Probability and Statistics for Finance

Chapter 17. Point Estimators

Information shapes a portfolio manager's or trader's perception of the true state of the environment such as, for example, the distribution of the portfolio return and its volatility or the probability of default of a bond issue held in a portfolio. A manager needs to gain information on population parameters to make well-founded decisions.

Since it is generally infeasible or simply too involved to analyze the entire population in order to obtain full certainty as to the true environment—for example, we cannot observe a portfolio for an infinite number of years to find out about the expected value of its return—we need to rely on a small sample to retrieve information about the population parameters. To obtain insight about the true but unknown parameter value, we draw a sample from which we compute statistics or estimates for the parameter.

In this chapter, we will learn about samples, statistics, and estimators. Some of these concepts we already covered in Chapter 3. In particular, we present the linear estimator, explain quality criteria (such as the bias, mean-square error, and standard error) and the large-sample criteria. In the context of large-sample criteria, we present the idea behind consistency, for which we need the definition of convergence in probability and the law of large numbers. As another large-sample criterion, we introduce the unbiased efficiency, explaining the best linear unbiased estimator or, alternatively, the minimum variance ...