Matlab: Chi-squared fit (chi2gof) to test if data is exponentially distributed

Question

I guess this is a simple question, but I can't sort it out. I have a vector, the first elements of which look like: ``` V = [31 52 38 29 29 34 29 24 25 25 32 28 24 28 29 ...]; ``` and I want to perform a `chi2gof` test in Matlab to test if `V` is exponentially distributed. I did: ``` [h,p] = chi2gof(V,'cdf',@expcdf); ``` but I get a warning message saying: ``` Warning: After pooling, some bins still have low expected counts. The chi-square approximation may not be accurate ``` Have I defined the `chi2gof` call incorrectly?

Accepted Answer

At 36 values, you have a very small sample set. From the second sentence of Wikipedia's article on the chi-squared test (emphasis added):

It is suitable for unpaired data from large samples.

Large in this case usually means around at least 100. Read about more assumptions of this test here.

Alternatives

You might try `kstest` in Matlab, which is based on the Kolmogorov-Smirnov test:

```
[h,p] = kstest(V,'cdf',[V(:) expcdf(V(:),expfit(V))])
```

Or try `lillietest`, which is based on the Lilliefors test and has an option specifically for exponential distributed data:

```
[h,p] = lillietest(V,'Distribution','exp')
```

In case you can increase your sample size, you are doing one thing wrong with `chi2gof`. From the `help` for the `'cdf'` option:

A fully specified cumulative distribution function. This can be a ProbabilityDistribution object, a function handle, or a function. name. The function must take X values as its only argument. Alternately, you may provide a cell array whose first element is a function name or handle, and whose later elements are parameter values, one per cell. The function must take X values as its first argument, and other parameters as later arguments.

You're not supplying any additional parameters, so `expcdf` is using the default mean parameter of `mu = 1`. Your data values are very large and don't correspond at all an exponential distribution with this mean. You need to estimate parameters as well. You the `expfit` function, which is basted on maximum likelihood expectation, you might try something like this:

```
[h,p] = chi2gof(V,'cdf',@(x)expcdf(x,expfit(x)),'nparams',1)
```

However, with only 36 samples you may not get a very good estimate for a distribution like this and still may not get expected results even for data sampled from a known distribution, e.g.:

```
V = exprnd(10,1,36);
[h,p] = chi2gof(V,'cdf',@(x)expcdf(x,expfit(x)),'nparams',1)
```

Matlab: Chi-squared fit (chi2gof) to test if data is exponentially distributed

Solution

Problem