What are the distributions on the positive k-dimensional quadrant with parametrizable covariance matrix? (bis)

April 3, 2012
By

(This article was first published on Xi'an's Og » R, and kindly contributed to R-bloggers)

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,

\dfrac{\alpha_{11}}{\beta_1} = \mu_1\,,\quad \dfrac{\alpha_{11}}{\beta_1^2} = \sigma_1^2\,,

\dfrac{\alpha_{21}\mu_1+\alpha_{22}}{\beta_2} = \mu_2\,,\quad \dfrac{\alpha_{21}^2\sigma^2_1}{\beta_2^2}+\dfrac{\alpha_{21}\mu_1+\alpha_{22}}{\beta_2^2} = \sigma^2_2\,,

and

\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

\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

\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

To leave a comment for the author, please follow the link and comment on their blog: Xi'an's Og » R.

R-bloggers.com offers daily e-mail updates about R news and tutorials on topics such as: Data science, Big Data, R jobs, visualization (ggplot2, Boxplots, maps, animation), programming (RStudio, Sweave, LaTeX, SQL, Eclipse, git, hadoop, Web Scraping) statistics (regression, PCA, time series, trading) and more...



If you got this far, why not subscribe for updates from the site? Choose your flavor: e-mail, twitter, RSS, or facebook...

Tags: , ,

Comments are closed.

Sponsors

Mango solutions



RStudio homepage



Zero Inflated Models and Generalized Linear Mixed Models with R

Quantide: statistical consulting and training

datasociety

http://www.eoda.de





ODSC

ODSC

CRC R books series





Six Sigma Online Training









Contact us if you wish to help support R-bloggers, and place your banner here.

Never miss an update!
Subscribe to R-bloggers to receive
e-mails with the latest R posts.
(You will not see this message again.)

Click here to close (This popup will not appear again)