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由 n 维立方体上随机变量确定的分布AI 解读

The distribution function of a random variable of the form $\sum^n_{i = 1} a_i Y_1 Y_2 \cdots Y_i$ where $a_i > 0$ and $0 \leq Y_i \leq 1$ is considered. A geometric argument is used to obtain the distribution function as a repeated integral. The result is used first to obtain the distribution function of a linear combination of variables defined over the simplex $X_i \geq 0, \sum^n_{i = 1} X_i \leq 1$. As a second application the distribution of certain quadratic forms over…

二元剂量响应数据下狄利克雷过程的后验分布AI 解读

The posterior distribution of a Dirichlet process from $N$ quantal responses at $r$ dosage levels has been recognized by Antoniak as a mixture of Dirichlet processes. The purpose of this paper is to develop a systematic procedure for computing finite-dimensional distributions of such mixtures which can be equivalently expressed as multivariate beta distributions with random parameter vectors. It is shown that the sequence of random parameter vectors of the updated beta…

子集选择的一类 Schur 程序与极小极大理论AI 解读

The problem of selecting a random subset of good populations out of $k$ populations is considered. The populations $\Pi_1, \cdots, \Pi_k$ are characterized by the location parameters $\theta_1, \cdots, \theta_k$ and $\Pi_i$ is said to be a good population if $\theta_i > \max_{1 \leq j\leq k}\theta_j - \Delta$, and a bad population if $\theta_i \leq \max_{1 \leq j \leq k} \theta_j - \Delta$, where $\Delta$ is a specified positive constant. A theory for a special class of…

内在统计推断（二）AI 解读

According to an invariance principle, for some models having a certain group structure, there is a uniquely defined prior representing ignorance, which is called the inner prior. It is shown that the corresponding posterior probability of a likelihood region has a simple frequency interpretation as a mean conditional confidence level. The central multivariate normal model is considered as an example.

固定样本量下向量参数极大似然估计的存在性与唯一性AI 解读

The maximum likelihood estimate is shown to exist and to be unique if a twice continuously differentiable likelihood function is constant on the boundary of the parameter space and if the Hessian matrix is negative definite whenever the gradient vector vanishes. The condition of constancy on the boundary cannot be completely removed, cf. Tarone and Gruenhage (1975). The theory is illustrated with several examples.

特征向量的渐近推断AI 解读

Asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors. These asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest. Applications of the general results to principal components analysis and canonical variate analysis are given.

乘法泊松模型的拟合优度检验AI 解读

A model for random $n \times k$ matrices $X$ is considered. The elements $X_{ij}$ are assumed to be independent and Poisson-distributed random variables with means $\alpha_i\exp(\gamma t'_{ij})$, where $t_{ij}$ are known $m$-dimensional vectors and $(\alpha, \gamma)$ is an unknown parameter. A goodness-of-fit test is proposed and an approximation is derived when the number $n$ of rows is large.

随机变量依赖关系的非参数度量AI 解读

In 1959 A. Renyi proposed a set of axioms for a measure of dependence for pairs of random variables. In the same year A. Sklar introduced the general notion of a copula. This is a function which links an $n$-dimensional distribution function to its one-dimensional margins and is itself a continuous distribution function on the unit $n$-cube, with uniform margins. We show that the copula of a pair of random variables $X, Y$ is invariant under a.s. strictly increasing…

Logistic 回归诊断AI 解读

A maximum likelihood fit of a logistic regression model (and other similar models) is extremely sensitive to outlying responses and extreme points in the design space. We develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit. The elements of the fitting process which constitute the usual output (parameter estimates, standard errors, residuals, etc.) will be used for this…

Jackknife 估计量的 Berry–Esseen 收敛速率AI 解读

Consider an ordinary estimation problem for an unknown parameter $\theta$. Let the estimator $\theta^\ast_n$ be the jackknife of a function of a $U$-statistic. Under mild assumptions, we demonstrate that $\sup_t |P\lbrack n^{1/2}(\theta^\ast_n - \theta)/S^\ast_n \leq t \rbrack - \Phi (t)| = O(n^{-p/2(p+1)})$, where $S^{\ast 2}_n$ is a jackknife estimator of the asymptotic variance of $n^{1/2}\theta^\ast_n, \Phi (t)$ is the standard normal distribution and $p$ is some…

正态性下保序回归的平方损失AI 解读

The maximum likelihood estimator $\hat{\mu}$ of a nondecreasing regression function has been studied in detail in the literature. However, little is known about its quadratic loss pointwise. This paper shows that the mean square error of $\hat{\mu}_i$ is less than that of the usual estimator $\bar{X}_i$ for each $i$ when $\bar{X}_1,\cdots, \bar{X}_k$ are independent normal variates.

渐近相对 Pitman 效率的若干性质AI 解读

A general approach to Pitman efficiency as a limit of the ratio of sample sizes is presented. The results can be used especially to derive the Pitman efficiency of tests based on asymptotically $\chi^2$-distributed statistics with different degrees of freedom.

相对于完全信息与完全无信息的统计信息度量AI 解读

Two measures of content of information in statistical experiments are considered. They are both based on Le Cam's notion of deficiency of one experiment with respect to another. The measures are: (i) The deficiency of the given experiment with respect to a totally informative experiment. (ii) The deficiency of a totally uninformative experiment with respect to the given experiment. We shall here discuss the interpretations of such measures, establish inequalities for them…

任意协方差结构下固定线性模型可估函数的无偏与最小方差无偏估计AI 解读

Consider a general linear model for a column vector $y$ of data having $E(y) = X \alpha$ and $\operatorname{Var}(y) = \sigma^2H$, where $\alpha$ is a vector of unknown parameters and $X$ and $H$ are given matrices that are possibly deficient in rank. Let $b = Ty$, where $T$ is any matrix of maximum rank such that $TH = \phi$. The estimation of a linear function of $\alpha$ by functions of the form $c + a'y$, where $c$ and $a$ are permitted to depend on $b$, is investigated.…

基于朴素估计量的失效率函数大样本估计与一致置信带AI 解读

In this paper we consider a naive estimator of the failure rate function and smooth it using any band-limited window. We show that this smoothed estimate is equivalent to estimates obtainable from the sample hazard function, as in Rice and Rosenblatt (1976). We obtain the asymptotic distribution of the global deviation of the smoothed estimate from the failure rate function, which can then be used to construct uniform confidence bands.

平稳点过程谱参数的估计AI 解读

This paper considers the two approaches for estimating the parameters specifying the spectral density of the counting process of a stationary point process, namely the frequency domain and the time domain approaches. The relation between the two is clarified; consistency and asymptotic normality of the estimates are established. Finally the special case of a rational spectral density is considered in some detail.

不连续型 Lévy 过程的渐近推断AI 解读

We establish contiguity of certain families of probability measures indexed by $T$, as $T \rightarrow \infty$, for classes of stochastic processes with stationary, independent increments whose sample paths are discontinuous. Many important consequences pertaining to properties of tests and estimates then apply. A new expression for the Radon-Nikodym derivative of these processes is obtained.

条件指数族与渐近推断的表示定理AI 解读

Conditional exponential families of Markov processes are defined and a representation of the score function martingale is established for the important conditionally additive case. This result unifies those obtained separately for different examples and provides the key to asymptotic normality results for the maximum likelihood estimate.

位置估计量的尾部行为AI 解读

Let $X_1, \cdots, X_n$ be a sample from a population with density $f(x - \theta)$ such that $f$ is symmetric and positive. It is proved that the tails of the distribution of a translation-invariant estimator of $\theta$ tend to 0 at most $n$ times faster than the tails of the basic distribution. The sample mean is shown to be good in this sense for exponentially-tailed distributions while it becomes poor if there is contamination by a heavy-tailed distribution. The rates of…

利用广义矩空间方法最大化M估计量的方差AI 解读

The problem considered is that of optimizing a function of a finite number of linear functionals over an infinite dimensional convex set $S$. It is shown that under some reasonably general conditions the method of moment spaces can be used to reduce the problem to one of optimizing over a simple finite dimensional set (generally a set of convex combinations of extreme points of $S$). The results are applied to finding the maximum asymptotic variance of M-estimators over…

隐式定义函数的随机逼近AI 解读

Let $S$ be a set, $R$ the real line, and $M$ a real function on $R \times S$. Assume there exists a real function, $f$, on $S$ such that $(x - f(s))M(x, s) \geq 0$ for all $x$ and $s$. Initially neither $M$ nor $f$ are known. The goal is to estimate $f$. At time $n, s_n$ (a value in $S$) is observed, $x_n$ (a real number) is chosen, and an unbiased estimator of $M(x_n, s_n)$ is observed. This problem has applications, for example, to process control. In a previous paper the…

检测小幅变化的非参数控制图AI 解读

We consider sequential observation of independent random variables $X_1,\cdots, X_N$ whose distribution changes from $F$ to $G$ after the first $\lbrack N\theta \rbrack$ variables. The object is to detect the unknown change-point quickly without too many false alarms. A nonparametric control chart based on partial weighted sums of sequential ranks is proposed. It is shown that if the change from $F$ to $G$ is small, then as $N \rightarrow \infty$, the appropriately scaled…

随机差分方程的参数估计AI 解读

Let $Y_t$ satisfy the stochastic difference equation $Y_t = \sum^q_{i=1} \psi_{ti} \alpha_i + \sum^p_{j=1} \gamma_j Y_{t-j} + e_t,$ where the $\{\psi_{ti}\}$ are fixed sequences and (or) weakly stationary time series and the $e_t$ are independent random variables, each with mean zero and variance $\sigma^2$. The form of the limiting distributions of the least squares estimators of $\alpha_i$ and $\gamma_j$ depend upon the absolute value of the largest root of the…

非线性最小二乘估计的渐近理论AI 解读

For a linear regression model, the necessary and sufficient condition for the asymptotic consistency of the least squares estimator is known. An analogous condition for the nonlinear model is considered in this paper. The condition is proved to be necessary for the existence of any weakly consistent estimator, including the least squares estimator. It is also sufficient for the strong consistency of the nonlinear least squares estimator if the parameter space is finite. For…