Computing C90 ellipse area for two dimensional time series data I am trying to determine the area of the ellipse which contains the true mean of $(X_n, Y_n)_{1\leq n \leq N}$ with a probability of 90%. The 90% confidence ellipse area, or C90 area. The sample consists of N = 3000 $X,Y$ coordinate pairs. The problem is that definitions I've come across differ in how they end up calculating the actual area of this ellipse. After assuming this and that about the sample ( distribution etc ), the main steps are as follows :

*

*Determine the covariance matrix of the sample. In this case its a $2\times 2$ matrix


*Determine the eigenvalues of this matrix $\lambda_1, \lambda_2$


*The square roots of the eigenvalues correspond to the axes of the ellipse. Hence the area can be calculated as $C90 = \pi\chi^2\sqrt{\lambda_1\lambda_2}$, where $\chi^2=4.605$
Step 3 is the one I'm confused about. I've looked at a couple of sources and each seems to compute the area differently. Some sources also claim that this corresponds to the area of the prediction ellipse, not the confidence ellipse.
My question is whether this 3 step procedure to find the C90 area is correct, or is this the area of the prediction ellipse.
Edit : updated my definition of the C90 area.

The principle behind the $4.605$ value is that the sum of the square of two normal distributed variables (also known as a $\chi^2(\nu = 2)$ variable) is 90% of the time below $4.605$. The area of that circle is $4.605 \pi$.

The term $\sqrt{\lambda_1\lambda_2}$ relates to a stretching of the variables into an elipse.

What your $C90$ value refers to depends on what the values $\lambda_1$ and $\lambda_2$ refer to. It seems like these $\lambda$ are estimates of the population variance and in that case the interval relates to a prediction interval. It can also be that the $\lambda$ relate to the variance of the estimate and in that case the interval relates to a confidence interval.
This is similar to the difference between the variance of the population and the variance of the estimate of the mean of the population (these two differ by a factor $\sqrt{n}$).
