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动态模型中最小二乘估计的强一致性AI 解读

The least squares estimate of the parameter matrix $\mathbf{B}$ in the model $\mathbf{y}_t = \mathbf{B'x}_t + \mathbf{u}_t$, where $\mathbf{u}_t$ is an $m$-component vector of unobservable disturbances and $x_t$ is a $p$-component vector, converges to $\mathbf{B}$ with probability one under certain conditions on the behavior of $x_t$ and $\mathbf{u}_t$. When $\mathbf{x}_t$ is stochastic and the conditional expectation of $\mathbf{u}_t$ given $\mathbf{x}_s$ for $s \leqslant t$…

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多元方差分析中相互依赖的显著性检验AI 解读

It is shown that in the MANOVA situation, certain dependent tests of hypothesis may be combined to obtain an overall conservative significance level. The result holds whether one uses the Wilks, Roy, Lawley-Hotelling or Pillai test criterion. This generalizes a result due to Kimball in univariate ANOVA.

有序备择非参数检验统计量精确分布的一则说明AI 解读

The exact distribution of a nonparametric test statistic for ordered alternatives, the so-called $\\bar{\\chi}^2_{\\operatorname{rank}}$ statistic, is discussed. Two tables are given.

圆周数据 Mann–Whitney 型统计量的小样本分布AI 解读

The union-intersection method of test construction can be used to derive from the Mann-Whitney rank test a test statistic for testing whether two samples of observations from circular distributions come from the same population. The null distribution of this test statistic will be investigated here. The approach followed is to use the principle of inclusion-exclusion to obtain an expression for the number of partitions of a positive integer which satisfy certain conditions.…

渐进删失检验中若干分位数过程的弱收敛AI 解读

For progressive censoring schemes pertaining to a general class of (parametric as well as nonparametric) testing situations, one encounters a (partial) sequence of linear combinations of functions of order statistics where the coefficients are themselves stochastic variables. Weak convergence of such a quantile process to an appropriate Gaussian function is studied here, and the same is incorporated in the formulation of suitable (time-) sequential tests based on these…

Logistic函数的上下概率推断AI 解读

A general system of inference which leads to upper and lower posterior distributions based on sample data has been proposed by Dempster (1967). This general theory of inference is applied to the two-parameter logistic function, given the data from independent binomial populations. Inferences are developed for fixed regions about the two parameters and about interesting combinations of these parameters. The resulting upper and lower probabilities are generated by a random…

调查抽样中随机化与估计的极小极大方法AI 解读

We consider a finite set of units, a population. With each unit is associated a real value (unknown to us) and a label (identifying the unit). Based on the labels we may select a sample, i.e., a subset of the population, to estimate the mean of the real values. In simple random sampling (not necessarily of fixed size) the selection probabilities of all samples are not affected by a permutation of the labels. It is assumed that we have to choose both a sampling design and a…

优惠券收集问题的不变性原理：鞅方法AI 解读

For the coupon collector's problem, invariance principles for the partial sequence of bonus sums after $n$ coupons as well as the waiting times to obtain the bonus sum $t(\geqslant 0)$ are studied through a construction of a triangular array of martingales related to these sequences and verifying the invariance principles for these martingales.

利用δ序列进行概率密度估计AI 解读

Let $X_1, X_2, \cdots, X_n$ be i.i.d. random variables with common density function $f$. A method of density estimation based on "delta sequences" is studied and mean square rates established. This method generalizes certain others including kernel estimators, orthogonal series estimators, Fourier transform estimators, and the histogram. Rates are obtained for densities in Sobolev spaces and for densities satisfying Lipschitz conditions. The former generalizes some results…

关于若干递归概率密度估计量的讨论AI 解读

The density estimator, $f^\ast_n(x) = n^{-1}\sum^n_{j = 1}h^{-1}_jK((x - X_j)/h_j)$, as well as the closely related one $f^\dagger_n(x) = n^{-1}h_n^{-\frac{1}{2}}\sum^n_{j = 1}h_j^{-\frac{1}{2}}K((x - X_j)/h_j)$ are considered. Expressions for asymptotic bias and variance are developed. Using the almost sure invariance principle, laws of the iterated logarithm are developed. Finally, illustration of these results with sequential estimation procedures are made.

ARMA 过程最大似然估计量的强一致性AI 解读

The strong consistency of the maximum likelihood parameter estimation method is established for multivariate Gaussian stochastic processes possessing autoregressive moving average (ARMA) representations. The demonstration in this paper exploits the ergodic theorem together with results from linear prediction theory.

多项分布与多个独立二项分布的可容许及极小极大估计AI 解读

Admissible and minimax estimation is discussed for estimating the parameters in the (a) multinomial distribution and in (b) $k$ independent binomial distributions. In (a) the loss function is $\sum^n_0\lbrack\delta_i(x) - \theta_i\rbrack^2/\theta_i$, where $\theta_0, \cdots, \theta_k(\sum\theta_i = 1)$ are the parameters in the multinomial distribution, and the estimators are restricted to $\sum^k_0\delta_i(x) = 1$. In (b) the loss functions considered are the weighted sum…

多项分布概率的估计AI 解读

This paper deals with the estimation of the parameters (cell probabilities) of a multinomial distribution. For the binomial distribution it is known that the maximum likelihood estimator (MLE) is admissible with respect to the squared error loss. It is shown that the MLE is admissible also in the case of the multinomial distribution.

线性模型中 M 估计量的渐近行为AI 解读

This paper deals with $M$-estimators for the linear model $y_i = \mathbf{x}'_i\mathbf{\theta} + u_i1 \leqslant i \leqslant n$, where the $\mathbf{x}_i$ are fixed $p$-dimensional vectors, and the $u_i$ are i.i.d. random variables with distribution $F$. The estimators considered are solutions $\hat\mathbf{\theta}$ of the equation $\sum^n_{j = 1}\psi(y_j - \mathbf{x}'_i\hat{\mathbf{\theta}})\mathbf{x_j = 0}$ for some function $\psi$. Let $\mathbf{X}$ be the matrix whose $i$th…

动态规划与统计决策理论的联系AI 解读

The main aim of the present work is to establish connections between the theory of dynamic programming and the statistical decision theory. The paper deals with a nonMarkovian dynamic programming decision model that includes Markovian decision models and Markov renewal decision models as special cases. The analysis is based on the total cost criterion where the convergence condition on the expected total cost is such that the discounted and the negative (unbounded) case are…

交换矩阵：若干性质与应用AI 解读

The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$…

连续时间平稳马尔可夫过程的密度估计AI 解读

This paper deals with a general class of recursive estimates of the density function in a continuous-time stationary Markov process. Under the condition $G_2$ of Rosenblatt sufficient conditions for almost sure convergence of such estimates are given.

指数族的共轭先验AI 解读

Let $X$ be a random vector distributed according to an exponential family with natural parameter $\theta \in \Theta$. We characterize conjugate prior measures on $\Theta$ through the property of linear posterior expectation of the mean parameter of $X : E\{E(X|\theta)|X = x\} = ax + b$. We also delineate which hyperparameters permit such conjugate priors to be proper.

关于相依情形下稳健估计的进一步讨论AI 解读

Results of an earlier paper giving first order optimal robust $M$-estimators of a location parameter in certain dependent situations are extended to the case where the dependency can be modeled by a symmetric form of a $(2k + 1)$st order moving average scheme. It is also shown that the results can not be extended to finding second order optimal $M$-estimators. That is, no $M$-estimators can be optimal to the second order unless they explicitly adapt themselves to the assumed…

基于检验的识别与选择程序AI 解读

Let $X_1, \cdots, X_k$ be independent random variables with distributions $Q_1, \cdots, Q_k$ defined on a common range space $(\mathscr{X}, \mathscr{G})$. At the beginning it is assumed that the $Q_i$ are known but not the pairing with the $X_i$, and the goal is to identify the $X_i$ which comes from $Q_1$. First it is shown that every procedure based on a total ordering in $\mathscr{X}$ can be viewed as being based on (the $p$-value of) a test for deciding between $H_0: \{Q_1\}$…

两个变量均含测量误差时的直线斜率估计AI 解读

Let $X_i$ and $Y_i$ be random variables related to other random variables $U_i, V_i$, and $W_i$ as follows: $X_i = U_i + W_i, Y_i = \alpha + \beta U_i + V_i, i = 1, \cdots, n$, where $\alpha$ and $\beta$ are finite constants. Here $X_i$ and $Y_i$ are observable while $U_i, V_i$ and $W_i$ are not. This model is customarily referred to as the regression problem with errors in both variables and the central question is the estimation of $\beta$. We give a class of estimates for…