Theory of Estimation

Abstract

We know from our basic knowledge of statistics that one of the objectives in statistics is to better understand and model the underlying process which generates the data. This is known as statistical inference: we infer from information contained in a sample properties of the population from which the observations are taken. In multivariate statistical inference, we do exactly the same. The basic ideas were introduced in Section 4.5 on sampling theory: we observed the values of a multivariate random variable X and obtained a sample $\mathcal{X} = \{ x_i \} _{i = 1}^n$. Under random sampling, these observations are considered to be realizations of a sequence of i.i.d. random variables X₁, . . ., X_n where each X_i is a p-variate random variable which replicates the parent or population random variable X. In this chapter, for notational convenience, we will no longer differentiate between a random variable X_i and an observation of it, x_i, in our notation. We will simply write x_i and it should be clear from the context whether a random variable or an observed value is meant.