论总体与总体单位的联系

长期以来统计基本理论体系在论述总体与总体单位时,对总体的分类、总体与总体单位的联系等内容,有过不同的意见.对总体与总体单位的意义、总体的分类、总体在统计调查中的二元以及总体与总体单位的转换关系提出一些见解,它们与数学中表述的总体与总体单位定义没有相悖,对总体与总体单位的联系有明确的认识.     

论统计与领导决策

决策是人类基本活动之一,自古有之。古今中外,有领导者决策正确所带来的辉煌成就和因领导者决策失误所造成的灾难性后果的例子是枚不胜举的。随着当今人类社会的进步与发展,决策面临的因素和条件愈来愈复杂化。因此,作为一个领导者学习和掌握有关决策方面的知识是十分重要的。大量决策实践证明,统计与决策有着密不可分的关系。     

On longevity and life-line

Newrick et al. (1990) have claimed from 100 consecutive autopsies that there is a significant association between life-line and longevity. This study has received wide publicity in the popular press and in the T.V. media, and we re-examine their claim.     

论多主题分层抽样

一、分层方法在多主题分层抽样中,层的确定对多主题分层抽样效果的好坏起着决定作用。在多主题分层抽样中,抽样框有两种,即无序抽样框和有序抽样框。在这两种抽样框中,对多主题总体进行分层的方法是不同的。(一)无序抽样框的分层无序抽样框是指将多主题总体不按任何标志排队,而只是将多主题总体中的各个总体单位随机排列而形成的抽样框。我们采用聚类分析的方法来对多主题总体进行分层。     

On bagging and nonlinear estimation

We propose an elementary model for the way in which stochastic perturbations of a statistical objective function, such as a negative log-likelihood, produce excessive nonlinear variation of the resulting estimator. Theory for the model is transparently simple, and is used to provide new insight into the main factors that affect performance of bagging. In particular, it is shown that if the perturbations are sufficiently symmetric then bagging will not significantly increase bias; and if the perturbations also offer opportunities for cancellation then bagging will reduce variance. For the first property it is sufficient that the third derivative of a perturbation vanish locally, and for the second, that second and fourth derivatives have opposite signs. Functions that satisfy these conditions resemble sinusoids. Therefore, our results imply that bagging will reduce the nonlinear variation, as measured by either variance or mean-squared error, produced in an estimator by sinusoid-like, stochastic perturbations of the objective function. Analysis of our simple model also suggests relationships between the results obtained using different with-replacement and without-replacement bagging schemes. We simulate regression trees in settings that are far more complex than those explicitly addressed by the model, and find that these relationships are generally borne out.     

On Conditional and Partial Correlation

A graphical technique, similar in spirit to probability plotting, can be used to judge whether a Poisson model is appropriate for an observed frequency distribution. This “Poissonness plot” can equally be applied to truncated Poisson situations. It provides a type of robustness for detecting isolated discrepancies in otherwise well-behaved frequency distributions.     

On Sample Skewness and Kurtosis

It is well documented in the literature that the sample skewness and excess kurtosis can be severely biased in finite samples. In this paper, we derive analytical results for their finite-sample biases up to the second order. In general, the bias results depend on the cumulants (up to the sixth order) as well as the dependency structure of the data. Using an AR(1) process for illustration, we show that a feasible bias-correction procedure based on our analytical results works remarkably well for reducing the bias of the sample skewness. Bias-correction works reasonably well also for the sample kurtosis under some moderate degree of dependency. In terms of hypothesis testing, bias-correction offers power improvement when testing for normality, and bias-correction under the null provides also size improvement. However, for testing nonzero skewness and/or excess kurtosis, there exist nonnegligible size distortions in finite samples and bias-correction may not help.     

论统计分析研究

一、统计分析研究在整个统计工作中的地位和作用通常人们把统计工作分成三个阶段,即统计调查、统计整理和统计分析。统计分析研究,是全部统计工作不可缺少的重要组成部分,不做好统计分析工作就等于没有完成统计工作的任务。只有做好统计分析研究,统计工作才能起到较好的服务和监督作用。统计分析研究的重要性和作用,可以归纳为以下三点:     

On left-truncating and mixing Poisson distributions

ABSTRACT

The distributions obtained by left-truncating at k a mixed Poisson distribution, denoted kT-MP, and those obtained by mixing previously left-truncated Poisson distributions, denoted M-kTP, are characterized by means of their probability generating function. The main consequence is that every kT-MP distribution is a M-kTP distribution, but not the other way around.     

On Whiteman's and Storer's difference sets

In this paper, we prove that all Storer's difference sets are cyclic. We also prove that for p<1.8×10²⁵, Whiteman's difference sets exist if and only if (p,q)=(17,53) and (46817,140453).     

On principles and arguments to likelihood

Birnbaum (1962a) argued that the conditionality principle (C) and the sufficiency principle (S) implied the likelihood principle (L); he then argued (Birnbaum 1972) that C and a mathematical equivalence principle M implied L. Evans, Fraser, and Monette (1985a) gave reference details, and this paper gives proof that C alone implies L. The level of support by the profession for L is sharply less than that for S or even for C; thus the paradoxical nature of these results. In this regard, we elaborate on the Monette example (Fraser, Monette, and Ng 1984), which provides a strong case against L. We also examine closely the various proofs linking the principles and find that S and C can each be used operationally to suppress information otherwise deemed relevant. From another viewpoint this says that S and C can each be used in contexts that directly conflict with the original examples and motivations supporting them; the principles can thus be viewed as inappropriately used, or more strongly, as invalid. In either case, the result that C and S imply L or that C implies L can be regarded as noneffective in the context of discriminating applications. A resolution of the apparent anomalies can be obtained by allowing the statistical model to include ingredients additional to those usually present (particularly for subsequent use with conditionality), or alternatively by restricting the application of the principles to contexts where the conflicts would seem not to arise.     

On Maximum Depth and Related Classifiers

Abstract.  Over the last couple of decades, data depth has emerged as a powerful exploratory and inferential tool for multivariate data analysis with wide-spread applications. This paper investigates the possible use of different notions of data depth in non-parametric discriminant analysis. First, we consider the situation where the prior probabilities of the competing populations are all equal and investigate classifiers that assign an observation to the population with respect to which it has the maximum location depth. We propose a different depth-based classification technique for unequal prior problems, which is also useful for equal prior cases, especially when the populations have different scatters and shapes. We use some simulated data sets as well as some benchmark real examples to evaluate the performance of these depth-based classifiers. Large sample behaviour of the misclassification rates of these depth-based non-parametric classifiers have been derived under appropriate regularity conditions.     

On Parametric Bootstrapping and Bayesian Prediction

Abstract.  We investigate bootstrapping and Bayesian methods for prediction. The observations and the variable being predicted are distributed according to different distributions. Many important problems can be formulated in this setting. This type of prediction problem appears when we deal with a Poisson process. Regression problems can also be formulated in this setting. First, we show that bootstrap predictive distributions are equivalent to Bayesian predictive distributions in the second-order expansion when some conditions are satisfied. Next, the performance of predictive distributions is compared with that of a plug-in distribution with an estimator. The accuracy of prediction is evaluated by using the Kullback–Leibler divergence. Finally, we give some examples.     

On randomness, determinism and computability

This paper discusses the meaning and relationship of randomness and determinism. A fundamental development of chaotic dynamical systems is given with examples. Such systems are seen to exhibit randomness in the usual sense of unpredictability. The formal definition of randomness in terms of algorithmic incompressibility is also discussed. The role of recursion in computability and randomness is also discussed.     

基尼系数和帕累托曲线

一、基尼系数与帕累托曲线(一)基尼系数。这是本世纪初意大利经济学家,基尼提出的定量测度收入分配平均程度的方法。计算公式为:G=sum from i=1 to m(Pw_i)·Iw_i+2[Pw_1(1-S_1)+Pw_2(1-S_2)+……+Pw_(m-1)(1-S_(m-1))]-1式中:Pw_i为第i组的人数比重;Iw_i为第i组的收入比重;S_i为第i组的累计收入比重;i=1,2,……,m即整个资料分为m组。(二)帕累托曲线。意大利经济学家。帕累托通过大量统计资料的研究,发现收入分配曲线为双曲线,可用如下公式表示:x=A/(y-a)~b     

On predictive distributions and Bayesian networks

In this paper we are interested in discrete prediction problems for a decision-theoretic setting, where the task is to compute the predictive distribution for a finite set of possible alternatives. This question is first addressed in a general Bayesian framework, where we consider a set of probability distributions defined by some parametric model class. Given a prior distribution on the model parameters and a set of sample data, one possible approach for determining a predictive distribution is to fix the parameters to the instantiation with the maximum a posteriori probability. A more accurate predictive distribution can be obtained by computing the evidence (marginal likelihood), i.e., the integral over all the individual parameter instantiations. As an alternative to these two approaches, we demonstrate how to use Rissanen's new definition of stochastic complexity for determining predictive distributions, and show how the evidence predictive distribution with Jeffrey's prior approaches the new stochastic complexity predictive distribution in the limit with increasing amount of sample data. To compare the alternative approaches in practice, each of the predictive distributions discussed is instantiated in the Bayesian network model family case. In particular, to determine Jeffrey's prior for this model family, we show how to compute the (expected) Fisher information matrix for a fixed but arbitrary Bayesian network structure. In the empirical part of the paper the predictive distributions are compared by using the simple tree-structured Naive Bayes model, which is used in the experiments for computational reasons. The experimentation with several public domain classification datasets suggest that the evidence approach produces the most accurate predictions in the log-score sense. The evidence-based methods are also quite robust in the sense that they predict surprisingly well even when only a small fraction of the full training set is used.     

On three and four vicious walkers

We establish a reflection principle for three lattice walkers and use this principle to reduce the enumeration of configurations of three vicious walkers to the enumeration of configurations of two vicious walkers. More precisely, the reflection principle leads to a bijection between three walks (L₁, L₂, L₃) such that L₂ intersects both L₁ and L₃ and three walks (L₁, L₂, L₃) such that L₁ intersects L₃. Hence we find a combinatorial interpretation of the formula for the generating function for the number of configurations of three vicious walkers, originally derived by Bousquet-Mélou by using the kernel method, and independently by Gessel by using tableaux and symmetric functions. This answers a question posed by Gessel and Bousquet-Mélou. We also find a reflection principle for four vicious walks that leads to a combinatorial interpretation of a formula derived from Gessel's theorem.