Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 225 / 4 / i

Table of contents
- Solution i

Solution

Let

B

be the integral operator with kernel

β

. Since

Y = BX + ε

and the centered error is independent of

X

,

E [a_{k} b_{l}] = E [a_{k} ⟨ BX, u_{l} ⟩] .

β_{l k} = ⟨ B ϕ_{k}, u_{l} ⟩

and using

E [a_{k} a_{j}] = λ_{k} 1_{{j = k}}

gives

E [a_{k} b_{l}] = λ_{k} β_{l k} .

Expanding the function-on-function linear model kernel in the product basis therefore yields

β (t, s) = k = 1 \sum \infty l = 1 \sum \infty \frac{E [ a _{k} b _{l} ]}{λ _{k}} ϕ_{k} (s) u_{l} (t) .

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