# Spectral density, estimator of the

A function of the observed values of a discrete-time stationary stochastic process, used as an estimator of the spectral density . As an estimator of the spectral density one often uses quadratic forms

where the are complex coefficients (depending on ). It can be shown that the asymptotic behaviour as of the first two moments of an estimator of the spectral density is satisfactory, in general, if one considers only the subclass of quadratic forms such that when . This enables one to restrict attention to estimators of the spectral density of the form

where

is a sample estimator of the covariance function of the stationary process and the are suitably chosen weights. The estimator can be written as

where is the periodogram and is some continuous even function with of its Fourier coefficients specified:

The function is called a spectral window; one usually considers spectral windows of the form

where is some continuous function on such that

and as , but . Similarly, one considers coefficients of the form

and a function , called a lag window or covariance window. Under weak smoothness restrictions on the spectral density , or assuming that is mixing, it is possible to prove that for a wide class of spectral or covariance windows the estimator is asymptotically unbiased and consistent.

In the case of a multi-dimensional stochastic process, estimation of the elements of the matrix of spectral densities proceeds in a similar way using the corresponding periodogram . Instead of an estimator of the spectral density in the form of a quadratic form in the observations, one often assumes that the spectral density depends in a particular way on a finite number of parameters, and then one seeks estimators based on the observations of the parameters involved in this expression for the spectral density (see Maximum-entropy spectral estimator; Spectral estimator, parametric).

#### References

[1] | D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1975) MR0443257 Zbl 0321.62004 |

[2] | E.J. Hannan, "Multiple time series" , Wiley (1972) MR0279952 Zbl 0279.62025 Zbl 0211.49804 |

[3] | T.M. Anderson, "Statistical analysis of time series" , Wiley (1971) MR0283939 Zbl 0225.62108 |

#### Comments

#### References

[a1] | G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , Holden-Day (1960) MR0436499 MR0353595 MR0353594 MR0272138 Zbl 1154.62062 Zbl 0858.62072 Zbl 0363.62069 Zbl 0284.62059 Zbl 0276.62080 Zbl 0249.62009 |

[a2] | P.E. Caines, "Linear stochastic systems" , Wiley (1988) MR0944080 Zbl 0658.93003 |

[a3] | K.O. Dzhaparidze, "Parameter estimation and hypothesis testing in spectral analysis of stationary time series" , Springer (1986) MR0775857 MR0812272 Zbl 0584.62157 |

[a4] | L. Ljung, "System identification theory for the user" , Prentice-Hall (1987) MR1157156 Zbl 0615.93004 |

**How to Cite This Entry:**

Spectral density, estimator of the.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Spectral_density,_estimator_of_the&oldid=24129