# Test for unimodal symmetric distribution

I am trying to test whether data from a sample I have follows a t distribution with n degrees of freedom for a given n.

I am looking for something more powerful/recent than Kolmogorov-Smirnov test. It need not be a test for t- distributions specifically, it can be anything geared for unimodal and symmetric distributions.

Any references/Matlab code would be very helpful.

Thanks a lot.

PS: My sample has thousands of points so I don't need anything involving bootstrapping and so on.

• 1. Your title and the body of your post ask for very different things. Do you want a test for a $t_n$ or a test for a unimodal symmetric distribution? A beta(2,2) distribution and a double exponential distribution are both unimodal&symmetric but neither are like any $t_n$. Please amend your question. Jan 12, 2015 at 0:05
• 2. No test will tell you that your data are from some distribution. With large samples, good tests highly likely to tell you they are not -- even when the distribution is very very close to the one you're testing against (since when do real data exactly follow simple models?). $\hspace{9cm}$ 3. If you do persist with goodness of fit tests, do you have any particular kinds of alternatives you seek power against? Jan 12, 2015 at 0:08
• 4. I understand seeking more power, but why would recency be relevant? $\qquad$ 5. The arguments that establish the useful properties of bootstrapping are asymptotic. Bootstrapping is quite well suited to large samples and may often fail to achieve good properties in small samples. Jan 12, 2015 at 0:15
• 6. Is this a standard t, or are there unspecified location and scale parameters? Jan 12, 2015 at 0:21
• Hi Glen_b, yes I am testing goodness of fit against standard t distribution. I thought it might be too much to hope for a test geared precisely for the t-distribution, so was hoping for something geared for unimodal symmetric distribtuions. Jan 20, 2015 at 6:01

As all decent statistical and scientific software packages, MATLAB contains functionality for fitting distributions and testing goodness-of-fit (GoF). As far as I understand, without knowing data distribution's probability density function, the empirical procedure for GoF testing is two-fold.

The first step would be to fit data to distribution, for which you can use either MATLAB internal functions, or external ones (yours or contributed by the community): http://blogs.mathworks.com/pick/2012/02/10/finding-the-best. The second step is, obviously, to select and perform a proper GoF test, as discussed below.

More details and some nice MATLAB code examples can be found in this relevant discussion here on CV. I think that, in addition to the mentioned in the discussion Kolmogorov-Smirnov, Anderson-Darling, Shapiro-Wilk, Shapiro-Francia and Liliefors GoF tests, it is feasible/important to consider a chi-square GoF test. Along with many others, chi-square GoF test is included in MATLAB Statistics Toolbox: http://www.mathworks.com/help/stats/distribution-tests.html. However, if you don't have access to Statistics Toolbox, you can use either example code from the above-mentioned CV discussion, or the chi2test() function's code from this course notes document (see pages 7-13).

In regard to the statistical power of various GoF tests, I ran across an interesting research paper, comparing the statistical power of most of the above-mentioned GoF tests (with an unfortunate exception of chi-square GoF test), showed that Shapiro-Wilk is the most powerful test, followed by Anderson-Darling, Lilliefors and Kolmogorov-Smirnov, correspondingly.

Some statements pending further clarification:

1) Power is a property of a specific alternative -- or in the case of power functions, a specific collection of alternatives. You should not necessarily expect to have a goodness of fit test that has great power against all alternatives -- some tests have more power against some kinds of alternatives, others against other kinds; knowing something about the alternatives that most interest you will help identify a test which suits the kind of power properties you seek. The more specific you can be about alternatives of interest, the better the chances of finding a test with good power, generally speaking.

2) A test for a completely specified distribution which has good properties against a wide variety of alternatives that people tend to find interesting is the Anderson-Darling test. In particular, it is more sensitive to differences of distribution in the tails (at the expense of being slightly less sensitive in the middle) than the Kolmogorov-Smirnov and performs extremely well in power studies at the normal; I would expect it has similarly good power in this case.

3) If you seek a test where location and scale are unspecified, besides adapting the Anderson-Darling to that situation, you might also consider a test something like the Shapiro-Francia type test in the normal case -- based off a correlation between order statistics and $t_n$-quantiles (in particular, I'd suggest using $n(1-r^2)$).

4) you might be able to base a test off the likelihood itself, Fisher style.