All admissible linear estimators of a regression coefficient under a balanced loss function

Admissibility of linear estimators of a regression coefficient in linear models with and without the assumption that the underlying distribution is normal is discussed under a balanced loss function. In the non-normal case, a necessary and sufficient condition is given for linear estimators to be admissible in the space of homogeneous linear estimators. In the normal case, a sufficient condition is provided for restricted linear estimators to be admissible in the space of all estimators having finite risks under the balanced loss function. Furthermore, the sufficient condition is proved to be necessary in the normal case if additional conditions are assumed.     

Inferring Boolean functions via higher-order correlations

Both the Walsh transform and a modified Pearson correlation coefficient can be used to infer the structure of a Boolean network from time series data. Unlike the correlation coefficient, the Walsh transform is also able to represent higher-order correlations. These correlations of several combined input variables with one output variable give additional information about the dependency between variables, but are also more sensitive to noise. Furthermore computational complexity increases exponentially with the order. We first show that the Walsh transform of order 1 and the modified Pearson correlation coefficient are equivalent for the reconstruction of Boolean functions. Secondly, we also investigate under which conditions (noise, number of samples, function classes) higher-order correlations can contribute to an improvement of the reconstruction process. We present the merits, as well as the limitations, of higher-order correlations for the inference of Boolean networks.     

平衡损失下有限总体回归系数的线性可容许预测

本文研究了平衡损失函数下正态总体和非正态总体中有限回归系数的可容许预测.利用统计决策理论,获得了非正态总体中齐次线性预测为可容许预测的充分必要条件和在正态总体中齐次线性预测在一切预测类中可容许性的充要条件,推广了二次损失下的若干相关结果.     

Local Weighted Composite Quantile Estimating for Varying Coefficient Models

A generalization of classical linear models is varying coefficient models, which offer a flexible approach to modeling nonlinearity between covariates. A method of local weighted composite quantile regression is suggested to estimate the coefficient functions. The local Bahadur representation of the local estimator is derived and the asymptotic normality of the resulting estimator is established. Comparing to the local least squares estimator, the asymptotic relative efficiency is examined for the local weighted composite quantile estimator. Both theoretical analysis and numerical simulations reveal that the local weighted composite quantile estimator can obtain more efficient than the local least squares estimator for various non-normal errors. In the normal error case, the local weighted composite quantile estimator is almost as efficient as the local least squares estimator. Monte Carlo results are consistent with our theoretical findings. An empirical application demonstrates the potential of the proposed method.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

相关系数与相关性度量

研究了度量相关性的两个主要工具:线性相关系数和尾部相关系数.线性相关系数反映了变量间的线性相关性,这对于一般的椭圆型分布是合适的.但如果随机变量具有不对称的尾部变化特征时,要用尾部相关系数描述它们之间的相关性.通过相关函数C opu la,对沪深股市的尾部相关系数进行了定量分析.结果表明:沪深股市具有较强的相关性.     

A new algorithm for calculating two-dimensional differential transform of nonlinear functions

A new algorithm for calculating the two-dimensional differential transform of nonlinear functions is developed in this paper. This new technique is illustrated by studying suitable forms of nonlinearity. Three strongly nonlinear partial differential equations are then solved by differential transform method to demonstrate the validity and applicability of the proposed algorithm. The present framework offers a computationally easier approach to compute the transformed function for all forms of nonlinearity. This gives the technique much wider applicability.     

对有限域上复合变换的线性逼近

本文解决了有限交换群上复合函数的特征谱的计算问题,定义了有限交换群上函数的相关 系数的概念,并解决了有限域上复合函数与线性映射的相关系数的计算问题,从而建立了对密码算 法中的复合变换进行线性逼近的理论基础.     

Optimizing an objective function under a bivariate probability model

The motivation of this paper is to obtain an analytical closed form of a quadratic objective function arising from a stochastic decision process with bivariate exponential probability distribution functions that may be dependent. This method is applicable when results need to be offered in an analytical closed form without double integrals. However, the study only applies to cases where the correlation coefficient between the two variables is positive or null. A stochastic, stationary objective function, involving a single decision variable in a quadratic form is studied. We use a primitive of a bivariate exponential distribution as first expressed by Downton [Downton, F., 1970. Bivariate exponential distributions in reliability theory. Journal of Royal Statistical Society B 32, 408–417] and revisited in Iliopoulos [Iliopoulos, George., 2003. Estimation of parametric functions in Downton’s bivariate exponential distribution. Journal of statistical planning and inference 117, 169–184]. With this primitive, optimization of objective functions in Operations Research, supply chain management or any other setting involving two random variables, or calculations which involve evaluating conditional expectations of two joint random variables are direct. We believe the results can be extended to other cases where exponential bivariates are encountered in economic objective function evaluations. Computation algorithms are offered which substantially reduce computation time when solving numerical examples.     

Optimal estimation in random coefficient regression models

In linear regression models with random coefficients, the score function usually involves unknown nuisance parameters in the form of weights. Conditioning with respect to the sufficient statistics for the nuisance parameter, when the parameter of interest is held fixed, eliminates the nuisance parameters and is expected to give reasonably good estimating functions. The present paper adopts this approach to the problem of estimation of average slope in random coefficient regression models. Four sampling situations are discussed. Some asymptotic results are also obtained for a model where neither the regressors nor the random regression coefficients replicate. Simulation studies for normal as well as non-normal models show that the performance of the suggested estimating functions is quite satisfactory.     

Copula的局部变结构点诊断的实证研究

基于Copula函数对相关性研究的特有优势,构建了二元正态Copula模型,提出了在时变相关系数的基础上对局部变结构点的诊断方法.以上证煤炭指数及有色金属指数作为实证样本,研究了煤炭指数和有色金属的相关性发生显著变化的时刻,并分析其变化原因.本文的研究结果能更敏锐地捕捉金融市场的动向和指导风险投资.     

A heuristic algorithm for a chance constrained stochastic program

A chance constrained stochastic program is considered that arises from an application to college enrollments and in which the objective function is the expectation of a linear function of the random variables. When these random variables are independent and normally distributed with mean and variance that are linear in the decision variables, the deterministic equivalent of the problem is a nonconvex nonlinear knapsack problem. The optimal solution to this problem is characterized and a greedy-type heuristic algorithm that exploits this structure is employed. Computational results show that the algorithm performs well, especially when the normal random variables are approximations of binomial random variables.     

数据预处理对替代数据检验方法的影响

替代数据检验法是检验时间序列中是否存在确定性非线性成分的重要统计方法.通过研究差分和数据平滑运算对替代数据检验方法的影响,指出常用的线性滤波等数据预处理步骤破坏了序列的静态性质,从而会导致对零假设的错误拒绝.因此,建议应该直接利用原始时间序列而非应用了差分等非静态滤波运算后的时间序列生成替代数据,再进行假设检验,以免造成对零假设的错误拒绝.     

Correlation-Immune and Resilient Functions Over a Finite Alphabet and Their Applications in Cryptography

We extend the notions of correlation-immune functions and resilient functions to functions over any finite alphabet. A previous result due to Gopalakrishnan and Stinson is generalized as we give an orthogonal array characterization, a Fourier transform and a matrix characterization for correlation-immune and resilient functions over any finite alphabet endowed with the structure of an Abelian group. We then point out the existence of a tradeoff between the degree of the algebraic normal form and the correlation-immunity order of any function defined on a finite field and we construct some infinite families of t-resilient functions with optimal nonlinearity which are particularly well-suited for combining linear feedback shift registers. We also point out the link between correlation-immune functions and some cryptographic objects as perfect local randomizers and multipermutations.     

相关系数矩阵与多元线性相关分析

相关系数指度量两个随机变量间线性关系的无量纲指标,在研究了相关系数矩阵性质及其与多元随机变量线性相关性之间关系的基础上,提出多元线性相关系数的定义,用于衡量多个变鼋间线性相关强弱的无量纲指标.分析表明,所提多元线性相关系数能够较全面地反映变量间的线性相关强度.     

独立非同分布二元高斯三角阵最大值的渐近性及相关统计推断

考虑二元独立非同分布高斯随机向量三角阵列最大值分布的渐近性及相关统计推断.此高斯三角阵的第n列的第i个向量服从二元高斯分布,其相关系数为i/n的函数并单调连续.首先建立了此高斯三角阵最大值分布的一阶和二阶渐近展开式.其次,分析相关系数参数估计及估计量的渐近性质.最后,通过随机模拟说明了相关系数之参数估计的有效性,并将该二元非同分布三角阵列模型应用于实际数据,得到了满意的结果.     

A Graphical Goodness-of-Fit Test for Dependence Models in Higher Dimensions

This article introduces a graphical goodness-of-fit test for copulas in more than two dimensions. The test is based on pairs of variables and can thus be interpreted as a first-order approximation of the underlying dependence structure. The idea is to first transform pairs of data columns with the Rosenblatt transform to bivariate standard uniform distributions under the null hypothesis. This hypothesis can be graphically tested with a matrix of bivariate scatterplots, Q-Q plots, or other transformations. Furthermore, additional information can be encoded as background color, such as measures of association or (approximate) p-values of tests of independence. The proposed goodness-of-fit test is designed as a basic graphical tool for detecting deviations from a postulated, possibly high-dimensional, dependence model. Various examples are given and the methodology is applied to a financial dataset. An implementation is provided by the R package copula. Supplementary material for this article is available online, which provides the R package copula and reproduces all the graphical results of this article.     

Inferences based on a bivariate distribution with von Mises marginals

There is very little literature concerning modeling the correlation between paired angular observations. We propose a bivariate model with von Mises marginal distributions. An algorithm for generating bivariate angles from this von Mises distribution is given. Maximum likelihood estimation is then addressed. We also develop a likelihood ratio test for independence in paired circular data. Application of the procedures to paired wind directions is illustrated. Employing simulation, using the proposed model, we compare the power of the likelihood ratio test with six existing tests of independence.     

Making Use of Incomplete Observations for Regression in Bivariate Normal Model

Two estimates of the regression coefficient in bivariate normal distribution are considered: the usual one based on a sample and a new one making use of additional observations of one of the variables. They are compared with respect to variance. The same is done for two regression lines. The conclusion is that the additional observations are worth using only when the sample is very small.     

散乱数据带自然边界条件二元样条光顺及数值微分

考虑一种新的散乱数据带自然边界二元样条光顺问题.根据样条变分理论和Hilbert空间样条函数方法,构造出了显式的二元带自然边界光顺样条解,其表达式简单且系数可以由系数矩阵对称正定的线性方程组确定.证明了解的存在和唯一性,讨论了收敛性和误差估计.并由此得到一种新的基于散乱数据上的正则化二元数值微分的方法.最后,给出了一些数值例子对方法进行了验证.