2Estimation of Process Characteristics

2.1 Introduction

In this chapter, we deal with the problem of estimating the main characteristics of a stationary process from data.

We start with a study of the properties of the covariance function. For such function, we have already seen some basic features in Section 1.5. Now, we investigate its properties more in depth (Sections 2.2 and 2.3).

We pass then to the problems of estimating the mean (Section 2.4), the covariance (Section 2.5), and the spectrum (Section 2.6) from observations. The properties of the covariance function seen in Sections 2.2 and 2.3 turn out to be useful to assess the asymptotic features of such estimators.

2.2 General Properties of the Covariance Function

As previously seen in Section 1.5, the covariance function of a stationary process images, defined as


enjoys the following main properties:

  1. (i) images,
  2. (ii) images,
  3. (iii) images

We add now an important observation: these properties are not exhaustive, in the sense that there exist functions ...

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