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Question Suppose X1, ...... X50 ~ iid Uniform[0, θ].
Part I. Find MLE for θ and compute the distribution of MLE.
Part II Generate your own observations x1....x50 (with only one maximum and θ = 1), and based on them plot a histogram of the bootstrap samples (N = 1000). Compare the distribution of MLE and this histogram.
Part III This example shows that bootstrap does poorly. In fact, P(θ^MLE = 1) = 0 but P(θ*^ = θ^MLE) ≈ 0.636. Prove these two statements.
Hint: P(θ^* = θ^MLE) = 1-(1 -1/n)n, n = 50.
Note: it need R code.
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