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Correcting the largest observation

StatisticsHard

Latencies are independent and $Uniform (0, θ)$ . In a sample of $4$ , the largest is $8$ . The maximum likelihood estimate of $θ$ is the sample maximum, which runs low. Rescale that estimate so that it becomes unbiased, then apply it to this sample. What is the resulting estimate of $θ$ ? Give a number.

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Hint

The largest of $n$ uniform draws has mean $n \times θ / (n + 1)$ .

Solution

For $n = 4$ the maximum $M$ has density $4 \times \frac{m ^{3}}{θ ^{4}}$ on $(0, θ)$ , so $E [M] = \frac{4}{5} \times θ$ .
To remove the bias, multiply $M$ by $\frac{5}{4}$ .
The estimate is $\frac{5}{4} \times 8 = 10$ .

Answer: $10$

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