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多元方差分析三种主要检验的单调性AI 解读

Summary This paper presents a unified and relatively simple proof of the monotonicity property of the three well-known tests for the multivariate analysis of variance (MANOVA), viz., (i) Hotelling-Lawley trace criterion, (ii) Wilks's likelihood-ratio criterion, and (iii) Roy's largest root criterion. The last one has already been considered by Roy and Mikhail (1961). Their proof, however, has a gap at one place, which has been bridged in this paper.

多元正态分类的后验优势比AI 解读

Summary A class of Bayesian methods is presented for finding the posterior odds or probability that a particular observation belongs to one of k multivariate normal populations. Solutions are given for various assumptions on the parameters of the k populations as well as for known and unknown prior probabilities. An application to genetics is included.

正态分布方差的贝叶斯估计AI 解读

Summary Some new general forms of estimators of the variance of a normal distribution are derived using Bayesian methods, and the conditions under which they lead to previously proposed estimators are discussed.

局部条件充分性AI 解读

Summary The concept of sufficiency can be examined locally for a neighbourhood of a parameter point θ0 and it leads to an estimate that has minimum variance among locally-unbiased estimates (Fraser, 1964). This local estimate generates the natural global estimate for the case of the simple exponential model, but it does not do so for the important case of the general translation model. The global estimate, the Pitman estimate, for this translation model is rooted in…

局部无偏估计AI 解读

Summary A concept of local unbiasedness is derived from an investigation by Fisher (1956, p. 145) of consistency for a neighbourhood of a parameter point θ0. Connections are made with Cramér-Rao type inequalities and conditions obtained under which the local estimates propagate to provide globally-unbiased estimates.

排队系统的渐近行为AI 解读

SUMMARY The homogeneous process N(t) on the lattice of integers in continuous time is discussed. When the increment probability generating function ϵ(u) is analytic at u = 1, the transition functions γn(t) = p{N(t) = n | N(0) = 0} are Bessel-like functions with simple asymptotic properties. The asymptotic behaviour of a variety of M/G/1, G/M/1 and related queueing systems may be deduced from these properties. An appendix is devoted to a review of saddle point methods and…

高斯移动平均过程残差方差异质性的检验AI 解读

SUMMARY A large sample test for variance heterogeneity in the residuals of a Gaussian moving average is presented, based upon periodogram-ratios of the data. For long series, it is shown that neither the null-hypothesis nor alternative-hypothesis behaviour of the test statistic is dependent upon the coefficients of the moving average (and these need not be estimated in advance). The alternative-hypothesis behaviour of the test statistic is discussed in terms of the…

平稳点过程的高阶性质AI 解读

SUMMARY This paper is concerned with joint distributions, joint moments, and joint renewal functions in a stationary point process. The random variables considered are (a) the numbers of events in adjacent intervals of given lengths, located either arbitrarily in time or by an arbitrary event, and (b) the lengths of intervals measured to the various events, beginning either at an arbitrary point or at an arbitrary event. The different bivariate probability functions are…

无约束相关随机游走AI 解读

SUMMARY A particle performs an unrestricted correlated random walk along a straight line with a velocity constant in magnitude but variable in direction. This direction remains constant for every time interval of unit length, and with probability p it is the same as in the previous interval and with probability 1 − p it is reversed. The object of this paper is to obtain probabilities for the first passage to position r, the jth. return to the origin and cognate results.

多元正态分布参数的后验分布AI 解读

SUMMARY A class of Bayes posterior distributions is obtained for the parameters of the multivariate normal distribution. These are compared with their fiducial and confidence counterparts.

多元方差分析中的 Hotelling 广义 T²AI 解读

SUMMARY Hotelling’s generalized T2 is the sum of the roots of the determinantal equation |Z - θY/n| = 0, where Y and Z have central and non-central p-dimensional Wishart distributions with d.f. Q and n. Y and Z are obtained as the sum of products matrices for p-variate normal distributions N(0, σ1) and N(b(q), σ1). The distribution of this statistic is shown to be a monotonic increasing function of each of the population roots. An unbiased estimate of a linear function of…

Jordan 矩阵求逆程序在多元回归与距离分析中的应用AI 解读

SUMMARY When applied to a matrix of mean squares and products, Jordan’s procedure is rich in intermediate quantities of potentially useful application in multiple regression. It also supplies a convenient method of partitioning the multivariate distance criterion D2 into single degrees of freedom.

带线性约束的多元线性假设检验AI 解读

SUMMARY In this paper we consider a normal multivariate linear regression model and give a criterion for the simultaneous testing of a set of linear estimable parametric functions. We also derive the distribution of the criterion and obtain some allied results.

非线性回归中的置信区间估计AI 解读

SUMMARY In a recent paper, E. J. Williams presents a procedure for obtaining what are described as fiducial intervals and regions for the non-linear parameters in non-linear regression. The linear parameters are not included in the regions which are developed and the class of non-linear regression functions for which his results hold is not clearly delineated. In this paper, it is shown how some results from standard regression theory can be used to discuss these points; the…

基于加权似然积分的置信点公式AI 解读

SUMMARY Lindley’s problem, relating to the conditions under which there is formal mathematical equivalence between Bayesian solutions and confidence theory solutions, is discussed. A result concerning location parameters is extended. The corresponding asymptotic theory is developed and a modification of the problem is also discussed.

最大似然估计量的高阶矩AI 解读

SUMMARY For a distribution depending on a single parameter the first four sampling moments of the maximum-likelihood estimate to orders N−2, N−3, N−3 and N−4 respectively are given. Expressions for the measures of skewness γ1 and γ2 are also given. Several illustrative examples are included as a check on the heavy algebra. The paper extends earlier work by Haldane and Smith.

点过程的谱分析AI 解读

SUMMARY The spectral analysis of stationary point processes in one dimension is developed in some detail as a statistical method of analysis. The asymptotic sampling theory previously established by the author for a class of doubly stochastic Poisson processes is shown to apply also for a class of clustering processes, the spectra of which are contrasted with those of renewal processes. The analysis is given for two illustrative examples, one an artificial Poisson process,…

马尔可夫链状态频数的二次式与二阶二元链AI 解读

SUMMARY Let nr be the frequency of the state r in a segment of a regular ergodic Markov chain. It is shown how to obtain the asymptotic distribution of any quadratic expression in the nr’s. The method is especially convenient where the Markov chain is produced by a random motion on a circle. An explicit formula is given for the asymptotic variance of the number of 0’s in a binary Markov chain of order 2.

Tukey 非加性检验的功效AI 解读

SUMMARY The problem of testing for non-additivity in a two-way classification has been considered in this paper where the mean μii can be expressed as μ+αi+βi where αi and βi are the effects of the two ways of classification (i = 1, …,p;j = 1, …,q). Tukey (1949) has suggested a test for non-additivity. The power function of this test has been found in this paper for alternatives of the form μii = μ+αi+βi+cαi βi and numerical calculations made for the case of p = q = 6. For…

关于 Richards 所提最大似然估计方法的若干说明AI 解读

SUMMARY Richards (1961) suggested a method of maximum-likelihood estimation for the case of several parameters. Some gaps in the general theory given by him in support of the method are pointed out. Though the method is justifiable, the justification is shown to depend on the correct use of the implicit function theorem and is not connected with the joint asymptotic normality and the asymptotic efficiency of the maximum-likelihood estimates.

短马尔可夫序列的参数估计AI 解读

SUMMARY A parametric representation of a two-state Markov process is given which enables the maximum-likelihood estimates of the two parameters to be found without iteration.

关于渐近正态估计量序列渐近效率的一则说明AI 解读

SUMMARY This paper gives a rigorous yet quite elementary proof that under fairly general conditions, the asymptotic efficiency of an asymptotically normal estimator sequence does not exceed unity. Both estimation of a single parameter and simultaneous estimation of several parameters are considered.

关于列联表交互作用的 Plackett 检验AI 解读

SUMMARY Based on a transformation matrix the rows of which are orthogonal to each other, a matrix that comes naturally to analysis of variance users, Plackett (1962) has proposed a test for zero second-order interaction in an r × s × t contingency table which requires that the user invert t +1 matrices, each having (r − 1) (s − 1) rows and (r −1) (s −1) columns. Based on a transformation matrix the rows of which are not orthogonal to each other, we present herein a…

最小二乘法的逻辑基础AI 解读

SUMMARY Two alternative modes of developing the logic of the least squares theorem are presented. The first is based on the mean square error, the second on a notion of restricted sufficiency.