
mu = polyval(PolyCoef_3 , n);
sigma = exp(polyval(PolyCoef_4 , n));
gam = polyval(PolyCoef_7 , n);
newSWstatistic = -log(gam-log(1-W));
elseif n > 11
mu = polyval(PolyCoef_5 , newn);
sigma = exp(polyval(PolyCoef_6 , newn));
newSWstatistic = log(1 - W);
elseif n == 3
mu = 0;
sigma = 1;
newSWstatistic = 0;
end
%
% Compute the normalized Shapiro-Wilk statistic and its p-value.
%
NormalSWstatistic = (newSWstatistic - mu) / sigma;
% NormalSWstatistic is referred to the upper tail of N(0,1),
% Royston (1992, p. 119).
pValue = 1 - normcdf(NormalSWstatistic, 0, 1);
% Special attention when n = 3 (this is a special case).
if n == 3
pValue = 6/pi * (asin(sqrt(W)) - asin(sqrt(3/4)));
% Royston (1982a, p. 121)
end
end
%
% To maintain consistency with existing Statistics Toolbox hypothesis
% tests, returning 'H = 0' implies that we 'Do not reject the null
% hypothesis at the significance level of alpha' and 'H = 1' implies
% that we 'Reject the null hypothesis at significance level of alpha.'
%
H = (alpha >= pValue);