Page 487 - Probability and Statistical Inference

464 9. Confidence Interval Estimation

Figure 9.4.1. The Elliptic Confidence Regions Q from (9.4.3)
and Q* from (9.4.4)
Example 9.4.2 (Example 9.4.1 Continued) Suppose that X , ..., X are iid
1 10
2–dimensional normal, N (µ, Σ), random variables with We
2
fix α = .05, that is we require a 95% confidence region for µ. Now, one has
–2log(.05) = 5.9915. Suppose also that the observed value of ′ is
(1, 2). Then, the confidence region from (9.4.2) simplifies to

which should be elliptic with its center at the point (1, 2). The Figure 9.4.1 gives
a picture (solid curve) of the region Q which is the inner disk of the ellipse. The
horizontal (x) and vertical (y) axis respectively correspond to µ and µ .
2
1
Instead, if we had then one can check that a 95% con-
fidence region for µ will also turn out to be elliptic with its center at the point
(1, 2). Let us denote

The Figure 9.4.1 gives a picture (dotted curve) of the region Q* which is the
inner disk of the ellipse. !
Example 9.4.3 Suppose that X , ..., X are iid p–dimensional multivariate
1
n
normal, N (µ, Σ), random variables with n ≥ 2. Let us assume that the disper-
p
2
sion matrix Σ = σ H where H is a p.d. and known matrix but the scale multi-
plier σ (∈ ℜ ), is unknown. With fixed α ∈ (0, 1), we wish to derive a (1 – α)
2
+
confidence region for the mean vector µ.