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REVISION AND PRACTICE

Revision questions with answers

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Question 1 — What is the difference between maximum and supremum?

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  1. A maximum must belong to the set. A supremum is the least upper bound and may lie outside it.

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Question 2 — What is f∗(t)f^*(t) for f(u)=u22f(u)=\frac{u^2}{2}?

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  1. f∗(t)=t22f^*(t)=\frac{t^2}{2}

    the maximizing u equals t.

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Question 3 — What is f∗(t)f^*(t) for f(u)=uln⁡uf(u)=u\ln u?

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  1. f∗(t)=et−1f^*(t)=e^{t-1}

    with u≥0u\geq0 and natural logarithms.

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Question 4 — Why introduce T(x)T(x)?

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  1. The density ratio depends on x, so the maximizing scalar can depend on x; a function represents these pointwise choices.

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Question 5 — Which expectation uses generated samples?

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  1. The second term EPθ[f∗(T(X))]\mathbb{E}_{P_\theta}[f^*(T(X))].

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Question 6 — Why does restricting T\mathcal{T} produce a lower bound?

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  1. A supremum over a subset cannot exceed the supremum over the full admissible set.

Editorial note: Worked numerical examples and prerequisite detours are explanatory teaching additions. Figures 21 and 36 are reconstructed formula/diagram illustrations because the corresponding original image files were not present in the available working files.

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