Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 221 / 2 / b / Solution

Retaining the estimator exactly as printed, define the probability limit

c = \frac{E [ ZX ]}{E [ Z ^{2} ]}, W = Z - c X .

The empirical equations are linear in

(β, α)

. Their coefficient matrix converges to

M = (E [A W] E [A X] E [X W] E [X^{2}]) .

Therefore a sufficient condition, requiring neither parametric assumption, is

0 < E [Z^{2}] < \infty, det M = E [A W] E [X^{2}] - E [X W] E [A X] \neq = 0,

with finite moments sufficient for the weak law of large numbers. The empirical determinant then converges to a nonzero number, so the two linear equations have a unique solution with probability tending to one.

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