Moment Information for Probability Distributions,Without Solving the Moment Problem. I: Where is the Mode?

How much information does a small number of moments carry about the unknown distribution function? Is it possible to explicitly obtain from these moments some useful information, e.g., about the support, the modality, the general shape, or the tails of a distribution, without going into a detailed numerical solution of the moment problem? In this paper a theoretical result of Johnson and Rogers is generalized to be valid for all moment problems and is exploited to demonstrate that a few moments are able to provide us with valuable information about the position of the mode of an unknown (unimodal) distribution.     

Nonparametric maximum likelihood estimators for ar and ma time series

The problem of estimating the parameters of moving average or autoregressive time series is studied when the error distribution is completely unknown. Four nonparametric maximum likelihood estimators (NPMLE) are presented for this purpose. These estimators are compared with the classical moment and least squares estimators in a simulation study. The behavior of these NPMLEs is much better than the classical ones, suggesting that they should be used extensively when no parametric information is known in advance about the error distribution. An application of these estimators to coal mining accidents data is also included.     

Inference about parameters in Binomial-Poisson distribution with additional incomplete data

ABSTRACT

This article considers the distribution of Binomial-Poisson random vector which has two components and includes two parameters: one is the rate of a Poisson distribution, the other is the proportion in a Binomial distribution. The inference about the two parameters is usually made based on only paired observations. However, the number of paired observations is, in general, not large enough because of either technical difficulty or budget limitation, and so one can not make efficient inferences with only paired data. Instead, it is often much easier and not too costly to have incomplete observation on only one component independently. In this article we will combine both the paired complete data and unpaired incomplete data for estimating the two parameters. The performances of various estimators are compared both analytically and numerically. It is observed that fully using the unpaired incomplete data can always improve the inference, and the improvement is very significant in the case when there are only a few paired complete observations.     

A sequential discrimination procedure for two almost identically shaped wrapped distributions

The way of investigating a distribution knowing its interesting properties might be often inadequate when the shapes of two distributions are almost similar. In each of these circumstances, the accurate decision about the genesis of a random sample from any of the two parent distributions will be very much ambiguous even with the availability of the existing testing procedure of the circular data. A sequential discrimination procedure has been suggested which is also invariant to the sample size. The performance of the proposed discrimination procedure has been evaluated by checking its capability of detecting the genesis of the known samples from the two identically shaped wrapped distributions.     

Multinomial Logistic模型在区域高等教育资源结构优化中的应用

在区域确定的前提下,将各高等教育资源数据进行整理,建立Multinomial Logistic模型,分析各结构的相对发生比率以及各结构的最佳分布。分析该回归分析的多维发生比率,由此确定各种离散等级状态之间的调整方向及调整程度。对黑龙江省的各高等教育区域进行实证分析,结果表明,黑龙江省高等教育资源结构以教学型、教学研究型、研究教学型、研究型的比例来判断,高等教育效用有待于进一步挖掘。多维发生比率以及自变量对多维发生比率的变化影响这两个参数对区域高等教育资源结构优化起着关键作用。     

Hybrid regions for post-change mean in a sequence of normal variables

The hybrid bootstrap uses resampling ideas to extend the duality approach to the interval estimation for a parameter of interest when there are nuisance parameters. The confidence region constructed by the hybrid bootstrap may perform much better than the ordinary bootstrap region in a situation where the data provide substantial information about the nuisance parameter, but limited information about the parameter of interest. We apply this method to estimate the post-change mean after a change is detected by a stopping procedure in a sequence of independent normal variables. Since distribution theory in change point problems is generally a challenge, we use bootstrap simulation to find empirical distributions of test statistics and calculate critical thresholds. Both likelihood ratio and Bayesian test statistics are considered to set confidence regions for post-change means in the normal model. In the simulation studies, the performance of hybrid regions are compared with that of ordinary bootstrap regions in terms of the widths and coverage probabilities of confidence intervals.     

分位数回归技术综述

普通最小二乘回归建立了在自变量X=x下因变量Y的条件均值与X的关系的线性模型。而分位数回归（Quantile Regression）则利用自变量X和因变量y的条件分位数进行建模。与普通的均值回归相比,它能充分反映自变量X对于因变量y的分布的位置、刻度和形状的影响,有着十分广泛的应用,尤其是对于一些非常关注尾部特征的情况。文章介绍了分位数回归的概念以及分位数回归的估计、检验和拟合优度,回顾了分位数回归的发展过程以及其在一些经济研究领域中的应用,最后做了总结。     

The probability of degenerate maximum-likelihood estimates with randomly truncated survival data

We study the nonparametric maximum-likelihood estimator of the marginal distribution function in bivariate data with truncated sum (Woodroofe 1985), and calculate the probability that it will assign zero probability to a range of observed values. We show that in a simple artificial model with uniform intensity in both variables the probability may be expressed using recursive formulae, and that large-sample approximations valid under weak regularity conditions show good agreement with as few as 20 observations. In the uniform-intensity model, the probability of degenerate estimates is rather small, being just over 5% with a dataset of 20 observations and falling below 1% with 125 observations. However, in a model of transfusion-associated AIDS in 125 children and adults (Kalbfleisch and Lawless 1989), the probability of degenerate estimates is much larger—about 30% with 20 observations and 3% with the actual 125 observations. This is largely because both the infection and incubation density are increasing over time. Such a pattern makes the condition for degeneracy, i.e., relatively high observed intensity at extreme values of each variable, much more likely.     

金融系统复杂性问题研究述评

复杂性科学研究在国内外方兴未艾,从复杂性角度去审视,金融系统具有非线性、非周期性、动力系统性、初值敏感性和自组织等一系列复杂性表征,是一个开放的、动态的复杂巨系统。20世纪80年代中期以来,研究金融系统复杂性应用比较广泛的理论主要有混沌理论、分形理论、标度理论和突变理论等源自自组织非线性动力学、热力学领域的相关理论;在金融系统复杂性建模方法上,以主体建模（ABM）思想为指导,进一步提出了复杂适应系统（CAS）理论,基于“主体”设定一些“规则”运用计算机建模,对主体之间的复杂交互作用进行着重考察,克服了传统建模方法的诸多缺陷;在实证研究方面,主要分析了金融市场的时序分布规律和市场异象特征。但是,现阶段金融系统复杂性的研究仍有着明显的局限性,存在较大的进一步研究的空间。     

Prior distribution assessment for a multivariate normal distribution: an experimental study

A variety of methods of eliciting a prior distribution for a multivariate normal (MVN) distribution have recently been proposed. This paper reports an experiment in which 16 meteorologists used the methods to quantify their opinions about climatology variables. Our results compare prior models and show, in particular, that it can be better to assume the mean and variance of an MVN distribution are independent a priori, rather than to model opinion by the conjugate prior distribution. Using a proper scoring rule, different forms of assessment task are examined and alternative ways of estimating parameters are compared. To quantify opinion about means, it proved preferable to ask directly about the means rather than individual observations while, to quantify opinion about the variance matrix, it was best to ask about deviations from the mean. Further results include recommendations for the way parameters of the prior distribution are estimated.     

Using jackknife to correct bias of MLE for the truncated Pareto distribution

The Pareto distribution is a well-known probability distribution in statistics, which has been widely used in many fields, such as finance, physics, hydrology, geology and astronomy. However, the parameter estimation for the truncated Pareto distribution is much more complicated than that for the Pareto distribution. In this paper, we demonstrate that the bias of the maximum likelihood estimation for the truncated Pareto distribution can be significantly reduced by its jackknife estimation, which has a very simple form.     

A note on the Zhang omnibus test for normality based on the Q statistic

A discussion about the estimators proposed by Zhang (1999) for the true standard deviation σof a normal distribution is presented. Those estimators, called by Zhang q 1 and q 2 , are functions of the expected values of the order statistics from a standard normal distribution and they were the basis of the Q statistic used in the derivation of a new test for normality proposed by Zhang. Although the type I error and the power of the test was discussed by Zhang, no study was performed to test the reliability of q 1 and q 2 as estimators of σ. In this paper, it is shown that q 1 is a very poor estimator for σespecially when σis large. On the other hand, the estimator q 2 has a performance very similar to the well-known sample standard deviation S. When some correlation is introduced among the sample units it can be seen that the estimator q 1 is much more affected than the estimators q 2 and S.     

An approximated distribution of the gini's rank associatio coefficient

An approximate distribution is proposed for the Gini's rank association coefficient g which is, like Kendall's and Spearman's rank correlation coefficient, a statistic to test independence between two random variables. The purposed distribution can be simply transformed into a Student's T distribution; so, hypothesis testing is made much easier.     

Design-based confidence limits for a distribution function

A method for constructing confidence limits for a distribution function is proposed. This method is a simple modification of the common method based on a normal approximation to the distribution of the estimated distribution function. The methods differ in how the estimated standard errors are used. The coverage properties of the two methods are compared in a simulation study. Coverage probabilities for the proposed method are found to be much closer to the nominal levels, particularly in the tails of the population distribution.     

On nonparametric Bayesian inference for the distribution of a random sample

The nonparametric Bayesian approach for inference regarding the unknown distribution of a random sample customarily assumes that this distribution is random and arises through Dirichlet-process mixing. Previous work within this setting has focused on the mean of the posterior distribution of this random distribution, which is the predictive distribution of a future observation given the sample. Our interest here is in learning about other features of this posterior distribution as well as about posteriors associated with functionals of the distribution of the data. We indicate how to do this in the case of linear functionals. An illustration, with a sample from a Gamma distribution, utilizes Dirichlet-process mixtures of normals to recover this distribution and its features.     

Nonlinear quasi-bayesian theory and inverse linear regression

We generalize Wedderburn's (1974) notion of quasi-likelihood to define a quasi-Bayesian approach for nonlinear estimation problems by allowing the full distributional assumptions about the random component in the classical Bayesian approach to be replaced by much weaker assumptions in which only the first and second moments of the prior distribution are specified. The formulas given are based on the Gauss-Newton estimating procedure and require only the first and second moments of the distributions involved. The use of GLIM package to solve for the estimation problems considered is discussed. Applications are made to estimation problems in inverse linear regression, regression models with both variables subject to error and also to the estimation of the size of animal populations. Some numerical illustrations are reported. For the inverse linear regression problem, comparisons with ordinary Bayesianand other techniques are considered.     

A new generalized normal distribution: Properties and applications

Abstract

The most commonly studied generalized normal distribution is the well-known skew-normal by Azzalini. In this paper, a new generalized normal distribution is defined and studied. The distribution is unimodal and it can be skewed right or left. The relationships between the parameters and the mean, variance, skewness, and kurtosis are discussed. It is observed that the new distribution has a much wider range of skewness and kurtosis than the skew-normal distribution. The method of maximum likelihood is proposed to estimate the distribution parameters. Two real data sets are applied to illustrate the flexibility of the distribution.     

A new method for interval estimation of the mean of the Gamma distribution

The two parameter Gamma distribution is widely used for modeling lifetime distributions in reliability theory. There is much literature on the inference on the individual parameters of the Gamma distribution, namely the shape parameter k and the scale parameter θ when the other parameter is known. However, usually the reliability professionals have a major interest in making statistical inference about the mean lifetime μ, which equals the product θk for the Gamma distribution. The problem of inference on the mean μ when both parameters θ and k are unknown has been less attended in the literature for the Gamma distribution. In this paper we review the existing methods for interval estimation of μ. A comparative study in this paper indicates that the existing methods are either too approximate and yield less reliable confidence intervals or are computationally quite complicated and need advanced computing facilities. We propose a new simple method for interval estimation of the Gamma mean and compare its performance with the existing methods. The comparative study showed that the newly proposed computationally simple optimum power normal approximation method works best even for small sample sizes.     

Inferences from Cross-Sectional,Stochastic Frontier Models

Conventional approaches for inference about efficiency in parametric stochastic frontier (PSF) models are based on percentiles of the estimated distribution of the one-sided error term, conditional on the composite error. When used as prediction intervals, coverage is poor when the signal-to-noise ratio is low, but improves slowly as sample size increases. We show that prediction intervals estimated by bagging yield much better coverages than the conventional approach, even with low signal-to-noise ratios. We also present a bootstrap method that gives confidence interval estimates for (conditional) expectations of efficiency, and which have good coverage properties that improve with sample size. In addition, researchers who estimate PSF models typically reject models, samples, or both when residuals have skewness in the “wrong” direction, i.e., in a direction that would seem to indicate absence of inefficiency. We show that correctly specified models can generate samples with “wrongly” skewed residuals, even when the variance of the inefficiency process is nonzero. Both our bagging and bootstrap methods provide useful information about inefficiency and model parameters irrespective of whether residuals have skewness in the desired direction.     

Discrete approximations to continuous univariate distributions—an alternative to simulation

A method to replace a continuous univariate distribution with a discrete distribution that takes MN different values is analysed. Both distributions share the same r th moments for r =0, . . ., 2 N −1 and their corresonding distribution functions coincide at least at M +1 points. Several statistical and engineering examples are considered in which the discrete approximation may be used to avoid a simulation study that would be much more demanding computationally.