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# Assume a population X/Exp(λ) and the iid sample X1, ..., X25 of it. You are interested in the specific value of the population parameter λ so you estimated

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Exercise - Bootstrapped Hypothesis Testing

Assume a population X/Exp(λ) and the iid sample X1, ..., X25 of it. You are interested in the specific value of the population parameter λ so you estimated it by the following plug-in estimator λˆ =   1    which you want to test now.

1)  Write a function that tests the following hypothesis pair H0 : λ = λ0 vs. H1 : λ = λ0 by using the limiting distribution  of  λˆ.  The  inputs  of  your  function  should  be  the  sample,  λ0 and  α.  As  result  it should return either a 0 for “H0 not rejected” or a 1 for “H0 rejected”. Hint: You will need the parametric delta method!

2)   Now write a function that uses the bootstrapped bias-corrected mean and variance instead of the sample moments for testing the same hypothesis pair! The function should have the number of bootstrap repetitions b as an additional imput.

3)  λˆ−λ0Next write a third function that tests the hypothesis pair by bootstapping the test statistic  √Var(λˆ) directly and using its empirical quantiles. Hint: The function should center the sample to λ0 before the resampling!

4)   Conduct a Monte Carlo simulation with 103 repetitions, α = 0.05, λ0 = 1, and b = 1000 to calculate empirical rejection rates for different values of λ as in the table below:

5)    At last, plot the empirical power functions for the Monte Carlo simulation like given below:

Remarks: You may work in groups of max 5 people. Download an R notebook file from OLAT and adjust its name with your last name(s). Write comments for everything you do. Codes that are not written using the notebook file and/or that return error messages will not be evaluated.

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