Estimation of Time Series Parameters

Abstract

In the preceding chapters we have described some parametric models for random processes and time series. In all the introduced parametric models there are parameters β or γ of mean values and parameters ν of covariance functions which are unknown in practical applications and which should be estimated from the random process, or time series, data. By this data we mean a real vector x of realizations of a finite observation X _O = {X(t);t ∈ T _O } of a random process X(.) = {X(t);t ∈ T}. Usually X _O = (X(1),..., X(n))′ if X(.) is a time series and X _O = (X(t ₁),..., X(t _n ))>′ if X _O is a discrete observation of the random process X(.) with continuous time at time points t ₁ ,...,t _n . The length of observation n is some natural number. In this chapter we shall assume that t _i+1 — t _i = d; i = 1, 2,..., n-1, that is we have an observation X _O of X(.) at equidistant time points t ₁,...,t _n ∈ T. Next we shall omit the subscript O and we shall denote the finite observation of the length n of a time series or of a random process X(.) by the unique notation (Math) to denote its dependence on n. The vector X will be, in both cases, called the finite time series observation. The vector x = (x(1),...,x(n))′ where x(t) is a realization of X(t);t = 1,2,..., n will be called the time series data.