Spectral density, estimator of the

A function of the observed values $ X ( 1) \dots X ( N) $ of a discrete-time stationary stochastic process, used as an estimator of the spectral density $ f ( \lambda ) $. As an estimator of the spectral density one often uses quadratic forms

$$ \sum _ {s , t = 1 } ^ { N } b _ {s,t} ^ {(} N) X ( s) X ( t) , $$

where the $ b _ {s,t} ^ {(} N) $ are complex coefficients (depending on $ \lambda $). It can be shown that the asymptotic behaviour as $ N \rightarrow \infty $ 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 $ b _ {s _ {1} , t _ {1} } ^ {(} N) = b _ {s _ {2} , t _ {2} } ^ {(} N) $ when $ s _ {1} - t _ {1} = s _ {2} - t _ {2} $. This enables one to restrict attention to estimators of the spectral density of the form

$$ \widehat{f} _ {N} ( \lambda ) = \frac{1}{2 \pi } \sum _ {t = - N + 1 } ^ { N- } 1 e ^ {i t \lambda } b _ {N} ( t) B _ {N} ( t) , $$

where

$$ B _ {N} ( t) = \frac{1}{N} \sum _ { s= } 1 ^ { {N } - | t | } X ( s) X ( s + | t | ) $$

is a sample estimator of the covariance function of the stationary process $ X ( t) $ and the $ b _ {N} ( t) $ are suitably chosen weights. The estimator $ \widehat{f} _ {N} ( \lambda ) $ can be written as

$$ \widehat{f} _ {N} ( \lambda ) = \ \int\limits _ {- \pi } ^ \pi \Phi _ {N} ( x) I _ {N} ( x + \lambda ) d x , $$

where $ I _ {N} ( x) $ is the periodogram and $ \Phi _ {N} ( x) $ is some continuous even function with $ 2N- 1 $ of its Fourier coefficients specified:

$$ b _ {N} ( t) = \int\limits _ {- \pi } ^ \pi \Phi _ {N} ( x) e ^ {i t x } d x ,\ \ t = - N + 1 \dots N - 1 . $$

The function $ \Phi _ {N} ( x) $ is called a spectral window; one usually considers spectral windows of the form

$$ \Phi _ {N} ( x) = A _ {N} \Phi ( A _ {N} x ) , $$

where $ \Phi ( x) $ is some continuous function on $ ( - \infty , \infty ) $ such that

$$ \int\limits _ {- \infty } ^ \infty \Phi ( x) d x = 1 , $$

and $ A _ {N} \rightarrow \infty $ as $ N \rightarrow \infty $, but $ A _ {N} N ^ {-} 1 \rightarrow 0 $. Similarly, one considers coefficients $ b _ {N} ( t) $ of the form

$$ b _ {N} ( t) = K ( A _ {N} ^ {-} 1 t ) $$

and a function $ K ( x) $, called a lag window or covariance window. Under weak smoothness restrictions on the spectral density $ f ( \lambda ) $, or assuming that $ X ( t) $ is mixing, it is possible to prove that for a wide class of spectral or covariance windows the estimator $ \widehat{f} _ {N} ( \lambda ) $ is asymptotically unbiased and consistent.

In the case of a multi-dimensional stochastic process, estimation of the elements of the matrix of spectral densities $ f _ {k,l} ( \lambda ) $ proceeds in a similar way using the corresponding periodogram $ I _ {N} ^ {( k , l ) } ( \lambda ) $. 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

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