# Generalization of one-sample Kolmogorov-Smirnov test for non identically distributed data?

Is there a standard approach for testing goodness-of-fit to a probability distribution when the samples are independent but not identically distributed? In other words, given the data $\{\{x_1, y_1\}, \{x_2, y_2\},..., \{x_n, y_n\}\}$ and the candidate density function $f(x;y)$, what is an accepted approach for testing whether the $x_i$ are sampled from $X|y_i \sim f(x; y_i)$? What are the assumptions of said approach?

Edit

In the comments, whuber suggests it is more appropriate to call $y_i$ a parameter for the distribution on $x_i$ and requests more information about how $f(x; y)$ depends on $y$. NB, I'm interested in "standard methods" more than a custom solution, so the answer should not depend on the details of $f$. That said, in my application, $f$ is a mixture of $m$ Dirichlet-Multinomial distributions (so $x$ is discrete, I asked about continuous variables above in order to reference the K-S test). The vector $\{y_{i,1},y_{i,2},...,y_{i,m}\}$ is the mixture weighting. The parameter for each Dirichlet-Multinomial distribution is independent of $y_i$.

• You don't quite define a probability distribution: you need also to specify a distribution for the $x_i$. When you do, that will determine a single univariate distribution for $f(x,y)$ which can then be tested.
– whuber
Commented Apr 10, 2014 at 21:45
• Hmm. I'm not sure the KS is a good 'starting point' for thinking about this problem. I don't currently have a good notion of an especially good way to think about it without specifying the kinds of alternatives you want to be able to identify, though. Commented Apr 10, 2014 at 23:23
• @whuber Can you explain further? $f$ is the univariate density for each $x_i$ in my problem statement. What is missing? The joint density on $\{x, y\}$ is unknown, and unnecessary because y is given in the application.
– Ian
Commented Apr 11, 2014 at 13:52
• @Glen_b Thanks for thinking about the question? What other information can I provide to help you come up with a different starting point?
– Ian
Commented Apr 11, 2014 at 13:56
• I think I follow: because you state the $y_i$ are "given" (and presumably measured without error) then, in effect, the $y_i$ play the roles of parameters rather than random variables. (The notation "$X|y_i$" suggests otherwise, unfortunately.) Several methods to deal with this situation come to mind, depending on exactly how the $y_i$ control $f$. Could you provide some details of that?
– whuber
Commented Apr 11, 2014 at 13:57