Theory and Methods of Statistics

Exercises

For all the problems below, when there are observations X₁, …, X_n, it is understood that

$\begin{array}{ll}\hfill {\stackrel{-}{X}}_n& =n^{-1}\sum _{i=1}^n{X}_i,s_n^2={\left(n-1\right)}^{-1}\sum _{i=1}^n{\left(X_i-{\stackrel{-}{X}}_n\right)}^2,\hfill \\ \hfill X_{n:1}& =min\left(X_1,\dots ,X_n\right),X_{n:n}=max\left(X_1,\dots ,X_n\right).\hfill \end{array}$

3.1. Prove VI(a,b) of Theorem 3.2.5.

3.2. Prove X(b) of Theorem 3.2.5.

3.3. Prove Theorem 3.2.6, the Delta Method.

3.4. Prove the results stated in the examples on applications of the Delta Method.

3.5. Let X₁, …, X_n be iid with $\text{E}\left[X_i\right]=\mu \ne 0$ , $\text{Var}\left[X_i\right]={\sigma }^2$ and $\text{E}\left[X_i^4\right]<\infty$. Find the asymptotic ...