Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 205 / 3 / d

Solution

Write

R_{i} = (x_{i} - f (z_{i})) (y_{i} - g (z_{i})) = (ε_{i} + F_{i}) (ξ_{i} + G_{i}) .

Expanding

R_{i}^{2}

gives the leading term

ε_{i}^{2} ξ_{i}^{2}

and terms of the forms treated in parts b and c, together with their versions obtained by interchanging

(ε, F)

and

(ξ, G)

. For example, the pure error terms are

ε_{i}^{2} G_{i}^{2}

ξ_{i}^{2} F_{i}^{2}

, and

F_{i}^{2} G_{i}^{2}

, and each cross term is controlled by Cauchy-Schwarz from these. Hence

τ_{D}^{2} = \frac{1}{n} \sum_{i} R_{i}^{2} = \frac{1}{n} \sum_{i} ε_{i}^{2} ξ_{i}^{2} + o_{p} (1) p E (ε_{1}^{2} ξ_{1}^{2}) .

Assuming this limit is positive, the continuous mapping theorem yields

τ_{D} p {E (ε_{1}^{2} ξ_{1}^{2})}^{1/2}

. Combining this with the assumed convergence in distribution of

n τ_{N}

and applying the Slutsky theorem gives

T = \frac{n τ _{N}}{τ _{D}} d N (0, 1) .

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