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利用次序统计量矩集合刻画分布AI 解读

It is known that the set $M = \{\mu_{r,n} : r = 1, 2, \cdots, n; n = 1, 2, 3, \cdots\}$ of expectations of order statistics of samples from a distribution $F$ completely determines $F$ (see, for example, Hoeffding (1953)). In addition Chan (1967), Konheim (1971) and Pollak (1973) have shown that certain subsets of $M$ determine $F$. In this note conditions are derived which are sufficient for a proper subset of $M$ to determine $F$.

用矩母函数刻画指数族分布AI 解读

It is shown that if $\mathbf{T}$ has an unknown exponential family distribution with natural parameter $\mathbf{\theta}$, then $\mathbf{G(\theta)} = \mathbf{ET}$ uniquely specifies the moment generating function. The converse is proved, namely, if $\{\mathbf{T_\theta}\}$ is a family of random variables with moment generating functions of a certain form, then it must be an exponential family. Moreover, several necessary and sufficient conditions are given so that a function…

未知累积量下的 Edgeworth 展开AI 解读

In this paper a new method of approximating one distribution by another is introduced. The method is essentially a modification of the Edgeworth technique which eliminates the necessity of knowing the cumulants of the distributions involved.

关于若干贝叶斯非参数估计的一则说明AI 解读

With respect to a general quadratic loss function, the Bayes rule for the mean of a probability distribution of unknown form is obtained, in the class of linear functions of the sample. The associated Bayes risk is also obtained. A number of recent results in the literature are shown to be direct corollaries of this result, and applications are given for the empirical distribution function of the sample.

众数非参数估计的一致性AI 解读

Let $X$ be an absolutely continuous real-valued random variable with additional restrictions to be imposed later. Venter (1967) ("On estimation of the mode," Ann. Math. Statist. 37 1446-1455) estimated the mode of $X$ by a point from the shortest interval containing a specified number $r = r(n)$ of observations. Venter demonstrated that such an estimator is strongly consistent under appropriate conditions on the distribution of $X$ and on $r(n)$. It is the purpose of this…

使用离散数据的连续时间回归AI 解读

A general continuous time distributed lag model is considered. The problem is that of estimating the parameters of the kernel when, as is often the case, the available data consist not of a continuous record but of discrete observations recorded at regular intervals of time. Fourier transformation of the model and insertion of the computable, discrete Fourier transforms of the variables produce an approximate model which is of non-linear regression type and is relatively…

已知前 N 阶矩时随机概率的极小极大估计AI 解读

$X$ and $\Theta$ are random variables; for given $\Theta = \theta$, the conditional distribution of $X$ is binomial with parameters $N$ and $\theta$; the first $N$ moments of $\Theta$ are known. An estimate of $\Theta$ is made based on the observed value of $X$, the risk being defined in terms of squared error loss. It is shown that as conjectured by H. Robbins, the ratio of the Bayes risk to the minimax risk for all possible distributions of $\Theta$ uniformly tends to…

以最小误分类概率为目标的线性特征选择AI 解读

We describe an approach to linear feature selection for $n$-dimensional normally distributed observation vectors which belong to one of $m$ populations. More specifically, we consider the problem of finding a rank $k k \times n$ matrix $B$ which minimizes the probability of misclassification with respect to the $k$-dimensional transformed density functions when a Bayes optimal (maximum likelihood) classification scheme is used. Theoretical results are presented which, for…

位置参数族中的一致最小方差估计AI 解读

Let $x_1, \cdots, x_n$ be a sample of size $n$ of an $\operatorname{rv}$ with $\operatorname{df} F(x - \theta)$, where $F$ is known but $\theta$ unknown. In this paper we make a Fourier approach to the problem of existence of a statistic $g(x_1, \cdots, x_n)$ which is a uniformly minimum variance (UMV) estimator of its own mean value. We mention only some of the results. If $n = 1$ we find an NASC for an estimator $g(x_1)$ to be, in a restricted sense, UMV. This condition is…

一般卡方拟合检验的统一大样本理论AI 解读

We present a unified large-sample theory of general chi-squared tests of fit under composite null hypotheses and Pitman alternatives. The statistics are quadratic forms in the standardized cell frequencies, and we allow random cells, $k$-variate observations from not necessarily continuous distributions, and quite general estimates of unknown parameters. Generalizations of published results on a number of specific chi-squared tests follow.

若干不变序贯概率比检验停止规则的终止性、矩与指数有界性AI 解读

It is well known that Wald's SPRT terminates with probability one and in fact the stopping time is exponentially bounded for every distribution $P$ except in the trivial case where the $\log$ likelihood ratio vanishes with probability one. The results in the literature for invariant SPRT's, however, have been considerably less complete. In all the parametric problems studied, moment conditions of certain random variables have been assumed to prove termination, and the…

摘要暂缺

参数分解与基于显著性、似然及客观后验的推断AI 解读

The concepts of significance, likelihood, and objective posterior have wide ranges of application in statistics. For certain very simple applications--single location variable--there has been recognition that the three concepts produce equivalent numerical results, specifically the equality of observed level of significance, integrated likelihood extreme values, and integrated objective posterior extreme values. The most general model permitting the use of the three concepts…

有限参数随机过程族中最大似然估计一致性的一个注记AI 解读

We consider families of stochastic processes indexed by a finite number of alternative parameter values. For general classes of stochastic processes it is shown that maximum likelihood estimates convergence almost surely to the correct parameter value. This established by use of a submartingale property of the sequence of maximized likelihood ratios together with a technique first employed by Wald [24] in the case of independently identically distributed random variables.

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…

非独立同分布随机向量的约化经验过程收敛AI 解读

Any triangular array of row independent random vectors with continuous df's has a standard reduction to random vectors with values in the unit cube. The reduced empirical process belonging to the transformed random vectors is always relatively compact. Weak convergence to a (necessarily Gaussian) process holds iff the corresponding covariance kernel converges pointwise.

用秩检验多条回归直线是否重合AI 解读

For several linear regression lines $Y_{ij} = \alpha_i + \beta_i(x_{ij} - x{_i.}) + Z_{ij}, i = 1,\cdots, k; j = 1, \cdot, n_i$, a statistic for testing $\alpha_i = \alpha, \beta_i = \beta$ is constructed based on the simultaneous ranking of all the observations. The asymptotic properties of the criterion are also studied. The results are, however, not directly applicable to the general design model $Y_{ij} = \alpha_i + \beta_i x_{ij} + Z_{ij}$, unless it is assumed that the…

支持 Fisher 置换检验的论证AI 解读

The problem of statistical comparison of two distributions, continuous as well as discrete, is considered. Very slight and reasonable modifications of traditional parameteric models, e.g. `normal distributions with equal variances', are shown to result in permutation tests, only. Fisher's permutations test is shown to have optimum properties which mean a good merit for its practical use. Further, an accurate method of determining the $p$-value of Fisher's test is proposed.

若干非参数问题的近似贝叶斯解AI 解读

The problem of making inferences about real functions of a probability distribution of unknown form is examined in a Bayesian nonparameteric framework. With respect to a general quadratic loss function, Bayes estimates within the class of linear combinations of a given set of functions on the sample space are obtained for general functions on the distribution space. The result is then used to derive Bayes polynomial estimates of the moments of the distribution.

离散分布的后验概率上界与下界AI 解读

Dempster (1966) defines two sampling models which he labels structures of the first and second types. In this paper we consider the application of the type one model to the problem of finding upper and lower posterior probabilities when sampling from discrete univariate distributions whose suppower consists of the set of noinnegative integers. Specific results are obtained for the Poisson and geometric distributions.

有限总体中一类一致可容许估计量AI 解读

Let $C'$ be a class of sampling designs of fixed expected sample size $n$ and fixed inclusion probabilities $\pi_i$ and $C$ be the subclass of $C'$ consisting of designs of fixed size $n$ and inclusion probabilities $\pi_i$. Then it is established that the pair $(e^\ast, p^\ast)$ where $p^\ast \in C$ and $e^\ast(x, \mathbf{x}) = \sigma_{i \in s} b_i x_i, b_1 > 1$, and $\sigma^N_1 (b_i)^{-1} = E(n(s)) = n$, is strictly uniformly admissible among pairs $(e_1, p_1)$ where $p_1 \in C'$…

有限总体均值置信区间集的强可容许性AI 解读

In a previous paper the admissibility was proved of a generalized version of the confidence sets, commonly used in practice, which are based on the sample mean and the sample standard deviation. A stronger definition of admissibility is obtained, if instead of the length of the interval for each individual observable sample $s$, only the expected length for all samples together for each $x \in R_N$, is taken into consideration for defining the permissible alternatives to the…

多设计多元线性模型下一般线性假设的检验AI 解读

The generalization of the standard model for MANOVA which allows for the possibility of different "design" matrices for the variates is known as the multiple design multi variate linear model. For example, in multivariate regression analysis we might have polynomial models of different degree in the "independent" variates. In this paper, new tests are given for the general linear hypothesis under the multiple design multivariate model and in one case the corresponding…

正态分布变换的均值与中位数之 UMVU 估计量的关系AI 解读

Let $\eta = f(\mu)$ and $\theta = Ef(Y)$, where $f$ is a monotone function and $Y \sim N(\mu, \sigma^2)$. In this note we use the method of convolution transforms to show that the UMVU estimators of $\eta$ and $\theta$ based on a pair of independent sufficient statistics $T \sim N(\mu, \alpha \sigma^2)$ and $S^2 \sim \sigma^2 \chi^2_{(\nu)}$ are related to each other in a simple manner: the replacement $\alpha$ by $\alpha - 1$ in the expression of the UMVU estimator of $\eta$…