2 marks, −0.667StatisticsPoint and interval estimationMaximum likelihood estimationModerateabout 1 min

IIT JAM 2021 Question 16: Point and interval estimation

Let $X$ be uniform on $[a, b]$ . A sample of size five is $\{1.3, 0.8, 9.5, 20.2, 8.2\}$ . The Maximum Likelihood Estimates $\hat a$ and $\hat b$ are

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Answer and worked solution

Answer: (A)

The likelihood of a $U[a,b]$ sample is $1/(b-a)^n$ over the region that contains all observations, so it is maximised by making the interval as small as possible while still covering every data point.

That means $\hat a$ is the sample minimum and $\hat b$ the sample maximum: $\hat a = 0.8$ and $\hat b = 20.2$ . The answer is (A).

The trap: (D)

\hat a = 0;\ \hat b = 21

Widening the interval beyond the data (as in option D) lowers the likelihood; the MLE endpoints hug the observed minimum and maximum exactly.

Plain-text version

Let X be uniform on [a, b]. A sample of size five is {1.3, 0.8, 9.5, 20.2, 8.2}. The Maximum Likelihood Estimates â and b̂ are (A) â = 0.8; b̂ = 20.2 (B) â = 1.3; b̂ = 9.5 (C) â = 1.3; b̂ = 8.2 (D) â = 0; b̂ = 21

Also written as: Let X be uniform on [a, b]. A sample of size five is {1.3, 0.8, 9.5, 20.2, 8.2}. The Maximum Likelihood Estimates a_hat and b_hat are

Point and interval estimation has come up 28 times in 6 exams since 2008.

What this question tests

This question is on Maximum likelihood estimation, part of Point and interval estimation in Statistics.

Choosing the parameter value that makes the data most likely.

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