Parameter Estimate Example: MSDA 3ED Ch8 53(a)

Example) Mathematical Statistics and Data Analysis 3RD Edition Chapter 8, 54 (a)  

Let X1 X2...Xn be iid uniform on [0,  $\theta$]
a) Find the method of moments estimate of $\theta$ and its mean and variance. 



▷Solution (a)
As X1...Xn are uniformly distributed so $\Rightarrow f(X|\theta)=\frac{1}{\theta}$

$E(X)=\int_{0}^{\theta} \frac{x}{\theta}dx = \frac{\theta}{2}$,  $E(X^2)=\int_{0}^{\theta} \frac{x^2}{\theta}dx = \frac{\theta}{3}$ 

$Var(X)=E(X^2)-[E(x)]^2$ $=\frac{\theta ^2}{3}-\frac{\theta ^2}{4}=\frac{\theta^2}{12}$ 


$\rightarrow E(X)=\frac{\theta}{2}\rightarrow E(\bar{X})=E(X)=\frac{\theta}{2} \rightarrow \theta=2\bar{X}, \hat{\theta}=2\bar{X}$ 

$\rightarrow Var(\theta) = Var(2\bar{X})=4\cdot Var(\bar{X})=4\cdot \frac{Var(X)}{n}=4\cdot \frac{\theta ^2}{12n}=\frac{\theta ^2}{3n}$ 


$\therefore\hat{\theta}=2\bar{X}, Var(\hat{\theta })=\frac{\theta ^2}{3n}$