Testing optimality of experimental designs for a regression model with random variables

Tsukanov (Theor. Probab. Appl. 26 (1981) 173–177) considers the regression model $\text{E(}\text{y}\text{|}\text{Z}\text{)=}\text{Fp}\text{+}\text{Zq}$, $\text{D(}\text{y}\text{|}\text{Z}\text{)=σ}{}^2\text{I}{}_{\mathrm{n}}$, where $\text{y}\text{(n×1)}$ is a vector of measured values,$\text{F}\text{(n×k)}$ contains the control variables, $\text{Z}\text{(n×l)}$ contains the observed values, and $\text{p}\text{(k×1)}$ and $\text{q}\text{(l×1)}$ are being estimated. Assuming that $\text{Z}\text{=}\text{FL}\text{+}\text{R}$, where $\text{L}\text{(k×l)}$ is non-random, and the rows of $\text{R}\text{(n×l)}$ are i.i.d. $\text{N(}\text{0}\text{,Σ)}$, we extend Tsukanov's results by (i) computing E(detH_p), where H_p is the covariance matrix of p?, the l.s.e. of p, (ii) considering ‘optimality in the mean’ for the largest root criterion, (iii) discussing these equations when the matrix R has a left-spherical distribution.     

一类矩阵乘积迹的算术几何平均不等式

本文推广、改进了刘家学的结果。证明了ｔｒ（Ａ２）＋ｔｒ（Ｂ２）≥２ｔｒ（ＡＢ）这里Ａ、Ｂ是具有ｎ个实特征有值的ｎ阶复矩阵，且ｒ（［Ａ，Ｂ］）≤１，或（Ａ，［Ａ，Ｂ］）＝［Ｂ，［Ａ，Ｂ］］＝０     

一种新的积分方法──微分积分法

一种新的积分方法──微分积分法杨运平０引言积分运算作为微分运算的逆运算，由于其定义的＂非构造性＂，使积分运算远比微分运算复杂、困难得多，它没有固定的计算程序、要具体问题具体分析，有时一题还可多解。因此，积分运算要求具有很大的灵活性与很高的技巧。但最终...     

关于费马问题费马点的探讨

本文详尽地介绍了伟大教学家费马（Ｆｅｒｍａｔ）的生平业绩和广大数学家对费马问题（Ｆｅｒｍａｔ＇ｓｐｒｏｂｌｅｍ）的研究近况，并对费马点（Ｆｅｒｍａｔ＇ｓｐｏｉｎｔ）作了认真地探索和补充，使其更加完善．     

Convergence results for sequences of quadratic forms

Let (XI,)be a sequence of independent random variables, and let Q_n= where for each N,(an:,k)is a doubly indexed sequence of weights. The convergence and the rate of convergence of the sequence of quadratic forms {Q_n} are studied. These quadratic forms are linear sums of dependent variables; however, their convergence properties are similar to those of linear sums of independent variables provided the variables have finite rth absolute moments with 0 < r 2.while the rate of convergence has not been obtained for r< 2, it is shown to be different from that of linear sums.     

Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem

A vector merging problem is introduced where two vectors of length n are merged such that the k-th entry of the new vector is the minimum over of the -th entry of the first vector plus the sum of the first k – + 1 entries of the second vector. For this problem a new algorithm with O(n log n) running time is presented thus improving upon the straightforward O(n ²) time bound.The vector merging problem can appear in different settings of dynamic programming. In particular, it is applied for a recent fully polynomial time approximation scheme (FPTAS) for the classical 0–1 knapsack problem by the same authors.     

Statistical Inference on the Parametric Component in Partially Linear Spatial Autoregressive Models

A statistical test procedure is proposed to check whether the parameters in the parametric component of the partially linear spatial autoregressive models satisfy certain linear constraint conditions, in which a residual-based bootstrap procedure is suggested to derive the p-value of the test. Some simulations are conducted to assess the performance of the test and the results show that the bootstrap approximation to the null distribution of the test statistic is valid and the test is of satisfactory power. Furthermore, a real-world example is given to demonstrate the application of the proposed test.     

Variable selection in classification model via quadratic programming

Variable selection is an important decision process in consumer credit scoring. However, with the rapid growth in credit industry, especially, after the rising of e-commerce, a huge amount of information on customer behavior is available to provide more informative implication of consumer credit scoring. In this study, a hybrid quadratic programming model is proposed for consumer credit scoring problems by variable selection. The proposed model is then solved with a bisection method based on Tabu search algorithm (BMTS), and the solution of this model provides alternative subsets of variables in different sizes. The final subset of variables used in consumer credit scoring model is selected based on both the size (number of variables in a subset) and predictive (classification) accuracy rate. Simulation studies are used to measure the performances of the proposed model, illustrating its effectiveness for simultaneous variable selection as well as classification.     

An accurate,tractable, and analytically integrable polynomial expansion of the skewed Student’s t-distribution

An integrable polynomial expansion is derived for the skewed Student’s t-distribution. Analytic integration of the density function results in a polynomial expression for the cumulative distribution function too, which is easily manageable from a numerical and mathematical point of view and enables computing quantiles with minimum numerical effort.     

Refined solution to upper-bound problem for the expectations of order statistics from decreasing density on the average distributions

Rychlik [Metrika 77, 539–557, 2014] described sharp upper negative bounds for the expectations of low-rank order statistics, centered about the population mean and measured in the mean absolute deviation from the mean units, for the i.i.d. sequences with common distribution possessing decreasing density function on the average. The bounds coincide with the negatives of maximal values of complicated functions on the unit interval. Here, we provide more precise solutions to the maximization problems.