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关于分布强收敛的一则札记AI 解读

If a sequence of distributions converges weakly to a distribution and if the associated densities are unimodal then the sequence of density functions converges to the density of the limiting distribution. This stronger form of convergence is useful in the investigation of limiting joint distributions.

子抽样中的完全分离组AI 解读

We study several combinatorial problems related to the choice of groups of subsets to be used in the subsampling procedure of Hartigan (J. Amer. Statist. Assoc. 1969). The complete separation property is shown to guarantee asymptotic efficiency relative to $t$. Problems associated with the relative variance criterion are discussed.

存在干扰参数时的估计方程AI 解读

Estimating equations for a real parameter $\theta$ which indexes a family of densities $p(x, \theta)$ were considered in the note by Godambe (Ann. Math. Statist. 31 (1960) 1208-1211). An optimality property of the equation $\partial \log p/\partial \theta = 0$ among unbiased estimating equations was established. In this paper an analogous result is proved for estimation of a real parameter $\theta_1$ in the presence of a nuisance parameter $\theta_2$.

线性模型中的稳健估计AI 解读

The purpose of this paper is to present robust estimates for the regression parameters in the general linear model. We start with a family of $M$-estimators, and using the observations, we estimate the asymptotic efficiency of each member in the family. Then we choose the estimate in the family with greatest estimated asymptotical efficiency. We prove that this procedure has the same asymptotical efficiency as the member of the family with the greatest asymptotical…

多个总体回归的分类与估计AI 解读

Two problems, classifying an individual into one of several populations and estimating the regression in that population, are simultaneously treated as one problem. This can be viewed as a problem of a missing observation on a categorical variable. When all variables are jointly distributed multivariate normal, the maximum likelihood solution is the intuitively appealing one: classify the individual using the usual likelihood ratio procedure, then estimate the regression…

位置未知的指数分布尺度参数的替代估计量AI 解读

Let $X_1, X_2,\cdots, X_n$ be independent observations from an exponential distribution with an unknown location-scale parameter $(\mu, \sigma)$. Let $\bar{X} = n^{-1} \sum X_i$ and $M = \min X_i$. Under squared error loss the best location-scale equivariant estimator of $\sigma$ is $\bar{X} - M$, which agrees with the maximum likelihood estimator. Arnold (J. Amer. Statist. Assoc. 65 (1970) 1260-1264) and Zidek (Ann. Statist. 1 (1973) 264-278) have shown that $\bar{X} - M$ is…

位置参数的序贯秩检验AI 解读

In this paper, for the one and two sample location problems, a class of sequential tests based on robust rank order statistics is developed. The proposed tests terminate with probability one for square integrable score functions. Under more stringent regularity conditions and for local alternatives, the OC and ASN of the proposed tests are obtained, and the allied asymptotic relative efficiency results with respect to the sequential probability ratio and likelihood ratio…

逆线性回归问题的序贯解法AI 解读

In this note we apply the sequential theory developed by Chow and Robbins [1] and Gleser [3], [4] to the inverse linear regression problem. A two-stage sequential procedure has been proposed for the construction of a fixed-width confidence interval for $x$ (an unknown parameter). It is shown that the limiting probabilities of "correct decision" are equal to $P^\ast$ (pre-assigned).

随机序约束下随机变量的最大似然估计AI 解读

Brunk, Franck, Hanson and Hogg (1966) ("Maximum likelihood estimation of the distributions of two stochastically ordered random variables," J. Amer. Statist. Assoc. 61 1067-1080) found and studied maximum likelihood estimates of a pair of stochastically ordered distribution functions. In this paper we discuss a generalization of this problem in which we do not require the domain of these "distribution functions" to be the real line. We think of the order restriction we…

自回归随机过程中的回归实验设计AI 解读

This paper extends some recent results on asymptotically optimal sequences of experimental designs for regression problems in stochastic processes. In the regression model $Y(t) = \beta f(t) + X(t), 0 \leqq t \leqq 1$, the constant $\beta$ is to be estimated based on observations of $Y(t)$ and its first $m - 1$ derivatives at each of a set $T_n$ of $n$ distinct points. The function $f$ is assumed known as is the covariance kernel of $X(t)$, a zero-mean $m$th order…

一致的自回归谱估计AI 解读

We consider an autoregressive linear process $\{x_t\}$, a one-sided moving average, with summable coefficients, of independent identically distributed variables $\{e_t\}$ with zero mean and fourth moment, such that $\{e_t\}$ is expressible in terms of past values of $\{x_t\}$. The spectral density of $\{x_t\}$ is assumed bounded and bounded away from zero. Using data $x_1,\cdots, x_n$ from the process, we fit an autoregression of order $k$, where $k^3/n \rightarrow 0$ as $n \rightarrow \infty$…

截断分布平移参数的最大似然估计（二）AI 解读

Let $f$ be a density which vanishes for negative values of its argument and varies regularly with exponent $\alpha - 1$ at zero, where $1 < \alpha < 2$. Further, let $f_\theta$ denote $f$ translated by $\theta$. We find and study the asymptotic distribution of the MLE $\hat{\theta}_n$ based on a sample size $n$ as $n \rightarrow \infty$.

克拉美–拉奥型不等式的充要条件AI 解读

For a random variable $X$ with possible distributions indexed by a parameter $\theta$, and for real-valued $T = T(X)$ and $V = V(X, \theta)$ with $\operatorname{Var} T < \infty$ and $0 < \operatorname{Var} V < \infty$, Schwarz's inequality gives $\operatorname{Var} T \geqq \{\operatorname{Cov} (T, V)\}^2/\operatorname{Var} V$. Necessary and sufficient conditions are given for this inequality to be of Cramer-Rao type: $\operatorname{Var} T \geqq \{a_m(\theta)\}^2/\operatorname{Var} V$…

利用正交展开估计分布AI 解读

Let $f(x)$ be a continuous, strictly positive probability density function over an interval $\lbrack a, b\rbrack$ and $F(x)$ its associated $\operatorname{cdf}$. Suppose $\{\phi_i(x)\}^\infty_{i=0}$ is a complete orthonormal basis for $L_2\lbrack a, b\rbrack$ and that $f(x)$ and $\log f(x)$ have orthogonal series expansions, in the $\phi_i$'s, over $\lbrack a, b\rbrack$. Estimators for $f(x)$ and $F(x)$ are chosen from the canonical exponential family of distributions…

若干功效为一的检验的期望样本量AI 解读

Asymptotic approximations to the expected sample size are given for a class of tests of power one introduced in [10]. Comparisons are made with the method of mixtures of likelihood ratios, and an application is given to Breiman's gambling theory for favorable games.

φ混合序列的双样本经验过程弱收敛与Chernoff–Savage定理AI 解读

Using Pyke-Shorack (Ann. Math. Statist. (1968) 755-771) approach, based on weak-convergence properties of empirical processes, a Chernoff-Savage theorem concerning the asymptotic normality of two-sample linear rank statistics is proved for stationary $\phi$-mixing sequences $\{X_m\}$ and $\{Y_n\}$ of rv's. This main result (Theorem 4.1) is almost as strong as proved by Pyke and Shorack for sequences of independent rv's. The basic tool employed is the following new result…

随机删失下生命表与乘积极限估计的大样本研究AI 解读

Using the model of random censorship, a necessary and sufficient condition for the consistency of the standard (actuarial) life table estimate of a survival distribution is derived. We establish the asymptotic normality of this estimate, showing that Greenwood's variance formula is nearly correct. In the case of a continuous survival distribution we establish limiting normality for the product limit estimate and for the closely related cumulative hazard process. Some…

多个回归参数的秩得分比较AI 解读

For testing $\beta_i = \beta, i = 1,\cdots, k$, in the model $Y_{ij} = \alpha + \beta_iX_{ij} + Z_{ij} j = 1,\cdots, n_i$ a class of rank score tests is presented. The test statistic is based on the simultaneous ranking of all the observations in the different $k$ samples. Its asymptotic distribution is proved to be chi square under the hypothesis and noncentral chi square under an appropriate sequence of alternatives. The asymptotic efficiency of the given procedure…

Villegas 确定先验分布方法中的不一致性AI 解读

A recent paper by Villegas (1969) uses a fiducial type argument to lend support to the use of Jeffrey's invariant prior for the variance-covariance matrix in the non-informative situation. Sampling is assumed to be on a multi-normal random variable. Villegas proceeds by selecting a pivotal quantity which has a fixed distribution. Since there is nothing unique about this pivotal quantity, we note in this paper, that the Villegas fiducial approach could lead to other priors,…

随机区组设计中有序备择下的秩检验比较AI 解读

Two classes of rank tests are considered for ordered alternatives in a randomized block design with $k$ treatments and $n$ blocks: tests based on among-blocks rankings ($A$-tests) and tests based on within-blocks rankings ($W$-tests). Previous efficiency comparisons for fixed $k, n \rightarrow \infty$ under the normal distribution have indicated that $A$-tests are more sensitive. In the present paper it is shown that this behavior is not typical under other distributions.…

经验概率测度的大偏差AI 解读

Sanov's statement of first-order asymptotic behaviour of probabilities of large deviations of an empirical distribution function is here established for empirical probability measures, with attendant simplification of conditions. For the case of distribution functions, our theorem is strictly more general than a specialisation of results of Hoadley.

非线性正交多项式泛函的核估计AI 解读

Let $(X(t), Y(t))$ be a complex vector process stationary of order $k$ for any $k, k = 1,2,\cdots$, such that $Y(t)$ is expressed as a polynomial functional of degree 2 operating on $X(t)$. Then $Y(t)$ can be rewritten as a sum of orthogonal projections $G_j(K_j, Y(t)), j = 0, 1, 2$. It is shown that there is a set of functionals which approximate in mean square the projection $G_2(K_2, Y(t))$. Moreover, it is possible to determine the kernels associated with these…

不同谱的两个高斯过程之间的大样本判别AI 解读

We study the probability of error asymptotically for testing one Gaussian stochastic process against another when the mean vectors are zero and we have the choice between two given covariance matrices. It is shown that under certain conditions the probabilities of error form asymptotically a geometric progression with a ratio that is derived. The approach employs Laplace's method of approximating integrals and does not appeal to Fourier analysis; in this sense it can be said…

矩阵求导及其在自适应线性决策问题中的应用AI 解读

A theory of matrix differentiation is presented which uses the concept of a matrix of derivative operators. This theory allows matrix techniques to be used in both the derivation and the description of results. Several new operations and identities are presented which facilitate the process of matrix differentiation. The derivative theorems and new operations are then applied to the problem of determining optimal policies in a linear decision model with unknown coefficients,…