# Testing if data comes from a normal distribution with mean 0 and unknown variance in Matlab

## Question

I have a vector of data, and I want to test if it came from a normal distribution with mean zero and unknown variance. Do you know if there is matlab function or simple script for this? If you don't know anything matlab specific, then a name and reference for the specific test is fine and I will just implement it myself.

Also, if the specific test can return the confidence level instead of just answering yes-no at a given confidence level then that would be a benefit, but is not essential.

If I want to test if my data is from a normal distribution with mean 0 and variance 1 then I can use the Kolmogorov-Smirnov test. If I want if my data is from a normal distribution with unknown mean AND variance then I can use the Lilliefors test or the Jarque-Bera test. However, I want a fixed mean (= 0) and unknown variance.

## Naive approach

The naive approach is to take my data $D$, calculate the variance around zero $\sigma^2_0$ and then renormalize my data by this to get a dataset $D'$. Then I can perform the Kolmogorov-Smirnov test on this. However, it is not clear how one would justify this, especially since the KS tests specifically warns against testing against distributions with parameters estimated from the same data (renormalizing $D$ to $D'$ will be the same as testing against a normal distibution with mean zero and variance $\sigma^2_0$). Is this naive approach justified?

• You need to specify an alternative hypothesis that you want to test this normal distribution against. If your hypothesis is false, what will you put in its place? Put another way, what would be the next thing that you would try if this (as yet unspecified) test came back saying "reject the hypothesis - this data is not normally distributed". Unless you can answer this question, what is the point of going to all the trouble of a formal test? – probabilityislogic Aug 19 '11 at 13:18
• There is a body of theory that deals with the case where parameters of the distribution are estimated from the data before the KS test is applied. The reference is J. Durbin, Distribution theory for tests based on the sample distribution function, SIAM 1973. Alternatively, you could use the Anderson Darling test for normality which can deal with the case where $\mu$ is known and $\sigma$ is unknown. There is a Matlab implementation of the latter called AnDartest, available in File Exchange. – Hans Engler Aug 19 '11 at 13:39
• @probabilityislogic I don't care what distribution the data came from if it didn't come from the normal; thus I don't need any further tests. – Artem Kaznatcheev Aug 19 '11 at 14:05
• @Artem - if you don't care about alternatives, you probably don't need a formal test - just look at a histogram of the observations that are supposed to be normal. Your naive method should also be sufficient too. The problem with ignoring alternatives is that you can't tell what power your test has (i.e. how often the test correctly rejects the null) – probabilityislogic Aug 19 '11 at 16:24
• You can use Spiegelhalter's test, which is fairly powerful against fat-tailed alternatives and simple to code in Matlab: stackoverflow.com/questions/1882944/… Another advantage is that you need not specify the mean nor the standard deviation. It may not maintain the nominal rate at small sample sizes (less than 50 say), but should be OK otherwise. – shabbychef Aug 21 '11 at 4:09

You can use Spiegelhalter's test (1983, not the 'omnibus test' from 1977):

function pval = spiegel_test(x)
% compute pvalue under null of x normally distributed;
% x should be a vector;
% D. J. Spiegelhalter, 'Diagnostic tests of distributional shape,'
% Biometrika, 1983
xm = mean(x);
xs = std(x);
xz = (x - xm) ./ xs;
xz2 = xz.^2;
N = sum(xz2 .* log(xz2));
n = numel(x);
ts = (N - 0.73 * n) / (0.8969 * sqrt(n)); %under the null, ts ~ N(0,1)
pval = 1 - abs(erf(ts / sqrt(2)));    %2-sided test. if only Matlab had R's pnorm function ...


I include code to test this under the null and under a few alternatives:

% under H0:
pvals = nan(10000,1);
for tt=1:numel(pvals);
pvals(tt) = spiegel_test(randn(300,1));
end
mean(pvals < 0.05)


I get something like:

ans =

0.0512


Under some alternatives:

%under Ha (using a Tukey g-distribution)
g = 0.4;
pvals = nan(10000,1);
for tt=1:numel(pvals);
pvals(tt) = spiegel_test((exp(g * randn(300,1)) - 1)/g);
end
mean(pvals < 0.05)

%under Ha (using a Tukey h-distribution)
h = 0.1;
pvals = nan(10000,1);
for tt=1:numel(pvals);
x = randn(300,1);
pvals(tt) = spiegel_test(x .* exp(0.5 * h * x.^2));
end
mean(pvals < 0.05)


I get:

ans =

0.8494

ans =

0.8959


This test discards the knowledge that the mean must equal zero, so is perhaps less powerful than other tests. Spiegelhalter notes this test performs reasonably well for sample sizes greater than about 25, and is designed to test against symmetric alternatives (e.g. the Tukey h-distribution). It is less powerful against asymmetric alternatives.