In many contexts involving multivariate linear regression models, some or all of the independent (predictor) variables are measured with error. It is argued that if the goal is to assess the relationship of the dependent variables to the true predictor variables, then it is important to determine the reliability matrix Λ of the measurement X of the vector of true predictors. If Λ is singular, then the slope matrix B is not identifiable. If Λ is nearly singular, then B cannot…
Because families have varying numbers of offspring, the maximum likelihood estimators of the interclass, or parent-offspring, and intraclass, or offspring-offspring, correlations are difficult to compute (Srivastava, 1984). For this reason, alternative estimators have been proposed, which are not necessarily asymptotically efficient. However, maximum likelihood estimators of the inter- and intraclass correlations in families of constant size can be obtained in closed form,…
Confidence intervals are widely used in statistical practice as indicators of precision for related point estimators or as estimators in their own right. In the present paper it is shown that for some models, including most linear and nonlinear errors-in-variables regression models, and for certain estimation problems arising in the context of classical linear models, such as the inverse regression problem, it is impossible to construct confidence intervals for key…
Leon Jay Gleser · Raymond J. Carroll · Paul P. Gallo·刊期 1987-03-01 · 卷 15（1）·The Annals of Statistics已编入正刊
测量误差回归模型中最小二乘估计的极限分布AI 解读
It is well-known that the ordinary least squares (OLS) estimator $\hat{\beta}$ of the slope and intercept parameters $\beta$ in a linear regression model with errors of measurement for some of the independent variables (predictors) is inconsistent. However, Gallo (1982) has shown that certain linear combinations of $\beta$. In this paper, it is shown that under reasonable regularity conditions such linear combinations of $\hat{\beta}$ are (jointly) asymptotically normally…
The problem of finding classes of estimators which dominate the usual estimator $X$ of the mean vector $\mu$ of a $p$-variate normal distribution $(p \geq 3)$ under general quadratic loss is analytically difficult in cases where the covariance matrix is unknown. Estimators of $\mu$ in this case depend upon $X$ and an independent Wishart matrix $W$. In the present paper, integration-by-parts methods for both the multivariate normal and Wishart distributions are combined to…
In an errors-in-variables regression model, the least squares estimate is generally inconsistent for the complete regression parameter but can be consistent for certain linear combinations of this parameter. We explore the conjecture that, when the least squares estimate is consistent for a linear combination of the regression parameter, it will be preferred to an errors-in-variables estimate, at least asymptotically. The conjecture is false, in general, but it is true for…
Goodness-of-fit tests of Kolmogorov type reject a null hypothesis H 0 : F(x) = F*(x) whenever the graph of the sample cumulative distribution function crosses one of two boundary functions, G 1(x), G 2(x). The best-known example of a test of this type is the Kolmogorov—Smirnov test D. When the true cumulative distribution function F(x) is continuous, a number of algorithms are available for calculating the exact powers of such tests. In this article it is shown that such…
Goodness-of-fit tests of Kolmogorov type reject a null hypothesis H 0 : F(x) = F*(x) whenever the graph of the sample cumulative distribution function crosses one of two boundary functions, G 1(x), G 2(x). The best-known example of a test of this type is the Kolmogorov—Smirnov test D. When the true cumulative distribution function F(x) is continuous, a number of algorithms are available for calculating the exact powers of such tests. In this article it is shown that such…
It is demonstrated that maximum likelihood estimators exist for the intercept, slope and unknown error variance in Dolby's (1976) unreplicated ultrastructural model with known error variance ratio even when the ratio of the variability of the measured variables to the variability of the measurement errors is unknown. This corrects an assertion in Dolby's paper. Patefield's (1978) adjusted maximum likelihood estimators for this model are shown to be strongly consistent and…
In a multivariate "errors in variables" regression model, the unknown mean vectors $\mathbf{u}_{1i}: p \times 1, \mathbf{u}_{2i}: r \times 1$ of the vector observations $\mathbf{x}_{1i}, \mathbf{x}_{2i}$, rather than the observations themselves, are assumed to follow the linear relation: $\mathbf{u}_{2i} = \alpha + B\mathbf{u}_{1i}, i = 1,2,\cdots, n$. It is further assumed that the random errors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{u}_i, \mathbf{x}'_i = (\mathbf{x}'_{1i}, \mathbf{x}'_{2i}), \mathbf{u}'_i = (\mathbf{u}'_{1i}, \mathbf{u}'_{2i})$…
Let $X$ be an observation from a $p$-variate normal distribution $(p \geqslant 3)$ with mean vector $\theta$ and unknown positive definite covariance matrix $\sum$. We wish to estimate $\theta$ under the quadratic loss $L(\delta; \theta, \sum) = \lbrack\mathrm{tr}(Q\sum)\rbrack^{-1}(\delta - \theta)'Q(\delta - \theta)$, where $Q$ is a known positive definite matrix. Estimators of the following form are considered: $\delta_{k, h}(X, W) = \lbrack I - kh(X'W^{-1}X)\lambda_1(QW/n^\ast)Q^{-1}W^{-1}\rbrack X,$…
Estimation of the mean θ of a normal distribution N(θ, aθ2) with known coefficient of variation a 1/2 is treated as a decision problem with squared-error loss. It is shown that all estimators linear in the sample mean and sample standard deviation s, as well as the maximum likelihood estimator (MLE), are dominated in risk by the admissible, minimum risk, scale equivariant estimator . A class of Bayes estimators against inverted-gamma priors is constructed, and shown to…
This article provides a simple distributional representation for the noncentral Wishart distribution most useful when the noncentrality matrix is of less than full rank. It is shown that the representation leads to a method for simulating a noncentral Wishart matrix that has advantages over a procedure currently used.
This article provides a simple distributional representation for the noncentral Wishart distribution most useful when the noncentrality matrix is of less than full rank. It is shown that the representation leads to a method for simulating a noncentral Wishart matrix that has advantages over a procedure currently used.
A theory of sequential Bayes interval estimation procedures for a single parameter is developed for the case where the loss for using an interval $I$ is a linear combination of the length of $I$, the indicator of noncoverage of $I$, and the number of observations taken. A class of stopping rules $\{t(c): c > 0\}$ is shown to be asymptotically pointwise optimal (A.P.O.) and asymptotically optimal (A.O.) for the confidence interval problem as the cost $c$ per observation tends…
Let $\{X(t), t = 0, \pm 1, \pm 2,\cdots\}$ be a $p$-dimensional, zero mean, stationary Gaussian time series with matrix-valued covariance function $R(u) = (r_{jk}(u)), u = 0, \pm 1, \pm 2, \cdots; j, k = 1, 2,\cdots, p$. Let $F(\omega)$ be the spectral density matrix of the time series (if $F(\omega)$ exists), and assume that $F(\omega)$ is positive definite for all $\omega\in(0, 2\pi\rbrack$. Let $\hat{F}_X(\omega_l)$ be an estimator of $F(\omega_l)$ formed by averaging $(2n + 1)$…
Except in special cases, the distribution of a quadratic form Q in the normal random vector y cannot be expressed in closed form. However, bounds on the cumulative distribution of Q have been obtained by many authors. This article derives a new class of such bounds by means of a certain mixture representation for the distribution of Q. It is shown how the results apply to the problem of finding the distribution of ratios of independent quadratic forms in normal variates.
Except in special cases, the distribution of a quadratic form Q in the normal random vector y cannot be expressed in closed form. However, bounds on the cumulative distribution of Q have been obtained by many authors. This article derives a new class of such bounds by means of a certain mixture representation for the distribution of Q. It is shown how the results apply to the problem of finding the distribution of ratios of independent quadratic forms in normal variates.
Leon Jay Gleser·刊期 1968-11 · 卷 55（3）·Biometrika已编入正刊
相关系数集合相等性检验的若干渐近结果AI 解读
This note considers tests of the hypothesis H that the off-diagonal elements of a correlation matrix are equal. A test statistic T3 proposed by Aitkin, Nelson & Reinfurt (1968) is shown to be asymptotically equivalent, under H, to a member of a certain class of quadratic forms in the sample correlations. The asymptotic null distribution of each member of this class is obtained, the distribution of T3 and of statistics proposed by other authors following as a special case.…
The Nonexistence of 100(1−α)% Confidence Sets of Finite Expected Diameter in Errors-in-Variables and Related Models
It is well-known that the ordinary least squares (OLS) estimator β^ of the slope and intercept parameters β in a linear regression model with errors of measurement for some of the independent variables (predictors) is inconsistent. However, Gallo (1982) has shown that certain linear combinations of β. In this paper, it is shown that under reasonable regularity conditions such linear combinations of β^ are (jointly) asymptotically normally…
The problem of finding classes of estimators which dominate the usual estimator X of the mean vector μ of a p-variate normal distribution (p≥3) under general quadratic loss is analytically difficult in cases where the covariance matrix is unknown. Estimators of μ in this case depend upon X and an independent Wishart matrix W. In the present paper, integration-by-parts methods for both the multivariate normal and Wishart distributions are combined to…
In a multivariate "errors in variables" regression model, the unknown mean vectors u1i:p×1,u2i:r×1 of the vector observations x1i,x2i, rather than the observations themselves, are assumed to follow the linear relation: u2i=α+Bu1i,i=1,2,⋯,n. It is further assumed that the random errors ei=xi−ui,xi′=(x1i′,x2i′),ui′=(u1i′,u2i′)…
Let X be an observation from a p-variate normal distribution (p⩾3) with mean vector θ and unknown positive definite covariance matrix ∑. We wish to estimate θ under the quadratic loss L(δ;θ,∑)=[tr(Q∑)]−1(δ−θ)′Q(δ−θ), where Q is a known positive definite matrix. Estimators of the following form are considered: δk,h(X,W)=[I−kh(X′W−1X)λ1(QW/n∗)Q−1W−1]X,…
A theory of sequential Bayes interval estimation procedures for a single parameter is developed for the case where the loss for using an interval I is a linear combination of the length of I, the indicator of noncoverage of I, and the number of observations taken. A class of stopping rules {t(c):c>0} is shown to be asymptotically pointwise optimal (A.P.O.) and asymptotically optimal (A.O.) for the confidence interval problem as the cost c per observation tends…
Let {X(t),t=0,±1,±2,⋯} be a p-dimensional, zero mean, stationary Gaussian time series with matrix-valued covariance function R(u)=(rjk(u)),u=0,±1,±2,⋯;j,k=1,2,⋯,p. Let F(ω) be the spectral density matrix of the time series (if F(ω) exists), and assume that F(ω) is positive definite for all ω∈(0,2π]. Let F^X(ωl) be an estimator of F(ωl) formed by averaging (2n+1)…