Example) Mathematical Statistics and Data Analysis, 3ED, Chapter 8. Q53(a)
Let X_{1},X_{2},\cdots, X_{n} be iid uniform [0, \theta] Find the method of moments estimate of \theta and its mean and variance.
\triangleright Solution
f(X|\theta)=\frac{1}{\theta} for any x between 0 and \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^2}{3}
\rightarrow Var(X)=E(X^2)-[E(X)]^2=\frac{\theta^2}{3}-\frac{\theta^2}{4}=\frac{\theta^2}{12}
E(X)=\bar{X}=\frac{\theta}{2}\rightarrow \therefore \theta=2\bar{X} \rightarrow \hat{\theta}=2\bar{X}
\therefore E(\hat{\theta})=E(2\bar{X})=E(2\cdot \frac{1}{n} \sum x_{i})=2 \cdot \frac{1}{n} \cdot n \cdot \frac{\theta}{2}= \theta
\therefore Var(\hat{\theta})=Var(2\bar{X})=4Var(\frac{1}{n}\sum x_{i})= \frac {4}{n^2} \sum Var(X_{i}) = \frac {4}{n^2} \sum \frac{\theta^2}{12} = \frac{4}{n^2} \cdot \frac {n\theta^2}{12}=\frac{\theta^2}{3n}
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