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 likelihood function is therefore
N
Y
1
g (xα − ν)0 Λ−1 (xα − ν)
N/2
(det Λ)
α=1
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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Then:
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x̄ is an unbiased estimator of the mean µ = ν;
S is an unbiased estimator of
2 R
Λ,
Σ= E
p
where
R 2 = (X − ν)0 Λ−1 (X − ν)
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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By the Central Limit Theorem,
√
d
N (x̄ − µ) → Np (0, Σ)
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By the Law of Large Numbers, S is a consistent estimator of Σ.
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So for large N, we can make inferences about µ in the same way as
for a multivariate normal population.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Maximum Likelihood
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The mle for µ satisfies
PN
α=1 wα xα
µ̂ = P
N
α=1 wα
where wα is the data-dependent weight
g 0 (xα − µ̂)0 Λ−1 (xα − µ̂)
wα = −2 × g (xα − µ̂)0 Λ−1 (xα − µ̂)
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Generally no closed-form solution: iterative methods are used.
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The likelihood may have multiple local maxima, so care is needed.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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The mle for Σ is
Σ̂ =
N
1 X
wα (xα − µ̂)(xα − µ̂)0
N
α=1
for the same data-dependent weights.
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Anderson gives the limiting normal distribution of Σ̂, but not of µ̂.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Multivariate t-distribution
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For the multivariate t-distribution with m degrees of freedom,
g R
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2
=
Γ
Γ
m
2
m+p 2
(mπ)p/2
R2
1+
m
− m+p
2
The data-dependent weights are
wα =
m+p
m + Rα2
where
Rα2 = (xα − µ̂)0 Λ̂
−1
(xα − µ̂)
so data points far from µ̂ carry less “weight”.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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The information matrix for µ is
m+1
Λ−1
J=
m+3
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The distribution is regular, so the mle is asymptotically efficient, and
√
m+3
d
−1
N (µ̂ − µ) → Np 0, J
= Np 0,
Λ
m+1
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Note:
m+3
m+1
Λ=
m+3
m+1
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m−2
m
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Σ= 1−
6
Σ
m(m + 1)
Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
Thus the (generalized) asymptotic efficiency of x̄ is
p
6
1−
m(m + 1)
and the canonical asymptotic efficiencies are all
1−
6
m(m + 1)
1
2
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The canonical efficiencies are 0 for m = 2 (why?),
7
4
10 for m = 4, and 5 for m = 5.
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The canonical efficiencies are greater than 90% for m ≥ 8.
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for m = 3,
Statistics 784
Multivariate Analysis