Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 223 / 1 / a

Solution

Let

m = F^{- 1} (1/2)

. Since

F

has an everywhere positive density, it is continuous and strictly increasing, so

F (m) = 1/2

Φ (m) - ε \leq \frac{1}{2} \leq Φ (m) + ε .

- Φ^{- 1} (\frac{1}{2} + ε) \leq m \leq Φ^{- 1} (\frac{1}{2} + ε) .

The stated asymptotic-bias formula for the sample median therefore yields

F \in P_{ε}^{K} (Φ) \cap M sup b ({T_{n}}, F) \leq b_{1}, b_{1} = Φ^{- 1} (\frac{1}{2} + ε) .

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