# 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*.

## Keywords

Time Series Maximum Likelihood Estimation Covariance Function Observe Time Series Likelihood Equation## Preview

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