Unformatted text preview: known as the likelihood ratio.
−2 log Λ is known as the likelihood ratio test
statistic.
Tests based on −2 log Λ are called likelihood ratio
tests. Copyright c 2012 (Iowa State University) Statistics 511 21 / 30 Under the regularity conditions in Shao (2003)
mentioned previously, the likelihood ratio test
statistic −2 log Λ is approximately distributed as
central χ2f −kr under the null hypothesis, where kf
k
and kr are the dimensions of the parameter space
under the full and reduced models, respectively.
This approximation can be reasonable if n is
“sufﬁciently large.” Copyright c 2012 (Iowa State University) Statistics 511 22 / 30 Likelihood Ratio Tests and Conﬁdence Regions for a
Subvector of the Full Model Parameter Vector θ Suppose θ is k × 1 vector and is partitioned into
vectors θ 1 k1 × 1 and θ 2 k2 × 1, where k = k1 + k2
θ1
and θ =
.
θ2
Consider a test of H0 : θ 1 = θ 10 . Copyright c 2012 (Iowa State University) Statistics 511 23 / 30 ˆ
ˆ
Suppose θ is the MLE of θ and θ 2 (θ 1 ) maximizes
θ1
over θ 2 for any ﬁxed value of θ 1 .
θ2
θ 10
is approximately
ˆ
θ 2 (θ 10 )
χ21 under the null hypothesis by our previous
k
result when n is “sufﬁciently large.”
Then 2 ˆ
(θ ) − Copyright c 2012 (Iowa State University) Statistics 511 24 / 30 Also,
Pr 2 ˆ
(θ ) − θ1
ˆ 2 (θ 1 )
θ ≤ χ21 ,1−α
k ≈1−α which implies
Pr θ1
ˆ
θ 2 (θ 1 ) Copyright c 2012 (Iowa State University) 1
ˆ
≥ (θ ) − χ21 ,1−α
2k ≈ 1 − α. Statistics 511 25 / 30 Thus, the set of values of θ 1 that, when
maximizing over θ 2 , yield a maximized likelihood
within 1 χ21 ,1−α of the likelihood maximized over all
2k
θ , form a 100(1 − α)% conﬁdence region for θ 1 .
Such a conﬁdence region is known as a proﬁle
likelihood conﬁdence region because
θ1
ˆ 2 (θ 1 )
θ
is the proﬁle log likelihood for θ 1 .
Copyright c 2012 (Iowa State University) Statistics 511 26 / 30 A Warning
The normal and χ2 approximations mentioned in
these notes may be crude if sample sizes are not
sufﬁciently large.
The regularity conditions mentioned in these
notes do not hold if the true parameter falls on the
boundary of the parameter space. Thus, as an
2
example, testing H0 : σu = 0 is not covered by the
methods presented here. Copyright c 2012 (Iowa State University) Statistics 511 30 / 30...
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 Spring '14
 Statistics

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