Optimality of Hotelling’s T 2
Invariance
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Consider as usual a random sample x1 , x2 , . . . , xN from Np (µ, Σ).
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We wish to test the null hypothesis H0 : µ = 0.
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The statistics x̄ and
A=
N
X
(xα − x̄) (xα − x̄)0
α=1
are sufficient for µ and Σ, so a good test statistic must be a function
of them.
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T 2 has an invariance property: for any nonsingular C,
T 2 (x1 , x2 , . . . , xN ) = T 2 (Cx1 , Cx2 , . . . , CxN )
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Optimality of Hotelling’s T 2
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Since T 2 is a function of x̄ and A, we can also write
T 2 (x̄, A) = T 2 (Cx̄, CAC0 )
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T 2 is essentially the only such invariant function; if f (x̄, A) is
invariant, it must depend on x̄ and A only through x̄0 A−1 x̄:
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I
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First find nonsingular C1 such that C1 AC01 = I.
Then f (x̄, A) = f (C1 x̄, I).
Now find orthogonal C2 with first row proportional to (C1 x̄)0 , so that
 p
  √

(C1 x̄)0 (C1 x̄)
x̄0 A−1 x̄


 
0
0


 
C2 C1 x̄ = 
=



..
..




.
.
0
0
and f (x̄, A) = f (C2 C1 x̄, I), a function of only x̄0 A−1 x̄.
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Optimality of Hotelling’s T 2
Uniformly Most Powerful
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So it’s not surprising that
the T 2 test is the uniformly most powerful invariant test.
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A related characteristic of T 2 is that its power depends on µ and Σ
only through Nµ0 Σ−1 µ.
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This is essentially equivalent to invariance, so it’s also not surprising
that
the T 2 test is uniformly most powerful among tests with
power that depends on Nµ0 Σ−1 µ.
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Optimality of Hotelling’s T 2
Admissible
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Suppose that T and T ∗ are two tests of the null hypothesis
H0 : ω ∈ Ω0 against the alternative H1 : ω ∈ Ω1 .
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T ∗ is as good as T if
Pr{Reject H0 |T , ω} ≤ Pr{Reject H0 |T ∗ , ω} ,
∗
Pr{Reject H0 |T , ω} ≥ Pr{Reject H0 |T , ω} ,
ω ∈ Ω0 ,
ω ∈ Ω1 ,
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T ∗ is better than T if it is as good, and one inequality is strict for at
least one ω.
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T is admissible if there is no T ∗ that is better than T .
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A theorem of Stein’s about exponential families implies that
the T 2 test is admissible.
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Elliptically Contoured Distributions
Elliptically Contoured Distributions
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Suppose that x1 , x2 , . . . , xN is a random sample from the elliptical
density
1
√
g (x − ν)0 Λ−1 (x − ν)
det Λ
The sample mean x̄ and covariance matrix S are unbiased estimators
of the mean µ = ν, and
2 R
Σ= E
Λ,
p
respectively, where
R 2 = (X − ν)0 Λ−1 (X − ν)
and we assume that E R 2 < ∞.
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Elliptically Contoured Distributions
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The likelihood ratio test of the null hypothesis H0 : ν = 0 is, in
general, not available in closed form.
We can still use T 2 , but its distribution is also generally not available
in closed form.
d
Large samples: T 2 → χ2p as N → ∞.
That is,
Pr T 2 > χ2p (α) → α, 0 < α < 1.
Since
(N − 1)p
Fp,N−1 (α) → χ2p (α),
N −1
it’s also true that
(N − 1)p
2
Pr T >
Fp,N−1 (α) → α,
N −1
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0 < α < 1.
So in large samples, we can treat T 2 approximately as if the data
were normally distributed.
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