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Ok I've been trying to interpret what they actually want me to do here, I've been sititng with this exact question for 1,5 days now:

enter image description here

What confuses me is the sentence "Draw 1000 samples of this size, for each samle calculate a bootstrap CI".

Then I'm supposed to calculate how many of these confidence interval covers the theoretical mean, which in this case is $\theta = 4$.

Do they mean that I should, for $n=10$, calculate 1000 CI's? We're drawing from a gamma distribution, here is what I have done:

# CI for n = 10
n = 10
K = vector("numeric", 1000L)
S = vector("numeric", 1000L)  #Create vectors to contain information

for (i in 1:1000){
  x = sample(rgamma(n,k,gamma), n)  # For every iteration, compute the mean of the 
  K[i] = mean(x)                      # 10 samples and place it in index i in K.
}

S = sort(K)
Upper = S[975]
Lower = S[25]

So doing a CI now, it would just give me ONE CI for all these 1000 means, but I want 1000 CI's, don't I?

I'm not sure if I've interpreted this correctly :S

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From my understanding of the bootstrap, you're right, obtaining 1000 bootstrap samples for $\hat \theta$ will give you a single CI. Perhaps you're expected to repeat this many times over.

See this Q&A, the explanation here regarding the bootstrap CIs is really good.

On a side note, a note of caution:

  1. R's sample function has the replace variable set to FALSE by default, you need to sample with replacement while doing a bootstrap.
  2. Note that rgamma uses an $\alpha, \beta$ parameterization unless you explicitly state scale = $\theta$. I can't see where the k variable is defined, so I'm not sure if you're using the right parameterization.
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  • $\begingroup$ Hello, indeed, i realised this and set replace = TRUE. For the parameters, k = 2, gamma = 1/2. I'll take a look at tha tlink you've probided, thank you! $\endgroup$ – Fabled Oct 3 '18 at 10:12

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