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We have a model relationship between three random variables like this:

$$ U = C + S $$

I have a ton of measurements of realizations of $U$, as well as a ton of realizations of $C$. But the interesting thing is the density of $S$. The measurements are not paired, so I can not take the differences $U-C$.

One solution we tried was called deconvolution, which does exactly what we want, i.e. extract the density of a "noise" variable $S$. This approach brought us fine results. Alternatively, we wanted to try a parametric approach, specifically the method of moments.

My first question: Is the following algorithm correct, assuming gamma distributed $U$, $C$ and $S$? This should exactly reflect the method of moments, if I understood it correctly:

  1. Fit a gamma distribution to $U$ and $C$, and store the parameters $k_U$, $k_C$, $\theta_U$ and $\theta_C$, using the wikipedia notation.
  2. Compute the expected value and variance of $U$ and $C$, using $\mu = k\theta$ and $\sigma^2 = k\theta^2$. Denote them as $\mu_U$, $\sigma^2_U$, $\mu_C$ and $\sigma^2_C$.
  3. Assuming independence, compute $\mu_S = \mu_U - \mu_C$ and $\sigma^2_S = \sigma^2_U - \sigma^2_C$.
  4. Solve the equation system $\mu_S=k_S\theta_S$ and $\sigma_S^2 = k_S\theta_S^2$ for $\mu_S$ and $\theta_S$.

Then I have the two parameters, and thus the gamma density of $S$.

But the kernel density estimates of $U$ and $C$ are very different from the fitted gamma densities. So, I'd like to try this approach with Weibull distribution assumptions.

This leads to my second question: How would I proceed when I assume Weibull distributed $U$, $C$ and $S$? The formula for the mean and variance look very ugly over at Wikipedia. Are there analytical solutions or do I have to go R and do some numerical optimizing there?

Finally, if you are still there, my third question: Deconvolution fails as soon as $C$ and $S$ are correlated. With the method of moments, I think I can just see how the distribution of $S$ would change by assuming a correlation of, say, 0.5. I would do this by altering only the step 3. in my first question, and computing $\sigma^2_S = \sigma^2_U - \sigma^2_C - 2 Cov(U,C)$. Is that a valid approach?

chl
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Alexander Engelhardt
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    You suggest early in this question that deconvolution might *not* be "parametric." How do you accomplish that without parameterizing $S$? – whuber Nov 13 '12 at 20:41
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    Sorry, my partner is the expert on the deconvolution part. The R package "deamer" has a deconvolution function which takes a sample of U and a sample of C and produces a density for S. The function calls in his code look like `deamerSE2(U.sample, error=C.sample, from=from, to=to, grid.length=100000)`. From that I assumed we don't need distribution assumptions. – Alexander Engelhardt Nov 13 '12 at 21:03

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