Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 1 / d

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- Solution d

Solution

By the invariance property of maximum likelihood estimation,

h = 1 0^{θ /5} = e^{θ / α} .

The sampling distribution is

θ \sim N (θ, σ_{θ}^{2})

, where

σ_{θ}^{2} = C / K + H / N,

h

has a log-normal distribution with

lo g h \sim N (lo g h, \frac{σ _{θ}^{2}}{α ^{2}}) .

v = σ_{θ}^{2} / α^{2}

, its exact fractional bias and variance are

\frac{E h - h}{h} = e^{v /2} - 1, \frac{Var ( h )}{h ^{2}} = e^{v} (e^{v} - 1) .

Hence, to leading order,

\frac{E h - h}{h} ≃ \frac{σ _{θ}^{2}}{2 α ^{2}}, \frac{Var ( h )}{h ^{2}} ≃ \frac{σ _{θ}^{2}}{α ^{2}} .

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