Asymptotic Methods in Statistical Inference - Statistics Centre

116
• We say that a r.vec. X has the multivariate normal
(μ Σ) distribution if the c.f. is
£ exp ¡ t 0 X ¢¤ =exp n t 0 o
μ − t0 Σt
2 . Putting all
but one component of t equal to 0 yields the
consequence that then ∼ ³ 2 ´, where
2 = Σ . Calculating
h exp ³ t 0X´i
|t=0
= h exp ³ t 0 X´i |t=0 = [ ]
and
(
)
exp t 0 μ − t0 Σt
2
|t=0
=
yields [ ]= and similarly h i
=
. We call Σ the covariance matrix. Note that
Σ = h (X − μ)(X − μ) 0i . If Σ 0 then there
is a density
(x; μ Σ)
= (2) −2 |Σ| −12 exp
(
− (x − μ)0 Σ −1 (x − μ)
2
)