Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 208 / 2 / d

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- Solution d

Solution

Let

ℓ^{*}

be a maximizing index for the original sample. Since

Z_{i}^{'}

is at least the value of its

ℓ^{*}

th linear form,

(Z - Z_{i}^{'})_{+} \leq {(X_{i} - X_{i}^{'}) A_{i, ℓ^{*}}}_{+},

and hence

(Z - Z_{i}^{'})_{+}^{2} \leq (X_{i} - X_{i}^{'})^{2} A_{i, ℓ^{*}}^{2} .

The one-sided replacement form of the Efron–Stein inequality is

Var (Z) \leq \sum_{i} E (Z - Z_{i}^{'})_{+}^{2} .

For independent uniform signs,

E [(X_{i} - X_{i}^{'})^{2} ∣ X] = 2

. It follows that

Var (Z) \leq 2 E \sum_{i} A_{i, ℓ^{*}}^{2} \leq 2 max_{1 \leq ℓ \leq m} \sum_{i} A_{i, ℓ}^{2} .

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