# 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