Take the 2-minute tour ×
Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It's 100% free, no registration required.

I tried to search info regarding this fact but I don't really understand why most of the standard goodness of fit tests (e.g. Kolmogorov-Smirnov, Anderson-Darling, a part of the Chi-square test, perhaps!) work only with continuous distributions. Can someone help me? Thank You.

share|improve this question
2  
What do you mean by a "validation test"? Also, please note that the chi-square test of goodness of fit is specifically for discrete distributions; it can be applied to continuous distributions only by binning them (that is, turning them into discrete distributions). –  whuber Sep 15 '11 at 16:36
    
Sorry, with validation tests i mean goodness of fit tests! in order to decide if a data sample come from a specific distribution. Ok leave the Chi-square test and take into account the others. –  Maurizio Sep 15 '11 at 17:03
    
You seem to be creating an artificial problem with your exclusion of the chi-square test from consideration. Since the chi-square test works (well) with discrete distributions, what help do you need exactly? –  whuber Sep 15 '11 at 17:07
1  
In other words, i don't understand why different goodness of fit tests works only for continuous distributions. Ok that the chi-square test works well with discrete distributions, but why the other (for example Kolmogorov-Smirnov, Anderson-Darling, etc) don't work for the discrete ones? –  Maurizio Sep 15 '11 at 17:24
add comment

1 Answer

The reason for the KS test is that its generality, e.g. it's usefulness for non-parametric models comes from the definition of the test statistic under the assumption of the CDF being continuous.

Where we define the KS Statistic as

$$D_n(F) = \max\left(D_n^+(F), D_n^-(F)\right)$$

$$D_n^+(F) = \sup_{x \in \mathbb{R}} [F_n(x) - F(x)]$$

(and the reverse for $D_n^-(F)$).

Then under the null $D_n^+(F) = \max_{0 \le i \le n} \left( F_n(x_i) - F(X_{(i)}) \right)$

Recall that under the null $F(X_{(i)})$ is continuous uniform on $(0,1)$ so the distribution of $F$ doesn't matter.

So you can create your own K-S like test for any discrete distribution, but it won't be a generalized test.

Reference/Citation, Mathematical Statistics (Shao 2010)

share|improve this answer
1  
I suspect this good reply hasn't been upvoted more because it doesn't sufficiently emphasize the important point that the sampling distribution of the K-S statistic has been computed for all continuous distributions (because they become uniform under the probability transformation) while each discrete distribution leads to its own idiosyncratic distribution of the K-S statistic. –  whuber Oct 16 '11 at 16:52
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.