# Spectral function, estimator of the

estimator of the spectral measure

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 function $F(\lambda)$. As an estimator of this function one often uses an expression of the form

$$F_N(\lambda)=\frac{2\pi}{N}\sum_{-\pi\leq2\pi k/N\leq\lambda}I_N\left(\frac{2\pi k}{N}\right),$$

where $I_N(x)$ is the periodogram. Under fairly general smoothness conditions on $F(\lambda)$, or under mixing conditions on the random process $X(t)$, this estimator turns out to be asymptotically unbiased and consistent.

The above estimator of $F(\lambda)$ is a special case of an estimator

$$\frac{2\pi}{N}\sum_{-\pi\leq2\pi k/N\leq\pi}A\left(\frac{2\pi k}{N}\right)I_N\left(\frac{2\pi k}{N}\right)$$

of a function

$$I(A)=\int\limits_{-\pi}^\pi A(x)f(x)dx$$

of the spectral density $f(\lambda)$. In particular, many estimators of the spectral density (cf. Spectral density, estimator of the) reduce to this form, where the function $A(x)$ depends on the size $N$ of the sample and is concentrated about the point $x=\lambda$.

#### References

 [1] D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart & Winston (1975) [2] E.J. Hannan, "Multiple time series" , Wiley (1970)