[Rd] [Package car/data.ellipse]: confidence intervals off by factor sqrt(2)??? (PR#2584)

[Deepayan Sarkar]

On Wednesday 26 February 2003 04:23 pm, Volker Franz wrote:
> Hi John,
>
> >>>>> "JF" == John Fox <jfox@mcmaster.ca> writes:
>
>     JF> Dear Volker, If the data ellipse (or, in this case, circle) is
>     JF> scaled so that its shadows (projections) on the axes each
>     JF> includes 68% of the data (that is of the marginal distribution
>     JF> of each variable), then the ellipse will include less than 68%
>     JF> of the data (i.e., of the joint distribution of the two
>     JF> variables). Conversely, to include 68% of the data in the
>     JF> ellipse, the shadows of the ellipse have to be larger.
>     JF> Did I understand your point correctly?
>
> I am not sure. I will try to rephrase my initial request:
>
> Let X by a one--dimensional random variable (standard normal
> distribution; mean=0; std=1). The 68% confidence intervall of X will
> approximately be: [-1,1]. Now, if I combine X with a stochastically
> independent second random variable Y, the marginal distribution of X
> should not change. Therefore, the projections of the error ellipse on
> the X--axis should still be: [-1,1].


Why so ? Let Y be an independent copy of X (i.e., Y ~ N(0,1) too, independent 
of X). Then P(Y is in [-Inf , Inf]) = 1. Now, think of the 2-D confidence 
region [-1, 1] x [-Inf, Inf]. This will have (by independence of X and Y) 
probability 0.68.

Now, how can you expect an ellipse that will have the same X-range, that is a 
strict subset of this region, to still have joint probability 0.68 ?

Hope that helps,

Deepayan