View source for Spectral function, estimator of the

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''estimator of the spectral measure''

A function of the observed values $X(1),\dots,X(N)$ of a discrete-time [[Stationary stochastic process|stationary stochastic process]], used as an estimator of the [[Spectral function|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|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|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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.R. Brillinger,   "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston  (1975)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.J. Hannan,   "Multiple time series" , Wiley  (1970)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.E.P. Box,   G.M. Jenkins,   "Time series analysis. Forecasting and control" , Holden-Day  (1960)</TD></TR></table>