Question Type 3: Estimating the mean and variance using unbiased estimators Exercises

Given the following data values with frequencies: 4 (8 times), 5 (12 times), 6 (6 times), 7 (8 times), and 9 (2 times), compute the unbiased estimator for the population mean.

Using the same data set (4:8, 5:12, 6:6, 7:8, 9:2), compute the unbiased estimator for the population variance.

From the result of the previous question, compute the sample standard deviation.

For the given data set, compute the biased estimator of variance (using denominator $n$) and compare it with the unbiased estimator $S^2=1.8418$. What is the numerical bias?

Assume a distribution has parameters $a,b$ satisfying $E(X)=a+b$ and $\mathrm{Var}(X)=a$. Using the sample mean $\bar x=5.6111$ and sample variance $S^2=1.8418$ from the data, find the method-of-moments estimators for $a$ and $b$.

With the same setup, consider the estimator $\hat b=\bar X-\hat a.$ Determine whether $\hat b$ is unbiased for $b$.

Let $X_1,\dots,X_n$ be iid from a distribution with $E(X)=a+b$ and $\mathrm{Var}(X)=a$. Show that the estimator $\hat a=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2$ is unbiased for $a$.

Determine whether the estimator $\tilde b=\bar X-\frac{1}{n}\sum_{i=1}^n(X_i-\bar X)^2$ is unbiased for $b$. If it is biased, find its bias.

Show that the estimator $\hat b=\bar X-\tfrac{1}{n-1}\sum(X_i-\bar X)^2$ is consistent for $b$.

Find a constant $c$ such that the estimator $\hat b_c=\bar X-c\sum_{i=1}^n(X_i-\bar X)^2$ is unbiased for $b$.

Using the value of $c$ found above, derive the variance of the unbiased estimator $\hat b=\bar X-\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2$ in terms of $a$ and $n$.