Estimating Bias in R - r

Question

Write a simulation experiment to estimate the bias of the estimator λˆ= 1/ X¯ by sampling
using x=rexp(n,rate=5) and recording the values of 1/mean(x). You should find that the
bias is λ/n−1. Here we’ve used λ = 5 but the result will hold for any λ.
Here is my solution ( I dont get λ/n−1). Am I doing something wrong here?
set.seed(1)
lambda <- 5
x <- rexp(n= 1e5, rate = lambda )
samp.mean <- mean(x)
lam.est <- 1/samp.mean
lam.est ##4.986549
bias <- abs(lambda - lam.est)
bias ##0.01345146

Answer 1

To start with, there is a mistake in your formula. The bias of the lambda estimator is not lambda/n-1 but lambda/(n-1)!
Then note that in order to carry out this experiment correctly, it is not enough to calculate the estimated estimator once.
Do the experiment "n" times on the vector of size "nx".
lambda = 3
nx = 150
n = 1e5
set.seed(1)
out = vector("numeric", n)
for(i in 1:n){
out[i] = 1/mean(rexp(n= nx, rate = lambda))
}
lambda/(nx-1)
mean(out)
bias = abs((mean(out)-lambda))
As you can see for lambda = 3 and nx = 150 the expression lambda/(nx-1) is 0.02013423. And your estimated lambda is 3.019485.
lambda = 5
nx = 200
n = 1e5
set.seed(1)
out = vector("numeric", n)
for(i in 1:n){
out[i] = 1/mean(rexp(n= nx, rate = lambda))
}
lambda/(nx-1)
mean(out)
bias = abs((mean(out)-lambda))
However, for lambda = 5 and nx = 200, the expression lambda/(nx-1) is 0.02512563. And your estimated lambda is 5.024315.
Perform this experiment for other values of lambda and nx and you will find that the bias of this estimator is lambda/(n-1).