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多元正态均值的一族可容许极小极大估计量AI 解读

Let the $p$-component vector $X$ be normally distributed with mean $\xi$ and covariance $\sigma^2I$ where $I$ denotes the identity matrix and $\sigma$ is known. For estimating $\xi$ with quadratic loss, it is known that $X$ is minimax but inadmissible for $p \geqq 3$. We obtain a family of estimators which dominate $X$ and are admissible. These estimators are, therefore, both minimax and admissible.

曲线型多元回归中用于预测的变量选择AI 解读

A Bayesian formulation of the problem of analysing data from a curvilinear regression of $y$ on $x_1, x_2, \cdots, x_r$ in order to predict a future value of $y$ is considered. The problem is to obtain a criterion to decide which is the best subset of $x_1, x_2, \cdots, x_r$ to perform this prediction. Under very strict assumptions the criterion obtained is shown to use the same statistic as the orthodox (least squares) approach.

渐近有效的随机逼近：RM 情形AI 解读

Anbar (1971) and, independently, Abdelhamid (1971) have shown that if the density $g$ of the errors of estimates of function values is known, a transformation of observations leads to stochastic approximation methods which under mild conditions produce asymptotically efficient estimators (the first author considers the RM case, the second the RM and KW cases). This paper treats the RM case and shows that the same asymptotic result can be achieved without the knowledge of the…

期望一致性与广义贝叶斯推断AI 解读

In a previous paper, the authors introduced a new criterion of expectation consistency between probability distributions of discrete data given discrete parameter values and arbitrary posterior probability distributions for the parameter. It is here shown, under very weak assumptions, that expectation consistency implies that the posterior distributions are generalized Bayes. However when the posterior distributions are generalized Bayes, the implied prior distribution need…

截尾均值的渐近分布AI 解读

In this paper it is shown that in order for the trimmed mean to be asymptotically normal, it is necessary and sufficient that the sample be trimmed at sample percentiles such that the corresponding population percentiles are uniquely defined. (The sufficiency of this condition is well known.) In addition, the (non-normal) limiting distribution of the trimmed mean when this condition is not satisfied is derived, and it is shown that in some situations the use of the trimmed…

单调回归的一致性研究AI 解读

For each $t$ in some subset $T$ of $N$-dimensional Euclidean space let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of $t$. Let $t_1, t_2,\cdots$ be a sequence of points in $T$ and let $Y_1, Y_2,\cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. Estimators $\hat{m}_n(t; Y_1,\cdots, Y_n)$ of $m(t)$ which are monotone in each coordinate of $t$ and…

密度对数导数的估计AI 解读

Suppose $g$ is a probability density on $R = (-\infty, +\infty)$ with a continuous derivative and with $I(g) = \int (g'/g)^2g < + \infty$. Suppose $\{\varepsilon_n\}$ is a sequence of positive numbers converging to zero and $V_i, Z_i$ are random variables, $Z_i \rightarrow 0$ with probability one and $V_n$ is conditionally (given $V_1, \cdots, V_{n-1}, Z_1, \cdots, Z_n$) distributed according to $g$. Estimates $h_n$ of $g'/g$ are constructed, which are based on $V_1 + Z_1, \cdots, V_n + Z_n$…

纤维束强度的固定宽度置信带AI 解读

The present paper deals with the asymptotic theory of sequential confidence intervals of prescribed width $2d (d > 0)$ and prescribed coverage probability $1 - \alpha (0 < \alpha < 1)$ for the (unknown, per unit) strength of bundle of parallel filaments. In this context, certain useful convergence results on the empirical distribution and on the bundle strength of filaments are established and incorporated in the proofs of the main theorems. The results are the sequential…

独立但不同分布情形下似然函数的渐近分布AI 解读

Let $\Theta$ be an open subset of $R^k$ and for each $\theta \in \Theta$, let $X_1, \cdots, X_n$ be independent rv's defined on the probability space $(\mathscr{X}, \mathscr{A}, P_\theta)$, and let $p_{j, \theta}$ be the distribution of the rv $X_j$. Let $f_j(\bullet; \theta)$ be a specified version of the Radon-Nikodym derivative of $p_{j, \theta}$ with respect to a $\sigma$-finite measure $\mu$ and set $f_j(\theta) = f_j(X_j; \theta)$. Furthermore, for $\theta, \theta^\ast \in \Theta$…

多维保序中位数回归AI 解读

We consider the partial order on the unit square; $s_1 = (x_1, y_1) \ll s_2 = (x_2, y_2)$ if and only if $x_i \leqq y_i$ for $i = 1, 2$, and say that a real-valued function $f$ is isotone if $s_1 \ll s_2$ implies that $f(s_1) \leqq f(s_2)$. Suppose that for each point, $s$, in the unit square we have a distribution with median $m(s)$ and $m(s)$ is isotone. In this paper we propose an isotone estimator for $m$ which we denote by $\hat{m}$ and give an algorithm for computing $\hat{m}$…

摘要暂缺

截断参数未知的幂级数分布AI 解读

The minimum variance unbiased estimates are obtained when a sample is drawn from the power series distribution with unknown parameter $\lambda$ and unknown truncation parameter $\nu$. Unbiased tests of hypotheses are formulated.

次序统计量的矩AI 解读

Let $X_{(i\mid n)}$ be the $i$th smallest order statistic from a population with $\operatorname{pdf} f(x)$ and $\operatorname{cdf} F(x)$. When $\tilde{x}$ is the population median, $n$ is the sample size and $G(x) = F(x)(1 - F(x))$, the following are proved. \begin{equation*}\begin{split}E(X_{(n-i+1\mid n)}) + E(X_{(i\mid n)}) \\ = n(n - 2i + 1)\binom{n - 1}{i - 1} \sum^{\lbrack (n-2i+1)/2\rbrack}_{\nu = 0}\frac{(-1)^\nu}{n - 2i + 1 - \nu}\binom{n - 2i + 1 - \nu}{\nu} \\ \times \Psi_1(i - 1 + \nu),\end{split}\end{equation*}…

二次型的渐近分布及其拟合检验应用AI 解读

Let $Z_1, Z_2,\cdots$, be independent and identically distributed random variables and $\{c_{ijn}\}$ real numbers; put $T_n = \sum^n_{i,j = 1} c_{ijn}Z_iZ_j$. This paper gives conditions under which the distribution of $T_n - ET_n$ converges to the distribution of $\sum \Upsilon_m(Y_m^2 - 1)$ with $\{\Upsilon_m\}$ a real sequence and $Y_1, Y_2,\cdots$ independent $N(0, 1)$ random variables. The results are applied to the calculation of the asymptotic distributions of test…

具有依赖性的伯努利试验中的统计推断AI 解读

A model for Bernoulli trials with Markov dependence is developed which possesses the usual frequency parameter $p = P\lbrack X_i = 1\rbrack$ and an additional dependence parameter $\lambda = P\lbrack X_i = 1 \mid X_{i-1} = 1\rbrack$. Sufficient statistics for the model with $p$ and $\lambda$ unknown are found and an exact closed form expression for their small sample joint distribution is given. Large sample distribution theory is also given and small sample variances…

联合平稳高斯时间序列中循环二次型的近似AI 解读

Let $\{X(t), t = 0, \pm 1, \pm 2,\cdots\}$ be a $p$-dimensional, zero mean, stationary Gaussian time series with matrix-valued covariance function $R(u) = (r_{jk}(u)), u = 0, \pm 1, \pm 2, \cdots; j, k = 1, 2,\cdots, p$. Let $F(\omega)$ be the spectral density matrix of the time series (if $F(\omega)$ exists), and assume that $F(\omega)$ is positive definite for all $\omega\in(0, 2\pi\rbrack$. Let $\hat{F}_X(\omega_l)$ be an estimator of $F(\omega_l)$ formed by averaging $(2n + 1)$…

含三个及以上自变量的某些多元回归问题中最大似然估计的不可容许性AI 解读

Consider a multiple regression problem in which the dependent variable and (3 or more) independent variables have a joint normal distribution. This problem was investigated some time ago by Charles Stein, who proposed reasonable loss functions for various problems involving estimation of the regression coefficients and who obtained various minimax and admissibility results. In this paper we continue this investigation and establish the inadmissibility of the traditional…

某些样本函数极值的极限行为AI 解读

For a sequence of random variables forming an $m$-dependent stochastic process (not necessarily stationary), asymptotic distribution and other convergence properties of the extremum of certain functions of the empirical distribution are studied. In this context, it is shown that the asymptotic probability of the classical Kolmogorov-Smirnov statistic exceeding any positive real number provides an upper bound for the corresponding probability when the underlying random…

参数由样本估计时经验分布函数的弱收敛AI 解读

The weak convergence of the sample df is studied under a given sequence of alternative hypotheses when parameters are estimated from the data. For a general class of estimators it is shown that the sample df, when normalised, converges weakly to a specified normal process. The results are specialised to the case of efficient estimation.

容量下的极小极大检验与 Neyman–Pearson 引理AI 解读

Robust test problems between two approximately known simple hypotheses can be formalized as minimax test problems between two composite hypotheses. We show that if the composite hypotheses can be described in terms of alternating capacities of order 2 (in the sense of Choquet), then the minimax tests are ordinary Neyman-Pearson tests between a fixed representative pair of simple hypotheses; moreover, the condition is in a certain sense also necessary. All the neighborhoods…

频率学派推断理论中的若干困难AI 解读

A study of relationships between confidence regions being Bayesian, and the existence of some generalizations of Fisher's notion of relevant subsets. For a betting scheme introduced by Buehler, and for finite parameter space, it is shown that non-Bayesian procedures allow a winning strategy for a statistician's adversary. It is further shown, for finite parameter space, non-Bayesian procedures must admit conditional confidence levels bounded away from the unconditional…

指数族中参数非正与为正的贝叶斯序贯检验的渐近形状AI 解读

We show that, in a weak sense, as the cost of observation tends to zero, the shape of the continuation region of the Bayes solution for the exponential family problem given above is approximated by that of a corresponding problem for the Wiener process with drift. The approach is an extension of that used in Bickel and Yahav, Proc. Sixth Berkeley Symp. Math. Statist. Prob. (1971).

若干非参数问题的贝叶斯分析AI 解读

The Bayesian approach to statistical problems, though fruitful in many ways, has been rather unsuccessful in treating nonparametric problems. This is due primarily to the difficulty in finding workable prior distributions on the parameter space, which in nonparametric ploblems is taken to be a set of probability distributions on a given sample space. There are two desirable properties of a prior distribution for nonparametric problems. (I) The support of the prior…

非线性算子与充分性AI 解读

Let $(X, \mathscr{A})$ be a measurable space and $\mathscr{P} \mid \mathscr{A}$ a family of probability measures. Given a sufficient sub-$\sigma$-field, we define the sufficiency operator by assigning to each integrable function a conditional expectation which is independent of $P \in \mathscr{P}$. The operator thus defined is $\mathscr{P}$-a.e. linear, monotone, idempotent, and expectation invariant for every $P \in \mathscr{P}$. Not all of these properties are necessary to…