.
Watterson’s estimator θW of θ was defined in (13.34).
a. Calculate the variance of θW.
b. Show that it decays at a rate proportional to 1/ log n in large samples,
and comment on the practical implications of this result.
.
R can be used to simulate observations having the distribution
of θW.
a. Write a function to simulate an observation _ having the distribution of
Ln in (13.31). Exponential random variables T2, . . . , Tn can be generated
using rexp.
b. Given the value of _, generate a Poisson variable s having mean θ_/2.
Poisson random variables can be generated using rpois. Given the value
of s, calculate an observation from θW from (13.34).
c. For θ = 1, 5, 25 and n = 10, 25, 100 simulate observations from the distribution
of θW, and compare the results.