Question

R Programming

In this section, we will expand upon some of the ideas we have incorporated in the homework. For this entire problem, we will use the Normal distribution. For all the simulated datasets, we wil assume a mean of μ-10 and a standard deviation of σ-3. Consider the following estimators for u: 1. X 2. Q2, the sample median 3. (Q1 Qs), the average of the first and third sample quartiles 4. XI, the first observation As you have noticed in class, often the mean and variance of estimators is dependent on the size of the sample. Therefore, in this problem, we will let the sample size vary. Since this procedure should be fairly familiar, we will jump straight into the assignment:

Answer 1

R Program

#Question a.

n=25
rep=10000
m1=numeric(rep)
m2=numeric(rep)
m3=numeric(rep)
m4=numeric(rep)

for(i in 1:rep)
{
x=rnorm(n,10,3)
m1[i]=mean(x)
m2[i]=median(x)
m3[i]=(quantile(x,.25)+quantile(x,.75))/2
m4[i]=x[1]
}

#Question b
## Bias, Variance & MSE of first estimator

bias_m1=mean(m1)-10
var_m1=var(m1)
mse_m1=mean((m1-10)*(m1-10))
bias_m1
var_m1
mse_m1

## Bias, Variance & MSE of second estimator

bias_m2=mean(m2)-10
var_m2=var(m2)
mse_m2=mean((m2-10)*(m2-10))
bias_m2
var_m2
mse_m2

## Bias, Variance & MSE of third estimator

bias_m3=mean(m3)-10
var_m3=var(m3)
mse_m3=mean((m3-10)*(m3-10))
bias_m3
var_m3
mse_m3

## Bias, Variance & MSE of fourth estimator

bias_m4=mean(m4)-10
var_m4=var(m4)
mse_m4=mean((m4-10)*(m4-10))
bias_m4
var_m4
mse_m4

#Question c.

boxplot(m1,m2,m3,m4,names=c("Mean","Q2",".5(Q1+Q3)","X1"))

#Question d.
boxplot(m1,m2,m3,names=c("Mean","Q2",".5(Q1+Q3)"))

Question e

The boxplot shows unbiasedness of all the estimators as the central is more or less
at 10. However, for mean, the variability is least as the length of the box is smallest.
But highest variability is observed for the Median based on the plot.

R Output

> #Question b
> ## Bias, Variance & MSE of first estimator

CN Mean Q2 .5(Q1+Q3) X1

CN Mean Q2 .5(Q1+Q3)
>bias_m1
[1] 0.003531876
> var_m1
[1] 0.3510436
> mse_m1
[1] 0.351021
>
> ## Bias, Variance & MSE of second estimator
>
> bias_m2=mean(m2)-10
> var_m2=var(m2)
> mse_m2=mean((m2-10)*(m2-10))
> bias_m2
[1] 0.003000309
> var_m2
[1] 0.5450123
> mse_m2
[1] 0.5449668
>
> ## Bias, Variance & MSE of third estimator
>
> bias_m3=mean(m3)-10
> var_m3=var(m3)
> mse_m3=mean((m3-10)*(m3-10))
> bias_m3
[1] 0.003185415
> var_m3
[1] 0.4346715
> mse_m3
[1] 0.4346382
>
> ## Bias, Variance & MSE of fourth estimator
>
> bias_m4=mean(m4)-10
> var_m4=var(m4)
> mse_m4=mean((m4-10)*(m4-10))
> bias_m4
[1] 0.01779539
> var_m4
[1] 9.257936
> mse_m4
[1] 9.257327
>