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Wondering about the question I posted on Friday (on StackExchange, no satisfactory answer so far!), I looked further at the special case of the gamma distribution I suggested at the end. Starting from the moment conditions,
and
+%5Calpha_%7B22%7D%7D%7B%5Cbeta_2%7D+=+%5Csigma_%7B12%7D+%5Cmu_1%5Cmu_2&bg=000000&fg=B0B0B0&s=0 "\dfrac{\alpha_{21}(\sigma^2_1+\mu_1^2)+\alpha_{22}}{\beta_2} = \sigma_{12}+\mu_1\mu_2")
the solution is (hopefully) given by the system
%7D%7B%5Csigma%5E2_1+%5Cmu_1%5E2-+%5Cmu_1%7D%5Cbeta_2%5C%5C+%5Cdfrac%7B(%5Csigma_%7B12%7D+%5Cmu_1%5Cmu_2-%5Cmu_2)%5E2%7D%7B(%5Csigma%5E2_1+%5Cmu_1%5E2-+%5Cmu_1)%5E2%7D+%5Csigma_1%5E2+++%5Cdfrac%7B%5Cmu_2%7D%7B%5Cbeta_2%7D+=+%5Csigma%5E2_2&%5C%5C++%5Cend%7Bcases%7D&bg=000000&fg=B0B0B0&s=0 "\begin{cases} \beta_1 =\mu_1/\sigma_1^2&\\ \alpha_{11}-\mu_1\beta_1 =0&\\ \alpha_{22} = \mu_2\beta_2 - \alpha_{21}\mu_1&\\ \alpha_{21} = \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)}{\sigma^2_1+\mu_1^2- \mu_1}\beta_2\\ \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)^2}{(\sigma^2_1+\mu_1^2- \mu_1)^2} \sigma_1^2 + \dfrac{\mu_2}{\beta_2} = \sigma^2_2&\\ \end{cases}")
The resolution of this system obviously imposes conditions on those moments, like
%5E2%7D%7B(%5Csigma%5E2_1+%5Cmu_1%5E2-+%5Cmu_1)%5E2%7D+%5Csigma_1%5E2+%3E0&bg=000000&fg=B0B0B0&s=0 "\sigma^2_2 - \dfrac{(\sigma_{12}+\mu_1\mu_2-\mu_2)^2}{(\sigma^2_1+\mu_1^2- \mu_1)^2} \sigma_1^2 >0")
So I ran a small R experiment checking when there was no acceptable solution to the system. I started with five moments that satisfied the basic Stieltjes and determinant conditions
# basically anything mu=runif(2,0,10) # Jensen inequality sig=c(mu[1]^2/runif(1),mu[2]^2/runif(1)) # my R code returning the solution if any sol(mu,c(sig,runif(1,-sqrt(prod(sig)),sqrt(prod(sig)))))
and got a fair share (20%) of rejections, e.g.
> sol(mu,c(sig,runif(1,-sqrt(prod(sig)),sqrt(prod(sig))))) $solub [1] FALSE $alpha [1] 0.8086944 0.1220291 -0.1491023 $beta [1] 0.1086459 0.5320866
However, not being sure about the constraints on the five moments I am now left with another question: what are the necessary and sufficient conditions on the five moments of a pair of positive vectors?! Or, more generally, what are the necessary and sufficient conditions on the k-dimensional μ and Σ for them to be first and second moments of a positive k-dimensional vector?
Filed under: R, Statistics, University life
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