Theory of Statistics - George Mason University

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7.3 Likelihood Ratio Tests, Wald Tests, and Score Tests 529
1
2σ2 (X − f(Z, β))T (X − f(Z, β)) + 1
σ2 (Lβ − β0) T λ.
Differentiate and set = 0:
−J T
f(ˆβ) (X − f(Z, ˆ β)) + L T λ = 0
L ˆ β − β0 = 0.
J T
f(ˆβ) (X − f(Z, ˆ β)) is called the score vector. It is of length k.
Now V(X − f(Z, ˆ β)) → σ 2 In, so the variance of the score vector, and
hence, also of L T λ, goes to σ 2 J T f(β) Jf(β).
(Note this is the true β in this expression.)
Estimate the variance of the score vector with ˜σ 2 J T
f( ˜ β) J f( ˜ β) ,
where ˜σ 2 = SSE( ˜ β)/(n − k + r).
Hence, we use L T˜ λ and its estimated variance.
Get
It is asymptotically χ 2 (r).
This is the Lagrange multiplier form.
Another form:
Use J T
f( ˜ β) (X − f(Z, ˜ β)) in place of L T˜ λ.
Get
1
˜σ 2 (X − f(Z, ˜ β)) T J f( ˜ β)
1
˜σ 2 ˜ λ T
L J T
f( ˜ β) Jf( ˜ −1 β) L T˜ λ (7.35)
J T
f( ˜ β) Jf( ˜ −1 β) J T
f( ˜ β) (X − f(Z, ˜ β)) (7.36)
This is the score form. Except for the method of computing it, it is the
same as the Lagrange multiplier form.
This is the SSReg in the AOV for a regression model.
Example 7.10 an anomalous score test
Morgan et al. (2007) illustrate some interesting issues using a simple example
of counts of numbers of stillbirths in each of a sample of litters of laboratory
animals. They suggest that a zero-inflated Poisson is an appropriate model.
This distribution is an ω mixture of a point mass at 0 and a Poisson distribution.
The CDF (in a notation we will use often later) is
P0,ω(x|λ) = (1 − ω)P(x|λ) + ωI[0,∞[(x),
where P(x) is the Poisson CDF with parameter λ.
(Write the PDF (under the counting measure). Is this a reasonable probability
model? What are the assumptions? Do the litter sizes matter?)
If we denote the number of litters in which the number of observed stillbirths
is i by ni, the log-likelihood function is
Theory of Statistics c○2000–2013 James E. Gentle