function [skekurtest] = skekurtest(X,d,alpha)
% Hypotheses test concerning skewness and kurtosis.
%(This file consider both the g statistic and the beta estimates measures.)
%
% Syntax: function [skekurtest] = skekurtest(X,d,alpha)
%
% Inputs:
% X - data vector.
% d - direction of tests (1 = one-tailed; 2 = two-tailed)[default = 2].
% alpha - significance level (default = 0.05).
%
% Outputs:
% For an one-tailed hypothesis (d = 1)
% - Whether or not the skewness to left or rigth were met.
% - Whether or not the kurtosis to left or rigth were met.
% For a two-tailed hypothesis (d = 2)
% - Whether or not the skewness and kurtosis were met.
%
%
% Example: From the example 6.1 of Zar (1999, p.70-71), we are interested to ask
% whether the data are skewned to left or to the rigth and also whether
% they are leptokurtic or platykurtic (d = 1) with a significance
% level = 0.05.
%
% x Frequency
% ----------------------
% 63 2
% 64 2
% 65 3
% 66 5
% 67 4
% 68 6
% 69 5
% 70 8
% 71 7
% 72 7
% 73 10
% 74 6
% 75 3
% 76 2
% ----------------------
%
% Data matrix must be:
% X=[63;63;64;64;65;65;65;66;66;66;66;66;67;67;67;67;68;68;68;68;68;68;69;69;69;69;69;
% 70;70;70;70;70;70;70;70;71;71;71;71;71;71;71;72;72;72;72;72;72;72;73;73;73;73;73;73;
% 73;73;73;73;74;74;74;74;74;74;75;75;75;76;76];
%
% Calling on Matlab the function:
% skekurtest(X,1)
%
% Answer is:
%
% ------------------------------------
% E s t i m a t o r
% ------------------------------------
% g b
% ------------------------------------
% Skewness -0.3452 -0.3378
% ------------------------------------
% Test to assesing skewness z= -1.2292
% Probability associated to the z statistic = 0.1095
% With a given significance = 0.050
% The population distribution is not skewed to left.
%
% ------------------------------------
% E s t i m a t o r
% ------------------------------------
% g b
% ------------------------------------
% kurtosis -0.7182 2.2476
% ------------------------------------
% Test to assesing kurtosis z= 1.2762
% Probability associated to the z statistic = 0.1009
% With a given significance = 0.050
% The population distribution is leptokurtic.
%
% Created by A. Trujillo-Ortiz and R. Hernandez-Walls
% Facultad de Ciencias Marinas
% Universidad Autonoma de Baja California
% Apdo. Postal 453
% Ensenada, Baja California
% Mexico.
% atrujo@uabc.mx
%
% September 09, 2003.
%
% To cite this file, this would be an appropriate format:
% Trujillo-Ortiz, A. and R. Hernandez-Walls. (2003). skekurtest: Hypotheses test concerning skewness
% and kurtosis. A MATLAB file. [WWW document]. URL http://www.mathworks.com/matlabcentral/fileexchange/
% loadFile.do?objectId=3953&objectType=FILE
%
% References:
%
% Zar, J. H. (1999), Biostatistical Analysis (2nd ed.).
% NJ: Prentice-Hall, Englewood Cliffs. pp. 70-71,115-119.
%
if nargin < 3,
alpha = 0.05;
end;
if (alpha <= 0 | alpha >= 1)
fprintf('Warning: significance level must be between 0 and 1\n');
return;
end;
if nargin == 1,
d = 2;
alpha = 0.05;
end;
n = length(X);
[c,v]=hist(X,X); %record of data in a frequency table form
nc=find(c~=0);
c=[v(nc) c(nc)'];
x = c(:,1);
f = c(:,2);
s1 = f'*x;
s2 = f'*x.^2;
s3 = f'*x.^3;
s4 = f'*x.^4;
SS = s2-(s1^2/n);
v = SS/(n-1);
k3 = ((n*s3)-(3*s1*s2)+((2*(s1^3))/n))/((n-1)*(n-2));
g1 = k3/sqrt(v^3);
k4 = ((n+1)*((n*s4)-(4*s1*s3)+(6*(s1^2)*(s2/n))-((3*(s1^4))/(n^2)))/((n-1)*(n-2)*(n-3)))-((3*(SS^2))/((n-2)*(n-3)));
g2 = k4/v^2;
eg1 = ((n-2)*g1)/sqrt(n*(n-1)); %measure of skewness
eg2 = ((n-2)*(n-3)*g2)/((n+1)*(n-1))+((3*(n-1))/(n+1)); %measure of kurtosis
%Testing skewness
A = eg1*sqrt(((n+1)*(n+3))/(6*(n-2)));
B = (3*((n^2)+(27*n)-70)*((n+1)*(n+3)))/((n-2)*(n+5)*(n+7)*(n+9));
C = sqrt(2*(B-1))-1;
D = sqrt(C);
E = 1/sqrt(log(D));
F = A/sqrt(2/(C-1));
Zg1 = E*log(F+sqrt(F^2+1));
if d == 1;
if Zg1 <= 0;
P1 = normcdf(Zg1);
fprintf('------------------------------------\n');
disp(' E s t i m a t o r ')
fprintf('------------------------------------\n');
disp(' g b ')
fprintf('------------------------------------\n');
fprintf('Skewness %3.4f%12.4f\n', g1,eg1);
fprintf('------------------------------------\n');
fprintf('Test to assesing skewness z= %3.4f\n', Zg1);
fprintf('Probability associated to the z statistic = %3.4f\n', P1);
fprintf('With a given significance = %3.3f\n', alpha);
if P1 <= alpha;
disp('The population distribution is skewed to left.');
else
disp('The population distribution is not skewed to left.');
end;
disp(' ');
else (Zg1 > 0);
P1 = 1-normcdf(Zg1);
fprintf('------------------------------------\n');
disp(' E s t i m a t o r ')
fprintf('------------------------------------\n');
disp(' g b ')
fprintf('------------------------------------\n');
fprintf('Skewness %3.4f%12.4f\n', g1,eg1);
fprintf('------------------------------------\n');
fprintf('Test to assesing skewness z= %3.4f\n', Zg1);
fprintf('Probability associated to the z statistic = %3.4f\n', P1);
fprintf('With a given significance = %3.3f\n', alpha);
if P1 >= alpha;
disp('The population distribution is skewed to rigth.');
else
disp('The population distribution is not skewed to rigth.');
end;
end;
else (d == 2);
Zg1 = abs(Zg1);
P1 = 2*(1-normcdf(Zg1));
fprintf('------------------------------------\n');
disp(' E s t i m a t o r ')
fprintf('------------------------------------\n');
disp(' g b ')
fprintf('------------------------------------\n');
fprintf('Skewness %3.4f%12.4f\n', g1,eg1);
fprintf('------------------------------------\n');
fprintf('Test to assesing skewness z= %3.4f\n', Zg1);
fprintf('Probability associated to the z statistic = %3.4f\n', P1);
fprintf('With a given significance = %3.3f\n', alpha);
if P1 >= alpha;
disp('The population distribution is symmetrical.');
else
disp('The population distribution is not symmetrical.');
end;
end;
%Testing kurtosis
G = (24*n*(n-2)*(n-3))/((n+1)^2*(n+3)*(n+5));
H = ((n-2)*(n-3)*abs(g2))/((n+1)*(n-1)*sqrt(G));
J = ((6*(n^2-(5*n)+2))/((n+7)*(n+9)))*sqrt((6*(n+3)*(n+5))/((n*(n-2)*(n-3))));
K = 6+((8/J)*((2/J)+sqrt(1+(4/J^2))));
L = (1-(2/K))/(1+H*sqrt(2/(K-4)));
Zg2 = (1-(2/(9*K))-L^(1/3))/sqrt(2/(9*K));
if d == 1;
if Zg2 <= 0;
P2 = normcdf(Zg2);
fprintf('------------------------------------\n');
disp(' E s t i m a t o r ')
fprintf('------------------------------------\n');
disp(' g b ')
fprintf('------------------------------------\n');
fprintf('kurtosis %3.4f%12.4f\n', g2,eg2);
fprintf('------------------------------------\n');
fprintf('Test to assesing kurtosis z= %3.4f\n', Zg2);
fprintf('Probability associated to the z statistic = %3.4f\n', P2);
fprintf('With a given significance = %3.3f\n', alpha);
if P2 <= alpha;
disp('The population distribution is platykurtic.');
else
disp('The population distribution is not platikurtic.');
end;
disp(' ');
else (Zg2 > 0);
P2 = 1-normcdf(Zg2);
fprintf('------------------------------------\n');
disp(' E s t i m a t o r ')
fprintf('------------------------------------\n');
disp(' g b ')
fprintf('------------------------------------\n');
fprintf('kurtosis %3.4f%12.4f\n', g2,eg2);
fprintf('------------------------------------\n');
fprintf('Test to assesing kurtosis z= %3.4f\n', Zg2);
fprintf('Probability associated to the z statistic = %3.4f\n', P2);
fprintf('With a given significance = %3.3f\n', alpha);
if P2 >= alpha;
disp('The population distribution is leptokurtic.');
else
disp('The population distribution is not leptokurtic.');
end;
end;
else (d == 2);
disp(' ');
Zg2 = abs(Zg2);
P2 = 2*(1-normcdf(Zg2));
fprintf('------------------------------------\n');
disp(' E s t i m a t o r ')
fprintf('------------------------------------\n');
disp(' g b ')
fprintf('------------------------------------\n');
fprintf('kurtosis %3.4f%12.4f\n', g2,eg2);
fprintf('------------------------------------\n');
fprintf('Test to assesing kurtosis z= %3.4f\n', Zg2);
fprintf('Probability associated to the z statistic = %3.4f\n', P2);
fprintf('With a given significance = %3.3f\n', alpha);
if P2 >= alpha;
disp('The population distribution is mesokurtic.');
else
disp('The population distribution is not mesokurtic.');
end;
end;
