Question Type 4: Working with parameters in determining unbiased estimators Exercises

Show that the sample variance $S^2 = \displaystyle \frac{1}{n-1}\sum_{i=1}^n (X_i - \bar X)^2$ is an unbiased estimator for $a$.

Show that the sample mean $\displaystyle \bar X = \frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased estimator of the parameter $a + b$, given that $E(X_i)=a+b$.

Find the constant $c$ such that the estimator $c\,S_n^2$ is unbiased for $a$.

Determine the bias of the estimator $S_n^2 = \displaystyle \frac{1}{n}\sum_{i=1}^n (X_i - \bar X)^2$ for the parameter $a$, where $\mathrm{Var}(X_i)=a$.

Derive the variance of the sample mean $\bar X$ in terms of $a$ for i.i.d. observations with $\mathrm{Var}(X_i)=a$.

Show that the estimator $T = \bar X - \displaystyle \frac{(n-1)S^2}{n}$ is an unbiased estimator for $b$.

Find the constant $k$ such that $T_2 = \bar X - k\,S_n^2$ is an unbiased estimator of $b$.

Show that the estimator $T_1 = \bar X - S_n^2$ is not unbiased for $b$. Compute its bias.

Using the method of moments for a sample from the distribution with $E(X)=a+b$ and $\mathrm{Var}(X)=a$, derive the moment estimators $\hat a$ and $\hat b$ and state whether each is unbiased.

Provide one linear combination $T = \alpha\,\bar X + \beta\,S^2$ that is unbiased for $b$. Specify $\alpha$ and $\beta$.

Assuming $X_i\sim N(a+b,a)$ so that $\bar X$ and $S^2$ are independent, express the variance of the estimator $T = \bar X - \frac{(n-1)S^2}{n}$ in terms of $a$ and $n$.

Use the Cramér–Rao inequality to find a lower bound for the variance of any unbiased estimator of $b$ when $X_i\sim N(a+b,a)$ and $a$ is known.