# What is a quadravariate distribution?

Question: What is a quadravariate distribution?

Motivation: I found a reference to quadravariate distributions in Priestley (1981) Spectral Analysis and Time Series on p325, but no definition (the author seems to think it is common knowledge). 10 minutes of googling revealed that the term gets used in quite a few papers, but I am still unable to find a definition anywhere.

• I'd interpret it to be a distribution on 4 variables - the next in the series univariate, bivariate, trivariate, .... However, it's more often spelled quadrivariate, which might explain why you had trouble finding it. Googling quadrivariate turns up many hits. But for that matter, when I google quadravariate the second and fourth hits (that I get, your experience may be different) are consistent with this interpretation, so it's not hard to find. What's the name of the Priestly reference? Sep 6, 2013 at 1:51
• @Glen_b Thanks for the response. If you turn that into an answer I'll give you the tick. Re: Priestly reference, haven't I given the name in the question? Or are you asking after something different? Also, your explanation implies that the class of quadra(i)variate distributions is quite broad, ie one can have a parameter for location, scale, asymmetry, and fat-tails. Is this interpretation correct? Sep 6, 2013 at 2:11
• I meant 'the title of the work'. Sorry to be unclear. You seem to have misinterpreted the meaning of the term; it doesn't mean a distribution with four parameters, but one with four variables. I will clarify in my answer. Sep 6, 2013 at 2:19
• @Glen_b Oh! So it is just any joint distribution that is defined over 4 random variables? Well that is a lot simpler than I was imagining :-) Also, the Priestly reference in the question is a textbook, not a journal article, so what I have provided is the title. Here is the google book reference. Thanks again. I'll tick and upvote forthcoming answer. Sep 6, 2013 at 2:28
• Yes, from the couple of lines Google books will show me, that's exactly what is being discussed on p 325 - a 'well known' result on joint distributions defined over four random variables. (It won't show me the result though) Sep 6, 2013 at 2:55

## 1 Answer

I'd interpret it to be a distribution on 4 variables - the next in the series univariate, bivariate, trivariate, ....

It's a multivariate random variable with four components.

However, the word is more often spelled quadrivariate, which might explain why you had trouble finding it.

To clarify; consider a gamma random variable $X\sim\text{gamma}(\alpha,\beta)$. That's univariate random variable; we say that the distribution of $X$ is a univariate distribution.

Now consider independent Poisson random variates $Z\sim\text{Pois}(\zeta)$, $X_1\sim\text{Pois}(\lambda_1)$ and $X_2\sim\text{Pois}(\lambda_2)$, and let $Y_1=X_1+Z$ and $Y_2=X_2+Z$. Then the distribution of $\left(\begin{array}{c}Y_1\\Y_2\end{array}\right)$ is an example of a bivariate Poisson distribution (in this case, one which has a correlation between its two components). Correspondingly $(Z,X_1,Y_2)'$ would be trivariate.

Now consider four independent normally distributed random variables $X_1,X_2,X_3$, and $X_4$, with mean vector $\mu=(\mu_1,\mu_1,\mu_1,\mu_0)'$ and variance-covariance matrix $\Sigma=\sigma^2I$.

Then $\bf{X}$ $=(X_1,X_2,X_3,X_4)'$ is an example of a quadrivariate random variable.

So if for each observational unit I record four quantities, the observation on that unit is quadrivariate. The distribution of that quadrivariate random variable is a quadrivariate distribution.