12 Estimation

Chapter Overview¶

Where We’re Going¶

In this chapter, we will answer the question, “how can I estimate an unknown value from related data?”

We will:

1. Introduce the idea of an estimator, a function that accepts data and returns an estimate (e.g. a best guess) at an unknown parameter (see Section 12.1).

   • Show how to derive a Maximum Likelihood Estimator (MLE) for an unknown parameter

   • Use maximum likelihood estimation to estimate unknown parameters for binomial, geometric, and normal random variables.

   • Show that many, but not all, MLE’s are empirical averages

2. Define an empirical distribution and distinguish empirical/sample averages from expectations/population averages.

   • Discuss the use of empirical averages as estimators to unknown expectations, variances, and covariances.

3. Discuss properties of estimators (see Section 12.2). In particular, we will discuss the:

   • consistency,

   • bias,

   • variance,

   • accuracy of an estimator.