Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 3 / d / ii

Table of contents
- Solution ii

Solution

Assume

d < c

. Directly comparing the two piecewise densities gives

\frac{g _{1} ( x )}{g _{0} ( x )} = ⎩ ⎨ ⎧ d, r (x), c, r (x) \leq d, d < r (x) < c, r (x) \geq c .

Since

lo g r (x) = Δ (x - \frac{θ _{0} + θ _{1}}{2}),

we obtain

lo g \frac{g _{1} ( x )}{g _{0} ( x )} = [Δ (x - \frac{θ _{0} + θ _{1}}{2})]_{l o g d}^{l o g c} .

Thus the sample log-likelihood ratio is

n S_{n}

with truncation values

a = lo g d

and

b = lo g c

. Rejecting for large

S_{n}

is exactly the likelihood-ratio test between the two least-favorable contaminated distributions.

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