I have a problem which doesn't really fit in my mental model: I have a set of many measured samples defined as $$X_i = Y_i + Z_i $$ where $$Y \sim Logis(\mu,s) $$ $$ Z\sim N(0,\sigma^2) $$ $Y$ follows a logistic distribution, $Z$ follows a Normal (Gaussian) distribution with 0 mean. I would like to know how to estimate $s$ and $\sigma^2$.

Estimating $\mu$ is trivial, as $E(Z+Y)=E(Z)+E(Y)$, the variance should follow the same rule $V(Z+Y) = V(Z)+V(Y)$ for $Z$ and $Y$ independent.

$Z$ and $Y$ are treated as independent, $s$ is related to the variance of $Y$: $\sigma^2_Y$ with the following $$ \sigma^2_{\ \ Y} = \frac{s^2\pi^2}{3} $$

I think that I need some sort of system to extract $\sigma^2_{\ \ Y}$ and $\sigma^2_{\ \ Z}$ from my samples, but I'm kind of confused... I'm also trying to reason with moments, as $ \sigma^2=M''(0)-[M'(0)]^2 $, but I do not have any experience with that.

  • $\begingroup$ There are several thing that are unclear. What does the variable Z represent? You mention the difference between V(X+Y) when independent and when correlated. How are you modeling them, independent or correlated.? How did you get the formula for the variance of Y? $\endgroup$ – Michael R. Chernick Dec 9 '16 at 11:49
  • $\begingroup$ Thank you @MichaelChernick for pointing this out, I'm sorry I messed up the variables. I did just edited my question correcting the variable mismatch and removing the unclear statement. $Z$ and $Y$ are independent. The formula to get $\sigma^2$ of $Y$ was given with the data and it should be related to the fact that $Y$ is a Logistic Distribution $\endgroup$ – Dallatorre Dec 9 '16 at 12:57

The natural approach is to look at the likelihood of your data, instead of fishing for estimates outside a proper model. Hence, you should derive the probability distribution of the variable $X=Y+Z$, which has the density $$f_X(x)=\int f_Y(y) f_Z(x-y)\,\text{d}y$$ and proceed from there by maximum likelihood or Bayesian estimation. In case the integral does not allow for a closed form resolution, which seems very likely, you can resort to the EM algorithm, again naturally because of the missing variable aspect of your problem. If EM is similarly intractable, Bayesian analysis with MCMC algorithms may be the only solution based on the likelihood.

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