i.
Mardia’s kurtosis statistic is skewed and converges very slowly to the limiting normal distribution. The corresponding half-lengths of the axes are obtained by the following expression:\(l_j = \sqrt{\lambda_j\chi^2_{p,\alpha}}\)The plot above captures the more helpful hints of these axes within the ellipse.
Note that knowing that x2 = a alters the variance, though the new variance does not depend on the specific value of a; perhaps more surprisingly, the mean is shifted by
12
22
1
(
a
2
)
{\displaystyle {\boldsymbol {\Sigma }}_{12}{\boldsymbol {\Sigma }}_{22}^{-1}\left(\mathbf {a} -{\boldsymbol {\mu }}_{2}\right)}
; compare this with the situation of not knowing the value of a, in which case x1 would have distribution
N
q
(
1
,
11
)
{\displaystyle {\mathcal {N}}_{q}\left({\boldsymbol {\mu }}_{1},{\boldsymbol {\Sigma }}_{11}\right)}
. .