The periodogram is a widely used tool to analyze second order stationary time series. An attractive feature of the periodogram is that the expectation of the periodogram is approximately equal to the underlying spectral density of the time series. However, this is only an approximation, and it is well known that the periodogram has a finite sample bias, which can be severe in small samples. In this article, we show that the bias arises because of the finite boundary of observation in one of the discrete Fourier transforms which is used in the construction of the periodogram. Moreover, we show that by using the best linear predictors of the time series over the boundary of observation we can obtain a ‘complete periodogram’ that is an unbiased estimator of the spectral density. In practice, the ‘complete periodogram’ cannot be evaluated as the best linear predictors are unknown. We propose a method for estimating the best linear predictors and prove that the resulting ‘estimated complete periodogram’ has a smaller bias than the regular periodogram. The estimated complete periodogram and a tapered version of it are used to estimate parameters, which can be represented in terms of the integrated spectral density. We prove that the resulting estimators have a smaller bias than their regular periodogram counterparts. The proposed method is illustrated with simulations and real data.

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小样本时间序列的频谱方法：完整周期图方法

周期图是分析二阶平稳时间序列的广泛使用的工具。周期图的一个吸引人的特征是周期图的期望值大约等于时间序列的潜在频谱密度。然而，这只是一个近似值，众所周知，周期图具有有限的样本偏差，这在小样本中可能很严重。在本文中，我们表明偏差的产生是由于在周期图构造中使用的离散傅立叶变换之一中的观察边界有限。此外，我们表明，通过在观察边界上使用时间序列的最佳线性预测器，我们可以获得“完整周期图”，它是频谱密度的无偏估计器。在实践中，由于最佳线性预测因子未知，因此无法评估“完整周期图”。我们提出了一种估计最佳线性预测变量的方法，并证明由此产生的“估计完整周期图”比常规周期图具有更小的偏差。估计的完整周期图及其锥形版本用于估计参数，这些参数可以用积分谱密度表示。我们证明，由此产生的估计量比其常规周期图对应物具有更小的偏差。所提出的方法用模拟和真实数据来说明。估计的完整周期图及其锥形版本用于估计参数，这些参数可以用积分谱密度表示。我们证明，由此产生的估计量比其常规周期图对应物具有更小的偏差。所提出的方法用模拟和真实数据来说明。估计的完整周期图及其锥形版本用于估计参数，这些参数可以用积分谱密度表示。我们证明，由此产生的估计量比其常规周期图对应物具有更小的偏差。所提出的方法用模拟和真实数据来说明。