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最大均值估计的渐近最优序贯程序AI 解读

Interval estimation of the largest mean of $k$ normal populations $(k \geqq 1)$ with a common variance $\sigma^2$ is considered. When $\sigma^2$ is known the optimal fixed-width interval is given so that, to have the probability of coverage uniformly lower bounded by $\gamma$ (preassigned), the sample size needed is minimized. This optimal interval is unsymmetric for $k > 2$. When $\sigma^2$ is unknown a sequential procedure is proposed and its behavior is studied. It is…

正态分布对双指数备择的最强尺度与位置不变检验AI 解读

The most powerful scale and location invariant test of normality against the double exponential alternative is derived by the technique of integrating with respect to the scale and location transformation group. The resultant test is asymptotically equivalent to the likelihood ratio test of this hypothesis and to Geary's test (i.e. mean deviation over standard deviation) for all three test statistics are shown to have the same asymptotic normal distribution when the sampling…

存在并列值时的渐近一致最强符号检验AI 解读

A modification of the one-sided sign test, which was proposed by Putter (1955), is shown to be an asymptotic UMP test. In contrast to the usual practice, zeros are not omitted but taken into account.

非独立同分布情形下约化经验与分位数过程的收敛及其顺序统计量应用AI 解读

Any triangular array of row independent $\mathrm{rv}$'s having continuous $\mathrm{df}$'s can be transformed naturally so that the empirical and quantile processes of the resulting $\mathrm{rv}$'s are relatively compact. Moreover, convergence (to a necessarily normal process) takes place if and only if a simple covariance function converges pointwise. Using these results we derive the asymptotic normality of linear combinations of functions of order statistics of non-i.i.d. $\mathrm{rv}$…

协方差矩阵未知时多元正态均值的广义贝叶斯极小极大估计AI 解读

Let $\mathbf{X}$ be a $p$-variate $(p \geqq 3)$ vector normally distributed with mean $\mathbf{\theta}$ and covariance matrix $\Sigma$, positive definite but unknown. Let $A$ be a $p \times p$ Wishart matrix with parameters $(n, \Sigma)$, independent of $\mathbf{X}$. To estimate $\mathbf{\theta}$ relative to quadratic loss function $(\hat{\mathbf{\theta}} - \mathbf{\theta})'\Sigma^{-1}(\hat{\mathbf{\theta}} - \mathbf{\theta})$, we obtain a family of minimax estimators $\mathbf{\delta}(\mathbf{X}, \mathbf{A})$…

具有线性结构的协方差矩阵的渐近有效估计AI 解读

One or more observations are made on a random vector, whose covariance matrix may be a linear combination of known symmetric matrices and whose mean vector may be a linear combination of known vectors; the coefficients of the linear combinations are unknown parameters to be estimated. Under the assumption of normality, equations are developed for the maximum likelihood estimates. These equations may be solved by an iterative method; in each step a set of linear equations is…

广义二次损失下正态随机向量均值的极小极大估计AI 解读

An admissible minimax estimate for the mean of a normal random vector with known covariance is derived for a generalized quadratic loss function. This loss function is quadratic in both the estimation error and the unknown mean. The estimate is derived using the method of least favorable prior distributions. The decision rule is linear, and the least favorable prior distribution for the unknown mean is normal with zero mean. The covariance of this least favorable normal…

利用回归性质刻画总体分布AI 解读

In the present work, we study characterizations of populations obtained by using regression properties of one statistic on another. We extend some of the results of Lukacs and Laha by considering the cubic regression (polynomial regression of order 3) of a cubic statistic $S$ on a linear one $L$. This assumption of cubic regression is used to derive a third order non-linear differential equation in the characteristic function $h(t)$ of a set of $n$ independently and…

多元线性回归的经验贝叶斯方法AI 解读

We consider estimation (subject to quadratic loss) of the vector of coefficients of a multiple linear regression model in which the error vector is assumed to have 0 mean and covariance matrix $\sigma^2 I$ but is not assumed to take on a specific parametric form, e.g., Normal. The vector of coefficients is taken to be randomly distributed according to some unknown prior. Restricted minimax solutions are exhibited relative to equivalence classes on the space of all prior…

协方差稳定化变换AI 解读

After reviewing the asymptotic variance stabilizing transformations in one dimension, a generalization of these to multivariate cases is discussed. Results are given for the uniqueness of solutions when they exist, but unlike the one-dimensional case, covariance stabilizing transformations need not exist. In the two-dimensional case, a necessary and sufficient condition is given for the existence of solutions. It takes the form of a second-order partial differential equation…

条件概率积分变换及其复合假设卡方拟合优度检验应用AI 解读

It is shown that certain conditional distributions, obtained by conditioning on a sufficient statistic, can be used to transform a set of random variables into a smaller set of random variables that are identically and independently distributed with uniform distributions on the interval from zero to one. This result is then used to construct distribution-free tests of fit for composite goodness-of-fit problems. In particular, distribution-free chi-square goodness-of-fit…

备择假设下Kolmogorov–Smirnov型统计量的极限分布AI 解读

Let $X_1, X_2, \cdots$ be a sequence of independent and identically distributed random variables with the common distribution being uniform on [0, 1]. Let $Y_1, Y_2, \cdots$ be a sequence of i.i.d. variables with continuous $\operatorname{cdf}F(t)$ and with [0, 1] support. Let $F_n(t, \omega)$ denote the empirical distribution function based on $Y_1(\omega), \cdots, Y_n(\omega)$ and let $G_m(t, \omega)$ the empirical $\operatorname{cdf}$ pertaining to $X_1(\omega), \cdots, X_m(\omega)$…

校准的统计理论AI 解读

The kind of calibration problem considered may be roughly described as follows: There are two related quantities $\mathscr{U}$ and $\mathscr{V}$ such that $\mathscr{U}$ is relatively easy to measure and $\mathscr{V}$ relatively difficult, requiring more effort or expense; furthermore the error in a measurement of $\mathscr{V}$ is negligible compared with that for $\mathscr{U}$. A distinguishing feature of the problem is, that from a single calibration experiment, where…

指数族中Wald大样本检验统计量的非中心收敛AI 解读

It is shown that under very mild assumptions Wald's large-sample test statistic (quadratic form based on unrestricted maximum likelihood estimators) converges to noncentral chi-square under a sequence of local alternatives of the order $n^{-\frac{1}{2}}$, when the family of distributions is assumed to be of exponential type. This eliminates, for these families, the necessity of invoking the strict regularity conditions of Wald for the purpose of justifying the asymptotic…

简单线性秩统计量的中心化AI 解读

Hajek (1968) proved that under weak conditions the distribution of a simple linear rank statistic $S$ is asymptotically normal, centered at the mean $ES$. He left open the question whether under the same conditions the centering constant $ES$ may be replaced by a simpler constant $\mu$, as was found to be true in the two-sample case and under different conditions by Chernoff and Savage (1958) and Govindarajulu, LeCam and Rhagavachari (1966). In this paper it is shown that…

维度约束下估计量的收敛AI 解读

Consider independent identically distributed observations whose distribution depends on a parameter $\theta$. Measure the distance between two parameter points $\theta_1, \theta_2$ by the Hellinger distance $h(\theta_1, \theta_2)$. Suppose that for $n$ observations there is a good but not perfect test of $\theta_0$ against $\theta_n$. Then $n^{\frac{1}{2}}h(\theta_0, \theta_n)$ stays away from zero and infinity. The usual parametric examples, regular or irregular, also have…