Refer to Copas’s kernel smoother (11.1) for binary regression, with ????(u) =

exp(−u2∕2).

a. To describe how close this estimator falls at a particular x value to a corresponding smoothing in the population, use the delta method to show that an estimated asymptotic variance is

̃????(x)[1 − ̃????(x)]

∑

i ????

[√

2(x − xi)∕????

]

{ ∑

i ????[(x − xi)∕????]

}2 .

Explain why this decreases as ???? increases, and explain the implication.

b. As ???? increases unboundedly, explain intuitively to what ̃????(x) and this estimated asymptotic variance converge.