Calculating likelihood for a particular biological sample. Is is possible?

#1
Hi you all!

I've got a problem when trying to analyze some biological data in my PhD. Let's say that I'm interested in performing an analysis using observed values of a random variable (O), whose distribution is unknown, and their expected values (E) using a model like:

Ei = k1 * exp ( - k2 · xi )

Where k1 and k2 are known constants, Ei is the expected value of the variable in an observation i, and xi is the value of a variable in the observation i, that I also can calculate.

What I would like to do here is:

1) to calculate how likely is the data (O) given this model (E), that is: p(O|E). So I wonder if non-parametric likelihood (NPMLE) is what I have to try here. In that case, what would be a good start point? I know nothing about NPMLE.

I guess that, if it is possible to obtain a likelihood function from here, calculating an expression for maximize likelihood for a parameter is more or less straight forward.

2) Once done 1) and, since variable x depends also in another variable y, given two different scenarios, E1 (x1,y1) and E2 (x2,y2), it would be possible to calculate a p-value in order to evaluate differences in E1 and E2?

I would like to add some extra information about the analysis and variable I'm working with:

* There is no complete independence between observations. For example, observation i=1 can be independent of i=2, but i=1 and i=3 can be highly dependent. So some test like Pearson's chi squared test should be avoid to solve this task.
* Regression can be done, since log(Ei) = log(k1) - k2*x1, but I'm looking for an additional analysis that does not need log transformation.
 
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