Estimation of Time Series Parameters

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.