Stat 514

Exercise

Suppose that a particle of initial size y₀ is subjected to repeated impacts, that on each impact a proportion, X_i , of the particle is left, and that the X_i are modeled as independent random variables having the same distribution (iid). After the first impact the size of the particle is Y₁= X₁y₀, and after n impacts the size is:

Y_n = X_nX_n-1...X₂X₁y₀

We are interested in EY_n and in P( Y_n < 3), Following are data from an experiment in which a particle was subjected to 50 impacts. Using the data, estimate EY_n , P( Y_n < 3), and the standard error of the estimate of EY_n. Let the value of y₀ be 10,000,000 units.

Solution:

1. Non-parametric bootstrap

resample from X1,X2,...X50 for B times (Let B be some large number, say 1000), calculate the corresponding Yn for every bootstrap

sample. finally take the mean and standard deviation, that is the estimator of the EYn and the standard error of the estimate of EYn.

2. Another way of Non-parametric bootstrap

similar as 1 except now you will calculate the Yn by Y0(x_bar)^50. this method is not as good as 1 since it is biased.

3. Parametric bootstrap

Now you would assume a parametric family. for this problem, beta distribution would be a suitable one. you estimate the two parameters

a and b from the 50 samples you get. then sample say 1000 times from the beta(a,b) distribution and get the estimator by similar way.