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Unformatted text preview: interval for θi
is
ˆ
ˆ
θi ± z1−α/2 Σii ,
where z1−α/2 is the 1 − α/2 quantile of the N (0, 1)
ˆ
ˆ
distribution and Σii is element (i, i) of Σ. Copyright c 2012 (Iowa State University) Statistics 511 14 / 30 An approximate pvalue for testing H0 : θ = θ 0 is
ˆ
ˆ −1 ˆ
Pr[χ2 ≥ (θ − θ 0 ) Σ (θ − θ 0 )],
k
where χ2 is a central χ2 random variable with k
k
degrees of freedom. Copyright c 2012 (Iowa State University) Statistics 511 15 / 30 MULTIVARIATE DELTA METHOD
Suppose g is a function from IRk to IRm , i.e., g1 (θ ) g (θ ) 2 k
for θ ∈ IR , g(θ ) = . .
.
gm (θ )
for some functions g1 , . . . , gm .
Suppose g is differentiable with derivative matrix ∂ g (θ) 1
· · · ∂ gm (1θ)
∂θ
∂θ .1 ..
. .
.
D≡ .
.
.
.
∂ g1 (θ )
∂ gm (θ )
· · · ∂θk
∂θk
Copyright c 2012 (Iowa State University) Statistics 511 16 / 30 ˆ
Now suppose θ has mean θ and variance Σ. Then
Taylor’s Theorem implies
ˆ
ˆ
g(θ ) ≈ g(θ ) + D (θ − θ )
which implies
ˆ
ˆ
E[g(θ )] ≈ g(θ ) + D E(θ − θ ) = g(θ )
and
ˆ
ˆ
Var[g(θ )] ≈ Var[g(θ ) + D (θ − θ )] = D ΣD. Copyright c 2012 (Iowa State University) Statistics 511 17 / 30 ˆ·
If θ ∼ N (θ , Σ), it follows that
ˆ·
g(θ ) ∼ N (g(θ ), D ΣD).
In practice, we often need to estimate D by
ˆ
ˆ
replacing θ in D with θ to obtain D.
Similarly, we often need to replace Σ with an
ˆ
estimate Σ. Copyright c 2012 (Iowa State University) Statistics 511 18 / 30 LIKELIHOOD RATIO BASED INFERENCE Suppose we wish to test the null hypothesis that a
reduced model provides an adequate ﬁt to a dataset
relative to a more general full model that includes the
reduced model as a special case. Copyright c 2012 (Iowa State University) Statistics 511 20 / 30 Deﬁne Λ as
Reduced Model Maximized Likelihood
.
Full Model Maximized Likelihood Λ is...
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This document was uploaded on 03/27/2014.
 Spring '14
 Statistics

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