Question

Use R.

Generate a random sample with n=15 random observations from an exponential distribution with mean=1.

• Calculate the sample median, which is an estimator of the population median.
• Use bootstrap (nonparametric, with B=1000) methods to estimate the variance of the estimator for the population median.
• use the Monte Carlo method, e.g. generate 1000 samples of size 15 to estimate the true variance of the median estimator. Compare and comment on your results.

Answer 1

> n=15
> b=1000
> s=rexp(n,1);s #sample of size n=15 from exponential with mean=1
[1] 0.407867708 0.223045019 0.571697697 0.516058067 1.433474415 0.005044012 0.292488087
[8] 1.265140870 0.561035751 2.060778400 1.044908378 0.274859678 0.376427082 0.130451812
[15] 0.048147370
> sample_median=median(s);sample_median
[1] 0.4078677
> #bootsrap
> med=c()
> for(i in 1:b)
+ {
+ s=sample(s,n)
+ med[i]=median(s)
+ }
> var(med) #bootsrap estimate of variance of estimate of population median
[1] 0
>
> ##monte carlo
> med1=c()
> for(i in 1:1000)
+ {
+ s=rexp(15,1)
+ med1[i]=median(s)
+ }
> var(med1)
[1] 0.06725635