# How to check for bivariate Gaussianity without the use of regression?

What steps could be taken to check for bivariate Gaussianity without using regression based check? Can we somehow employ the use of definition of variogram measure for assessing spatial variability?

• You might like to review replies to What is the most surprising characterization of the Gaussian (normal) distribution?. For instance, the accepted answer suggests that if you standardize the marginals to $X$ and $Y$, you need only test that (a) $X$ and $Y$ have the same distribution and (b) $X-Y$ is independent of $X+Y$. Another reply by @Robin Girard suggests yet another test. – whuber Mar 31 '11 at 22:47
• It is difficult to see how the variogram applies. It summarizes a single multivariate observation and requires strong additional assumptions in order to estimate variances and covariances. – whuber Mar 31 '11 at 22:49
• @whuber: Thanks for the link. However, that question talks about univariate gaussianity, and nothing about bivariate gaussianity. – user1102 Mar 31 '11 at 22:53
• Good point. But it seems like those replies generalize in some ways. For instance, they suggest that one way to detect bivariate normality is to check various linear combinations of the two variables separately for normality. This reduces it to a set of univariate tests. – whuber Mar 31 '11 at 23:50
• @whuber: I guess what you are suggesting is regression, and I have stated in my question that without the use of regression – user1102 Mar 31 '11 at 23:53

## 1 Answer

I have recently come across this method that was displayed in Johnson and Wichern.

Let the data points that you want to test for bivariate normality be designated as $\{ x_{i} \}$. Next, compute the sample covariance matrix and deisgnate it as $S$.
For each observed point calculate $d_{j}^{2} = (x_{j} - \bar{x})^{T} S^{-1} (x_{j} - \bar{x})$. Order the values of the $d_{j}^{2}$ from low to high. The last mathematical step is to plot the pair $(q_{c,p}((j- \frac{1}{2}), d_{j}^{2})$, where $q_{c,p}((j- \frac{1}{2})/n)$ is the $100(j- \frac{1}{2})$ quantile of the chi-squared distribution. The plot should be a straight line if the data has a bivariate normal distribution.