Raising the bar

Chapter 8. Estimation: One or Two Populations

Unit 8.5. Questions for repetition

Ex. 1. Show that existence of one unbiased estimator of some parameter implies existence of many unbiased estimators of the same parameter.
Ex. 2. If you have two unbiased estimators of the same parameter, how do you construct a more efficient one?
Ex. 3. Define consistency and illustrate geometrically.
Ex. 4. Derive the confidence interval for the population mean when: (a) $latex \sigma$ is known and (b) $latex \sigma$ is unknown.
Ex. 5. Prove that the variable defined in Eq. (8.5) is distributed as $latex t_{n-1}$ stating all theoretical facts that you use.
Ex. 6. Derive the confidence interval for population proportion. Why do you use $latex z$ statistic in this case?
Ex. 7. If $latex T_n$ is a consistent estimator of $latex \tau$, we write $latex \textrm{plim}_{n \to \infty} T_n = \tau$ and say that the probability limit of $latex T_n$ is $latex \tau$. The whole issue of consistency and probability limits may seem like a dreadful topic but there are properties that are transparent and easy to apply.

$latex \qquad$(a) The probability limit preserves arithmetic operations in the sense that

$latex \qquad$$latex \textrm{plim}(X_n + Y_n) = \textrm{plim}\, X_n + \textrm{plim}\, Y_n,$

$latex \qquad$$latex \textrm{plim}(X_n Y_n) = \textrm{plim}\, X_n \cdot \textrm{plim}\, Y_n,$

$latex \qquad$$latex \textrm{plim}(X_n / Y_n) = \textrm{plim}\, X_n / \textrm{plim}\, Y_n$

$latex \qquad$(in the last case we have to assume that $latex \textrm{plim}\, Y_n \ne 0$).

$latex \qquad$(b) A common observation is parallelism between finite and large sample properties: usually finite sample properties of type

$latex \qquad$$latex E\bar X = \mu_X,\ Es_X^2 = \sigma_X^2$

$latex \qquad$have their analogs in terms of probability limits:

$latex \qquad$$latex \textrm{plim}\, \bar X = \mu_X,\ \textrm{plim}\, s_X^2 = \sigma_X^2$

$latex \qquad$(we proved the first of these equations in Consistency).

Use the above information to prove that $latex T_n$ defined in Example 8.1 is consistent, $latex \textrm{plim}\, T_n = \mu_X$.

Ex. 8. Assuming that the population variances are known, derive the confidence interval for the difference between two means in case of (a) matched pairs and (b) independent samples.
Ex. 9. In the context of two independent samples with unknown but equal variances: (a) calculate $latex V(\bar X - \bar Y)$, define the pooled estimator and find its mean and (b) derive the confidence interval.
Ex. 10. Give the decision tree for testing the difference between two means showing your choices when: (a) $latex \sigma$ is known/unknown, (b) the samples are similar (matched) or independent, (c) in case of unknown $latex \sigma$, equal or unequal variances and (d) the parent population is general or Bernoulli. In all cases give the expression for $latex ME$.

Previous: Difference between two population proportions | Next: Concepts of hypothesis testing

Want the whole book (all 14 chapters, exercises, and the index) as a single PDF? Available on Lulu for $1.99.