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二元分布的单调依赖函数AI 解读

A new characterization of monotonic dependence is given here proceeding in a natural way from the consideration of a type of dependence weaker than quadrant dependence. More precisely, each bivariate distribution of $(X, Y)$ is transformed onto a pair of functions $^\mu{X, Y}$ and $^\mu{Y, X}$ defined on the interval $0 < p < 1$ and taking values from [-1, 1], with $^\mu{X, Y}(p)$ being a suitably normalized expected value of $X$ under the condition that $Y$ exceeds its $p$th…

序列相关检验的一项稳健性性质AI 解读

This paper shows that the UMP or UMPU tests for serial correlation, derived under the assumption of a normal distribution, are quite robust against departure from normality. In fact, the tests are still UMP or UMPU in much broader classes of distributions and the null distributions remain unchanged under these classes. The results will be applied to a linear model.

模型错误时似然比检验的表现AI 解读

Let the random variable $X$ have a distribution depending on a parameter $\theta \in \Theta$. Consider the problem of testing the hypothesis $H: \Theta_0 \subseteqq \Theta$ based on a sequence of observations on $X$. The likelihood ratio test for $H$ is constructed by selecting a model for the unknown distribution of $X$. In this paper the asymptotic performance of the likelihood ratio test is studied when the model is incorrect, that is, when the probability distribution of…

广义组检验程序AI 解读

A person wishes to determine which, if any, of $n = \Pi^k_{j=1} a_j$ i.i.d. random variables, $X(i_1,\cdots, i_k), i_j = 1,\cdots, a_j$, lie in some specified set $A$. Such observations will be called unsafe. It is assumed that the density of the $X$'s is known and that $Y_j(i_1,\cdots, i_j)$, the sum of all the $X$'s whose first $j$ indices are $i_1,\cdots, i_j$, can be measured as easily as the individual $X$'s. In this paper, search procedures of the following form are…

对数线性模型与期望格频数较小的频数表AI 解读

In the case of frequency data, traditional discussions such as Rao (1973, pages 355-363, 391-412) consider asymptotic properties of maximum likelihood estimates and chi-square statistics under the assumption that all expected cell frequencies become large. If log-linear models are applied, these asymptotic properties may remain applicable if the sample size is large and the number of cells in the table is large, even if individual expected cell frequencies are small.…

对齐秩次过程的两端固定维纳过程近似及应用AI 解读

For independent random variables distributed symmetrically around an unknown location parameter, aligned rank order statistics are constructed by using an estimator of the location parameter based on suitable rank statistics. The sequence of these aligned rank order statistics is then incorporated in the construction of suitable stochastic processes which converge weakly to some Gaussian functions, and, in particular, to tied-down Wiener processes in the most typical cases.…

K 样本线性秩统计量两种排序方法的渐近等价性AI 解读

Two methods of ranking $K$ samples for rank tests comparing $K$ populations are considered. The first method ranks the $K$ samples jointly; the second ranks the $K$ samples pairwise. These procedures were first suggested by Dunn (1964), and Steel (1960), respectively. It is shown that both ranking procedures are asymptotically equivalent for rank-sum tests satisfying certain nonrestrictive conditions. The problem is formulated in terms of multiple comparisons, but is…

概率密度的 Hermite 级数估计性质AI 解读

An unknown density function $f(x)$, its derivatives, and its characteristic function are estimated by means of Hermite functions $\{h_j\}$. The estimates use the partial sums of series of Hermite functions with coefficients $\hat{a}_{jn} = (1/n) \sum^n_{i=1} h_j(X_i)$ where $X_1\cdots X_n$ represent a sequence of i.i.d. random variables with the unknown density function $f$. The integrated mean square rate of convergence of the $p$th derivative of the estimate is $O(n^{(p/r) + (5/6r)-1})$…

稳健估计量在真实数据中有效吗？AI 解读

Most studies of robust estimators of location parameters have relied upon mathematical theory, computer simulated data, or a combination of these. This paper presents a comparison of the performances of eleven estimators using real data sets. Twenty sets of data from 1761 determinations of the parallax of the sun, from 1798 measurements of the mean density of the earth, and from circa 1880 measurements of the speed of light, are employed in the study, with the current values…

有限总体估计量的一则札记AI 解读

Two previous results of the author, regarding the admissibility of the sample mean as estimator of the population mean in the class of all estimators for arbitrary sampling designs with the squared error as loss function, and for a wide class of loss functions, are shown to remain valid in the more general case when the population means of many variates are simultaneously under estimation.

序贯程序的一致性保证：分布类上的均匀性AI 解读

An extension of a central limit theorem for a mean of a random number of observations is given. A natural application occurs in the area of fixed-width confidence intervals. We provide an example which shows that the standard procedure does not preserve the intended coverage probability uniformly over nontrivial sets of distribution functions. The major weak convergence result is used to provide conditions for and a simple proof of such uniformity. The results are also shown…

向量值二元失效率的非参数估计AI 解读

Let $\mathbf{X} = (X_1, X_2)'$ be a bivariate random vector distributed according to an absolutely continuous distribution function $F(\mathbf{x})$ which has first partial derivatives. Let $\bar{F}(\mathbf{x}) = P(X_1 > x_1, X_2 > x_2).$ The vector-valued bivariate failure rate is defined as $\mathbf{r}(\mathbf{x}) = (r_1(\mathbf{x}), r_2(\mathbf{x}))',$ where $r_i(\mathbf{x}) = -\partial \ln \bar{F}(\mathbf{x})/\partial x_i (i = 1, 2)$. In this paper, we propose a smooth…

基于 Cramér–von Mises 统计量正交分量线性组合的检验渐近性质AI 解读

Let $X_1, X_2, \cdots, X_n$ be independent identically distributed random variables defined on the unit interval. The generalized $j$th orthogonal component is defined as $V_{nj} = n^{-\frac{1}{2}} \sum^n_{i = 1} d_j(X_i),$ where $\{1, d_1, d_2, \cdots\}$ is an orthonormal basis for $\mathscr{L}_2(\lbrack 0, 1\rbrack).$ These statistics are a generalization of the orthogonal components of the Cramer-von Mises statistic [2]. Linear combinations of the $V_{nj}$ are applied to…

与经验分布函数线性界有关的分布AI 解读

$X_1, \cdots, X_n$ are i.i.d. Uniform (0, 1) rv's with empirical df $\Gamma_n$ and order statistics $0 < U_1 < \cdots < U_n < 1.$ Define random variables $U_\ast, i_\ast$ (for $n \geqq 2$) by $\max_{1\leqq i \leqq n - 1} \frac{U_{i + 1}}{i} = \frac{U_{i_\ast} + 1}{i_\ast}, U_\ast = U_{i_\ast + 1};$ $i_\ast + 1$ is the (random) index of the order statistic at which the maximum is achieved and $U_\ast$ is the value of that order statistic. The distributions of $(U_\ast, i_\ast)$…

次序统计量伴随变量的一般分布理论AI 解读

Let $(X_i, Y_i) (i = 1, 2, \cdots, n)$ be $n$ independent $\mathrm{rv}$'s from some bivariate distribution. If $X_{r:n}$ denotes the $r$th ordered $X$-variate, then the $Y$-variate $Y_{\lbrack r:n\rbrack}$ paired with $X_{r:n}$ is termed the concomitant of the $r$th order statistics. The exact and asymptotic distribution theory of $Y_{\lbrack r:n\rbrack}$ and of its rank are studied. The results obtained are applied to a prediction problem in a Round Robin tournament.

Several stopping times which arise from problems of sequential estimation may be written in the form $t_c = \inf\{n \geqq m: S_n < cn^\alpha L(n)\}$ where $S_n, n \geqq 1,$ are the partial sums of i.i.d. positive random variables, $\alpha > 1, L(n)$ is a convergent sequence, and $c$ is a positive parameter which is often allowed to approach zero. In this paper we find the asymptotic distribution of the excess $R_c = ct_c^\alpha - S_{t_c}$ as $c \rightarrow 0$ and use it to…

选择包含最优总体的子集：贝叶斯方法AI 解读

The problem of selecting a subset of $k$ populations $\pi_1, \cdots, \pi_k,$ which contains the "best" population, is considered. The unknown values $\theta_1, \cdots, \theta_k$ are the characteristics associated with $\pi_1, \cdots, \pi_k$ and the unknown population associated with $\theta_{\lbrack k\rbrack} = \max_i\theta_i$ is called the "best." It is assumed that, given $\mathbf{\theta} = (\theta_1, \cdots, \theta_k)$ the $\operatorname{pdf}$ of the independent random…

贝叶斯序贯估计AI 解读

For fixed $\theta$, let $X_1, X_2, \cdots$ be a sequence of independent identically distributed random variables having density $f_\theta(x)$. Using a sequential Bayes decision theoretic approach we consider the problem of estimating any strictly monotone function $g(\theta)$ when the error incurred by a wrong estimate is measured by squared error loss and the sampling cost is $c$ units per observation. A heuristic stopping rule is suggested. It is shown that the excess risk…

若干变分结果及其在多重推断中的应用AI 解读

Let $(\mathbf{M}, \mathbf{T}, \mathbf{S})$ be random matrices such that $\mathbf{M}$ and $\mathbf{S}$ are Hermitian positive definite almost everywhere. Let $\mathbf{M}_{(t)} = \lbrack m_{ij}; 1 \leqq i, j \leqq t\rbrack, \mathbf{S}_{(t)} = \lbrack s_{ij}; 1 \leqq i, j \leqq t\rbrack$ and $\mathbf{T}_{(r,s)} = \lbrack t_{ij}; 1 \leqq i \leqq r, 1 \leqq j \leqq s\rbrack$, and define $Q(r, s) = P\lbrack G((\mathbf{M}_{(r)})^{-\frac{1}{2}}\mathbf{T}_{(r,s)}(\mathbf{S}_{(s)}) ^{-\frac{1}{2}}) \leqq c\rbrack$…

特殊容量下的最不利分布对AI 解读

The least favorable pair (LFP) that Huber (1965), (1968) wrote down when he considered minimax test problems between neighborhoods of single probability measures $P_0, P_1$ defined in terms of $\varepsilon$-contamination and total variation is a canonical but only one possible choice of an LFP. We treat these neighborhoods by means of special capacities. The minimax test statistic is obtained by explicitly solving a minimization program, all LFP's are characterized by their $(P_0 + P_1)$…

大观测值下后验分布的近似行为AI 解读

Let $X$ be a real valued random variable with a family of possible distributions belonging to a one parameter exponential family with the natural parameter $\theta \in(\underline{\theta}, + \infty)$. Let $g$ be a prior probability density for $\theta$ with unbounded support. Under some additional assumptions it is shown that for large values of $x$ the posterior distribution of $\theta$ given $X = x$ is approximately normally distributed about its mode. If $\delta_g$ denotes…

置信水平、后验概率与Buehler例子AI 解读

A confidence level is sometimes treated as a probability and accordingly given substantial "confidence." And with certain statistical models a confidence level can also be an objective posterior probability (Fraser and MacKay, 1975). An instance involving such a probability with a binary error was discussed at the Symposium on Foundations of Statistical Inference, University of Waterloo, 1970 (Godambe and Sprott, 1971, page 49). An example by Buehler (ibid., page 337) of a…

基于样本和的功效为一的序贯检验AI 解读

This paper studies the properties of open-ended power-one tests of $H_0: \theta \leqq \theta_0$ versus $H_1: \theta > \theta_0$ or of $H: \theta = \theta_0$ versus $K: \theta \neq \theta_0$ based on sample sums stopped at moving boundaries. The behavior of the expected sample size is analyzed and certain asymptotic results as $\theta \rightarrow \theta_0$ are obtained in the case of a location parameter and also in the case of an exponential family of distributions.

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…