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自回归模型中的典型相关分析与降秩回归AI 解读

When the rank of the autoregression matrix is unrestricted, the maximum likelihood estimator under normality is the least squares estimator. When the rank is restricted, the maximum likelihood estimator is composed of the eigenvectors of the effect covariance matrix in the metric of the error covariance matrix corresponding to the largest eigenvalues [Anderson, T. W. (1951). Ann. Math. Statist. 22 327-351]. The asymptotic distribution of these two covariance matrices under…

一般条件下降秩回归估计量的渐近分布AI 解读

In the regression model $\mathbf{Y} = \eta + \mathbf{BX} + \mathbf{Z}$ with $\mathbf{Z}$ unobserved, $\mathscr{E}\mathbf{Z} = \mathbf{0}$ and $\mathscr{E}\mathbf{ZZ}' = \mathbf{\Sigma}_{ZZ}$, the least squares estimator of $\mathbf{B}$ is $\hat{\mathbf{B}} = \mathbf{S}_{YX}\mathbf{S}_{XX}^{-1}$. If the rank of $\mathbf{B}$ is known to be $k$ less than the dimensions of $\mathbf{Y}$ and $\mathbf{X}$, the reduced rank regression estimator of $\mathbf{B}$, say $\mathbf{B}_k$,…

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谱分布的拟合优度检验AI 解读

The spectral distribution function of a stationary stochastic process standardized by dividing by the variance of the process is a linear function of the autocorrelations. The integral of the sample standardized spectral density (periodogram) is a similar linear function of the autocorrelations. As the sample size increases, the difference of these two functions multiplied by the square root of the sample size converges weakly to a Gaussian stochastic process with a…

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广泛因子分析模型的渐近卡方检验AI 解读

Three types of asymptotic $\chi^2$ goodness-of-fit tests derived under the normal assumption have been used widely in factor analysis. Asymptotic behavior of the test statistics is investigated here for the factor analysis model with linearly or nonlinearly restricted factor loadings under weak assumptions on the factor vector and the error vector. In particular the limiting $\chi^2$ result for the three tests is shown to hold for the factor vector, either fixed or random…

多元方差分量秩检验的分位点AI 解读

In the balanced one-way multivariate analysis of variance with random effects the likelihood ratio criterion for testing that the rank of the covariance matrix of the effect vector is not greater than a specified number is considered. The limiting distribution of −2 times the logarithm of the criterion is the distribution of the sum of squares of the positive characteristic roots of a certain random symmetric matrix. This paper presents exact and simulated quantiles of this…

多元方差分量秩检验的分位点AI 解读

In the balanced one-way multivariate analysis of variance with random effects the likelihood ratio criterion for testing that the rank of the covariance matrix of the effect vector is not greater than a specified number is considered. The limiting distribution of −2 times the logarithm of the criterion is the distribution of the sum of squares of the positive characteristic roots of a certain random symmetric matrix. This paper presents exact and simulated quantiles of this…

一般条件下因子分析估计量的渐近正态分布AI 解读

Asymptotic properties of estimators for the confirmatory factor analysis model are discussed. The model is identified by restrictions on the elements of the factor loading matrix; the number of restrictions may exceed that required for identification. It is shown that a particular centering of the maximum likelihood estimator derived under assumed normality of observations yields an asymptotic normal distribution that is common to a wide class of distributions of the factor…

多元方差分量模型的极大似然估计与似然比准则AI 解读

Maximum likelihood estimators are obtained for multivariate components of variance models under the condition that the effect covariance matrix is positive semidefinite with a maximum rank. The rank of the estimator is random. The estimation procedure leads to a likelihood ratio test that the rank of the effect matrix is not greater than a given number against the alternative that the rank is not greater than a larger specified number. Linear structural relationship models…

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联立方程系统中单方程估计量的比较AI 解读

Comparisons of estimators are made on the basis of their mean squared errors and their concentrations of probability computed by means of asymptotic expansions of their distributions when the disturbance variance tends to zero and alternatively when the sample size increases indefinitely. The estimators include k -class estimators (limited information maximum likelihood, two-stage least squares, and ordinary least squares) and linear combinations of them as well as…

方向参数的最佳不变估计AI 解读

Let $X$ be an $n \times k$ random matrix whose coordinates are independently normally distributed with common variance $\sigma^2$ and means given by $EX = e\mu' + \theta\lambda',$ where $e$ is the vector in $R^n$ having all coordinates equal to $1, \theta \in R^n,$ and $\mu, \lambda \in R^k$ with $\sum^k_{j = 1} \lambda^2_j = 1.$ The problem is to estimate $\lambda$, say by $\hat{\lambda},$ with loss function $1 - (\lambda'\hat{\lambda})^2$ when $\mu, \theta,$ and $\sigma^2$…

线性统计关系的估计AI 解读

This paper on estimating linear statistical relationships includes three lectures on linear functional and structural relationships, factor analysis, and simultaneous equations models. The emphasis is on relating the several models by a general approach and on the similarity of maximum likelihood estimators (under normality) in the different models. In the first two lectures the observable vector is decomposed into a "systematic part" and a random error; the systematic part…

斜率系数最大似然估计量的精确分布与近似分布AI 解读

Summary When the errors are normally independently distributed with equal variance, the maximum likelihood estimator of the slope of a linear functional relationship is the slope of the line minimizing the sum of squared deviations orthogonal to the line. The exact density and distribution of this estimator are obtained. Approximate distributions are obtained, and their accuracies are discussed.

有限信息极大似然估计量分布函数的计算AI 解读

T. W. Anderson, Naoto Kunitomo, Takamitsu Sawa, Evaluation of the Distribution Function of the Limited Information Maximum Likelihood Estimator, Econometrica, Vol. 50, No. 4 (Jul., 1982), pp. 1009-1027

含误差成分动态模型的估计AI 解读

Observations on N cross-section units at T time points are used to estimate a simple statistical model involving an autoregressive process with an additive term specific to the unit. Different assumptions about the initial conditions are (a) initial state fixed, (b) initial state random, (c) the unobserved individual effect independent of the unobserved dynamic process with the initial value fixed, and (d) the unobserved individual effect independent of the unobserved…

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动态模型中最小二乘估计的强一致性AI 解读

The least squares estimate of the parameter matrix $\mathbf{B}$ in the model $\mathbf{y}_t = \mathbf{B'x}_t + \mathbf{u}_t$, where $\mathbf{u}_t$ is an $m$-component vector of unobservable disturbances and $x_t$ is a $p$-component vector, converges to $\mathbf{B}$ with probability one under certain conditions on the behavior of $x_t$ and $\mathbf{u}_t$. When $\mathbf{x}_t$ is stochastic and the conditional expectation of $\mathbf{u}_t$ given $\mathbf{x}_s$ for $s \leqslant t$…

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自回归过程的重复测量AI 解读

Estimation of parameters and tests of hypotheses are studied in first-order autoregressive processes where the process is observed several times over a given time interval. The process may be homogeneous (i.e., the parameters may be constant over time) or inhomogeneous (time-varying parameters). Sufficient statistics under normality are obtained for various cases and several tests of hypotheses are given.

自回归移动平均模型的时域与频域估计AI 解读

The autoregressive moving average model is a stationary stochastic process $\{y_t\}$ satisfying $\sum^p_{k=0} \beta_ky_{t-k} = \sum^q_{g=0} \alpha_g\nu_{t-g}$, where the (unobservable) process $\{v_t\}$ consists of independently identically distributed random variables. The coefficients in this equation and the variance of $v_t$ are to be estimated from an observed sequence $y_1, \cdots, y_T$. To apply the method of maximum likelihood under normality the model is modified…