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自回归谱估计量的偏差AI 解读

Bias of the least squares estimator of the log of the spectral density of an autoregression attenuates the peaks of the estimator. Under the assumption of an autoregressive generating process of known finite order, we obtain an expression for the order 1/T bias, where T is the sample length of the observed series. This approximation is a sum of several simple functions of the unknown coefficients. When the spectral density has sharp peaks, one of these functions dominates…

自回归谱估计量的偏差AI 解读

Bias of the least squares estimator of the log of the spectral density of an autoregression attenuates the peaks of the estimator. Under the assumption of an autoregressive generating process of known finite order, we obtain an expression for the order 1/T bias, where T is the sample length of the observed series. This approximation is a sum of several simple functions of the unknown coefficients. When the spectral density has sharp peaks, one of these functions dominates…

自回归估计量偏差的不动点刻画AI 解读

Least squares estimators of the coefficients of an autoregression of known, finite order are biased to order $1/T$, where $T$ is the sample length, unless the observed time series is generated by a unique model for that order. The coefficients of this special model are the fixed point of a linear mapping defined by the bias of the least squares estimator. Separate results are given for models with known mean and unknown mean. The "fixed point models" for different orders of…

自回归系数估计量的偏差AI 解读

This article presents simple expressions for the bias of estimators of the coefficients of an autoregressive model of arbitrary, but known, finite order. The results include models both with and without a constant term. The effects of overspecification of the model order on the bias are described. The emphasis is on least-squares and Yule-Walker estimators, but the methods extend to other estimators of similar design. Although only the order T -1 component of the bias is…

谱估计中的若干贝叶斯考虑AI 解读

This paper considers the problem of incorporating prior information about the shape and smoothness of a specrtal density into the formation of a spectral estimate. Two types of finite dimensional parameters are considered, the spectral ordinates at a specified collection of frequencies and the amount of power in each of a set of frequency bands. A method which is conditional on the .asymptotic distribution of periodogram averages is proposed. A formal procedure applies…

谱估计中的若干贝叶斯考虑AI 解读

This paper considers the problem of incorporating prior information about the shape and smoothness of a specrtal density into the formation of a spectral estimate. Two types of finite dimensional parameters are considered, the spectral ordinates at a specified collection of frequencies and the amount of power in each of a set of frequency bands. A method which is conditional on the .asymptotic distribution of periodogram averages is proposed. A formal procedure applies…

平稳协方差矩阵及其逆的近似与 ARMA 应用AI 解读

Approximation of the covariance matrix $\mathbf{\Sigma}$ of $T$ consecutive observations from a second-order stationary process with continuous positive spectral density $f(\lambda) = \lbrack\sigma^2/(2\pi)^2\rbrack|\sum^\infty_{j=0}\delta_je^{i\lambda j}|^2$ is considered. If $\mathbf{\Sigma}^\ast$ is the covariance matrix corresponding to a process with spectral density $1/\lbrack(2\pi)^2f(\lambda)\rbrack$, then $\mathbf{\Sigma}^\ast - \mathbf{\Sigma}^{-1} \geqq 0$. A…

移动平均与自回归过程协方差矩阵的近似逆AI 解读

Let $\mathbf{\Sigma}$ denote the convariance matrix of a vector $\mathbf{x} = (x_1, \cdots, x_T)'$ of $T$ successive observations from a stationary process $\{ x_t\}$ with continuous positive spectral density $f(\lambda)$. Let $\mathbf{\Gamma}$ be the $T \times T$ matrix with elements $\gamma(s, t) = (2\pi)^{-2} \int^\pi_{-\pi} e^{i\lambda (s-t)} f^{-1}(\lambda) d\lambda$. The properties of $\mathbf{\Gamma}$ considered as an approximate inverse of $\mathbf{\Sigma}$ are…

自回归移动平均过程协方差矩阵的逆AI 解读

SUMMARY The problem of inverting the covariance matrix ZT of T successive observations from an autoregressive-moving average process of order (p, q) is discussed. The inverse is obtained for a moving average process (p = 0). An explicit expression is given for the elements of E-1 for a process of order (0, 2).

自回归移动平均过程协方差矩阵的逆AI 解读

The problem of inverting the covariance matrix ΣT of T sucoessive observations from an autoregressive-moving average process of order (p, q) is discussed. The inverse is obtained for a moving average process (p = 0). An explicit expression is given for the elements of Σ−1T for a process of order (0, 2).

一阶移动平均过程协方差矩阵的逆AI 解读

Three distinct expressions are noted for the elements of the inverse of the covariance matrix ΣT of T successive observations from a stationary first order moving average process. An observation of Arato (1961) shows that the problem of finding the exact inverse can be reduced to the problem of finding the covariance determinant |∑T|⁠. Approximate expressions for ∑T-1 are then obtained by approximating various forms of the determinant. The resulting approximate inverses are…

一阶移动平均过程协方差矩阵的逆AI 解读

Three distinct expressions are noted for the elements of the inverse of the covariance matrix ΣT of T successive observations from a stationary first order moving average process. An observation of Arato (1961) shows that the problem of finding the exact inverse can be reduced to the problem of finding the covariance determinant |∑T|⁠. Approximate expressions for ∑T-1 are then obtained by approximating various forms of the determinant. The resulting approximate inverses are…