Question

5. Let X1,.. ., Xn be a random sample from Uniform(0,0) with an unknown endpoint θ > 0, we want to estimate the parameter θ (a) Find the method of moments estimator (MME) of θ. (b) Find the MLE θ of θ (c) (R) Set the sample size as 25, do a simulation in R to compare these two esti- mators in terms of their bias and variance. Include a side-by-side boxplot that compares their sampling distributions

Answer 1

Souuim 2. 2. Thas if-for estimator foro is T x) we mst have 3 mm-(1)一4 (2) e 2 X 2

(b) MLE Since' angeinvol ves paramete enside rden tatistics random ample the den s is minimum and for this

R Codes

>n=25 > theta = 2 > MME = MLE = C() > for(i in 1:100) + x = runi f (n ,0 , theta) + MME[1] = 2*mean(x) + MLE[i] = max (x) head (MLE) [1] 1.967708 1.986284 1.871534 1.989326 1.887240 1.988021 > head(MME) [1] 1.927163 2.210313 1.694916 2.287035 2.185816 2.403628 > boxplot (MME,MLE)

1.6 1.8 2.0 2.2 2.4

> var(MLE)
[1] 0.004005521
> var(MME)
[1] 0.04614172

Variation in MLE is less as compare with MME.

> bias = MLE - MME
> mean(bias)
[1] -0.08384014