## Info

2n k1

The sample periodogram is a natural estimator of the spectrum, but it has a statistical flaw. With the sample variance and the T — 1 autocovariances, we are estimating T parameters with T observations. The periodogram is, in the end, T transformations of these T estimates. As such, there are no "degrees of freedom"; the estimator does not improve as the sample size increases. A number of methods have been suggested for improving the behavior of the estimator. Two common ways are truncation and windowing [see Chatfield (1996, pp. 139-143)]. The truncated estimator of the peri-odogram is based on a subset of the first L < T autocovariances. The choice of L is a problem because there is no theoretical guidance. Chatfield (1996) suggests L approximately equal to 2^fT is large enough to provide resolution while removing some of the sampling variation induced by the long lags in the untruncated estimator. The second mechanism for improving the properties of the estimator is a set of weights called a lag window. The revised estimator is

where the set of weights, {wu, u = 0,..., L}, is the lag window. One choice for the weights is the Bartlett window, which produces

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