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基于密度—分位数函数的最优间距选择AI 解读

In this paper design techniques for continuous parameter time series regression analysis are employed to develop a general approach to optimal spacing selection for the linear estimation of location and scale parameters by sample quantiles from uncensored or censored samples. The spacings derived from this approach are asymptotically optimal in the sense that they result in near optimal asymptotic relative efficiencies for large values of $k$, the number of spacing elements.…

某些时间序列模型中最优设计的唯一性与最终唯一性AI 解读

Using the results of Barrow, et al., and Chow on the optimal placement of knots in the approximation of functions by piecewise polynomials, we show the uniqueness or "eventual uniqueness" of optimal designs for certain time series models considered by Sacks and Ylvisaker, and Wahba. In addition, the limiting behavior (as the sample size increases) of the variance of the BLUE of the regression coefficient is characterized in terms of the density defining the design, and the…

准确且简约的正态线性回归模型的可容许选择AI 解读

Let $M_0$ be a normal linear regression model and let $M_1,\cdots, M_K$ be distinct proper linear submodels of $M_0$. Let $\hat k \in \{0,\cdots, K\}$ be a model selection rule based on observed data from the true model. Given $\hat k$, let the unknown parameters of the selected model $M_{\hat k}$ be fitted by the maximum likelihood method. A loss function is introduced which depends additively on two parts: (i) a measure of the difference between the fitted model $M_{\hat k}$…

相关扰动回归中最小二乘估计量的强一致性AI 解读

This note considers, under minimal assumptions, the strong consistency of least squares estimates in regression with correlated errors.

删一法方差估计AI 解读

Tukey's jackknife estimate of variance for a statistic $S(X_1, X_2, \cdots, X_n)$ which is a symmetric function of i.i.d. random variables $X_i$, is investigated using an ANOVA-like decomposition of $S$. It is shown that the jackknife variance estimate tends always to be biased upwards, a theorem to this effect being proved for the natural jackknife estimate of $\operatorname{Var} S(X_1, X_2, \cdots, X_{n-1})$ based on $X_1, X_2, \cdots, X_n$.

高斯与最小二乘法的发明AI 解读

The most famous priority dispute in the history of statistics is that between Gauss and Legendre, over the discovery of the method of least squares. New evidence, both documentary and statistical, is discussed, and an attempt is made to evaluate Gauss's claim. It is argued (though not conclusively) that Gauss probably possessed the method well before Legendre, but that he was unsuccessful in communicating it to his contemporaries. Data on the French meridian arc are…

将渐近形状方法应用于非指数族AI 解读

Fortus' generalization of asymptotic shapes of optimal testing regions for composite hypotheses does away with the restriction to exponential families originally imposed by us. Here we survey his work critically, and suggest some improvements that may be crucial for its practical applicability to parametric problems, and point out its shortcomings for nonparametric ones.

一般线性假设中参数的有效非参数估计AI 解读

Asymptotically universally efficient estimators of the parameters in the general linear hypothesis are proposed. These estimators are based on ranks of residuals (or their absolute values); they are analogous to the linearized rank estimators proposed by Kraft and van Eeden in the sense that they are obtained by replacing, in these estimators, the function generating the scores by certain estimators of this function. Finally, it is shown that estimators of the score…

密度及其导数估计量的精确渐近行为AI 解读

For an integer $p \geq 0$, Singh has proposed a class of kernel estimators $\hat{f}^{(p)}$ of the $p$th order derivative $f^{(p)}$ of a density $f$. This paper examines the detailed asymptotic behavior of these estimators. In particular, asymptotically equivalent expressions for the bias $(E\hat{f}^{(p)} - f^{(p)})$, the mean squared error $E(\hat{f}^{(p)} - f^{(p)})^2$ and the error $(\hat{f}^{(p)} - f^{(p)})$ are obtained, which in turn give exact rates of convergence of…

偏序集索引随机变量之和与回归函数积分的估计AI 解读

The estimator proposed by Brunk for the indefinite integral of a regression function defined on the unit cube in $\beta$ dimensional Euclidean space is studied. It is shown to be strongly uniformly consistent if the errors satisfy a first moment type of condition and an almost sure rate of convergence of order $O((n/\log_2n)^{-1/2})$ is obtained.

单调回归估计量的渐近行为AI 解读

An estimator for a monotone regression function was proposed by Brunk. He has shown that if the underlying regression function has positive slope at a point, then, based on $r$ observations, the difference of the regression function and its estimate at that point has a nondegenerate limiting distribution if this difference is multiplied by $r^{1/3}$. To understand how the behavior of the regression function at a point influences the asymptotic properties of the estimator at…

简单回归模型中的 Brown–Mood 型中位数估计量AI 解读

In the context of simple linear regression we investigate a class of estimators for the parameters which contains the Brown-Mood estimators. We derive the asymptotic efficiency properties of the class of slope estimators and find the most efficient slope estimator. This estimator is shown to have efficiency properties analogous to those of other median estimators for slope found in the literature. The equations which define the optimal estimators are median analogues of the…

The convolution result of Block and Savits is generalized.

多元 Behrens–Fisher 判别分析中的矩阵变量与特征根分布AI 解读

In this paper it is shown that some distributions of the matrix variates and latent roots arising in the multivariate Behrens-Fisher discriminant problem can be explicitly expressed in terms of the invariant polynomials with two matrix arguments, due to A. W. Davis, extending the zonal polynomials of matrix argument.

两个似不相关回归方程之间的独立性检验AI 解读

When the error terms in two different regression equations are correlated, Zellner proposed an alternative estimator for the coefficients of each equation based on an estimated covariance matrix between the two error terms. However, since an estimated covariance matrix is used, the OLSE seems better than Zellner's estimator when the correlation of the two equations is close enough to zero. This paper considers the problem of testing the independence between two regression…

一般线性模型中异方差的渐近最优检验AI 解读

Some plausible models for violations of homoscedasticity and linearity in the general linear model have been proposed by Tukey and Anscombe. Bickel has provided robust tests for such violations. In this paper Bickel's tests are shown to be asymptotically optimal as well.

使用δ序列估计多元密度函数AI 解读

This paper studies some asymptotic properties of density estimates $\hat{f}$ of $f$ based on $d$-variate delta sequences. The mean-square consistency, almost sure consistency, and asymptotic normality of $\hat{f}$ have been obtained as corollaries to the $L_1$ convergence properties of these delta sequences. Estimators based on kernel functions, orthogonal series, and some histogram methods can be obtained as special cases of $\hat{f}$.

点态密度估计的渐近最优核AI 解读

Kernel estimation of $f(0)$ is considered where $f$ is a density in some class $\mathscr{F}$ of $d$-dimensional densities, described in terms of a Taylor series expansion. A sequence of kernels which asymptotically minimizes the maximum mean square error of estimation over $\mathscr{F}$ is given. The shape of the kernel is fixed, the size of the window depends on $f(0)$, and an easily computed estimate is obtained to efficiently adapt the sequence to the unknown value of $f(0)$…

不变序贯概率比检验的渐近最优性AI 解读

It is well known that Wald's SPRT for testing simple hypotheses based on i.i.d. observations minimizes the expected sample size both under the null and under the alternative hypotheses among all tests with the same or smaller error probabilities and with finite expected sample sizes under the two hypotheses. In this paper it is shown that this optimum property can be extended, at least asymptotically as the error probabilities tend to 0, to invariant SPRTs like the…

一种时间序列递推的二阶性质AI 解读

In this work we prove the asymptotic efficiency of a certain time series recursion for the parameters of an ARMAX time series model.

最优自回归谱估计AI 解读

An asymptotic lower bound is obtained for the integrated relative squared error of autoregressive spectral estimate when the order of autoregression is selected. The bound is attained in the limit by the same selection as has been proposed for prediction.

方差分量模型中的非负最小偏差不变估计AI 解读

In a general variance component model, nonnegative quadratic estimators of the components of variance are considered which are invariant with respect to mean value translations and have minimum bias, analogously to estimation theory of mean value parameters. Here the minimum is taken over an appropriate cone of positive semidefinite matrices, after having made a reduction by invariance. Among these estimators, which always exist, the one of minimum norm is characterized.…

稳健估计中的局部渐近极小极大性与可容许性AI 解读

For a particular pseudoloss function, local asymptotic minimaxity and admissibility in the sense of Hajek and Le Cam are studied when probability measures are replaced by certain capacities ($\epsilon$-contamination, total variation). A minimax bound for arbitrary estimator sequences is established, admissibility of minimax estimators is proved, and it is shown that minimax estimators must necessarily have an asymptotic expansion in terms of a truncated logarithmic…

单样本与双样本秩检验对粗大误差的稳健性AI 解读

The classical nonparametric hypotheses of symmetry and equality of distribution functions are extended to hypotheses of approximate symmetry and approximate equality, by allowing for gross errors, which is indispensable for practical applications. It is shown that for these hypotheses one- and two-sample rank statistics maintain their distribution freeness, which now refers to their stochastically extreme laws. These laws are evaluated asymptotically, also under similarly…