Stat610: Intermediate Theory of Statistics
Homework 7, Additional Problems (10/17/06)
Instructor: Jianhua Huang Editor: Jose Montes Valarde
TA: Seokho Lee
Problem A1.
Let pθ,η(x) (viewed as a probability density) be a parametric model, where θis the parameter of interest and
ηis a nuisance parameter. When ηis not free, but is a function of θ, we say that we have a submodel of the
original model. This submodel can be denoted as pθ,η(θ)(x). Compute the (expected) Fisher information of
θin this submodel. Find a η(θ) that minimizes the Fisher information in the submodel among all possible
functions η(θ). The corresponding submodel is called the hardest submodel. Can you relate the Fisher
information in the submodel to that in the original model?
Solve the problem first by assuming θand ηare one-dimensional. Then extend your results to multi-
dimensional case.
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