Chapter I - Statistics II - Sampling Distributions

This is a set of exercises for Statistics II, Chapter II — Point Estimators, from the Lisbon Accounting and Business School (ISCAL–IPL).

The exercise set consists of True/False questions (where you should determine the truth value and justify your choice) and multiple choice questions (where you should select the correct option and justify it appropriately).

Questions

Question 1

Consider the estimators \(T_1,T_2\) and \(T_3\) of parameter \(\theta\) such that:

\[ \begin{aligned} E(T_1)=\theta,\qquad V(T_1)=9\\ E(T_2)=3\theta,\qquad V(T_2)=3\\ \end{aligned} \] \[ T_3=\frac{T_2}{3} \]

  1. \(T_2\) is not an unbiased estimator for \(\theta\) and \(Bias(T_2)=2\theta\)

  2. \(T_1\) and \(T_3\) unbiased estimators for \(\theta\).

  3. \(T_1\) is a more efficient estimator than \(T_3\).

Question 2

Let \((X_1,X_2,\ldots,X_n)\) a sample with size \(n\geq 2\) from a population \(X\) uniformly distributed between \([a,0]\) (with \(a<0\)). Let \(\hat{a}=2\bar{X}\) an estimator for the population parameter \(a\).

We have \(E(\hat{a})=a\) \(\forall x<0\).

Question 3

Consider a random sample \((X_1,X_2,\ldots,X_n)\), with \(n\in\mathbb{N}\), taken from a population \(X\) with mean \(\mu\) and variance \(\sigma^2\). Consider the estimator \(T_1\) for \(\mu\): \[ T_1=X_1+\frac{1}{n-1} \sum_{i=2}^n X_i \]

  1. This estimator is biased for \(\mu\) and its bias is equal to \(\mu\).

  2. Consider an alternative estimator, \(T_2\), for \(\mu\): \[ T_2=\frac{1}{2}T_1 \]

\(T_2\) is a consistent estimator.

Question 4

Consider a population \(X\) with distribution depends on some parameter \(\theta\in\mathbb{R}\), unknown. From the population you know that \[ E(X)=\theta-2\quad\text{and}\quad V(X)=1 \]

From a random sample \((X_1,X_2,\ldots,X_n)\) with \(n\geq 2\), you got two estimators \(\theta^*\) and \(\hat{\theta}\), from which you know the following:

\[ \begin{aligned} \theta^*=\bar{X}+2\\ E(\hat{\theta})=\theta\\ V(\hat{\theta})=\frac{2}{n} \end{aligned} \]

  1. In relation to the estimators \(\theta^*\) and \(\hat{\theta}\):
  1. Just \(\theta^*\) is unbiased
  2. None is unbiased
  3. Both are unbiased
  4. Just \(\hat{\theta}\) is unbiased

  1. \(\theta^*\) is more efficient than \(\hat{\theta}\)

Question 5

The following estimators are proposed for the variance of a Normal population:

\[ S^2=\frac{\sum_{i=1}^n(X_i-\bar{X})^2}{n}\qquad S'^2=\frac{\sum_{i=1}^n(X_i-\bar{X})^2}{n-1} \]

  1. Knowing that \(S'^2\) is an unbiased estimator for \(\sigma^2\), we can say that \(S^2\) is a biased estimator for \(\sigma^2\), with bias equal to \((-1/n)\sigma^2\).

  2. Two estimates for \(\sigma\), \(\sigma_1^*\) and \(\sigma_2^*\), from 117 samples for \(X\), knowing that the same samples satisfy the following:

\[ \sum_{i=1}^{117} x_i=115\qquad \sum_{i=1}^{117}x_i^2=801 \]

Are, respectively, equal to:

Question 6

If \(\hat{\theta}\) is an unbiased estimator for the parameter \(\theta\), unknown, from a population \(X\),

  1. \(\hat{\theta}\) is the mean of the sampling distribution for \(\theta\).
  2. \(\theta\) is the mean of the sampling distribution for \(\hat{\theta}\).
  3. \(V(\hat{\theta})=\frac{V(\theta)}{n}\)
  4. \(\hat{\theta}=\theta\)