Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 28K / c

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- Solution c

Solution

Ignoring the bounded second-order remainder, this is the linear regression model

Y_{i} - g (x_{0}) = h Z_{i} β + ε_{i}, β = g^{'} (x_{0}) .

The ordinary least squares estimate through the known zero intercept is

β = \frac{\sum _{i} ( h Z _{i} ) ( Y _{i} - g ( x _{0} ))}{\sum _{i} ( h Z _{i} ) ^{2}} = \frac{1}{N} i = 1 \sum N \frac{Z _{i} ( Y _{i} - g ( x _{0} ))}{h},

because

Z_{i}^{2} = 1

. This is the randomized symmetric finite-difference derivative estimator.

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