Accurate confidence intervals in regression analyses of non-normal data

A linear model in which random errors are distributed independently and identically according to an arbitrary continuous distribution is assumed. Second- and third-order accurate confidence intervals for regression parameters are constructed from Charlier differential series expansions of approximately pivotal quantities around Student’s t distribution. Simulation verifies that small sample performance of the intervals surpasses that of conventional asymptotic intervals and equals or surpasses that of bootstrap percentile-t and bootstrap percentile-|t| intervals under mild to marked departure from normality.     

Finite Groups with Some S-quasinormally Embedded Subgroups

Suppose that G is a finite group and H is a subgroup of G. H is said to be S-quasinormal in G if it permutes with every Sylow subgroup of G; H is said to be S-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. We investigate the influence of S-quasinormally embedded subgroups on the structure of finite groups. Some recent results are generalized.     

联合检测过程位置参数和尺度参数的非参数Cusum控制图

用于检测生产过程的多数传统控制图都假定过程的受控分布是已知的,并假定数据服从正态分布。然而在很多情况下,由于没有足够的数据来估计过程的分布,这种假定就变得不现实,而非参数控制图却不需要任何关于分布的特殊形式的假定。另外,多数的已有控制图都是使用两个单独的均值与方差控制图来同时检测生产过程.本文中,我们提出一个新的基于Cramer-von-Mises(CvM)检验的非参数累积和控制图(称为CvM图)来同时检测过程位置参数和尺度参数。文中给出了基于不同受控平均运行长度(ARL)下的CvM图的控制限,通过步长的均值、方差及分位数来研究控制图的性能表现。最后用一个实例来说明CvM图的实际应用。     

On the Polar Decomposition of the Product of Two Operators and Its Applications

Let T = U|T| and S = V|S| be the polar decompositions. In this paper, we shall obtain the polar decomposition of TS as TS = UWV|TS|, where |T||S^*| = W||T||S^*|| is the polar decomposition. Next, we shall show that TS = UV|TS| is the polar decomposition if and only if |T| commutes with |S^*|. Lastly, we shall apply this result to binormal and centered operators. We shall obtain characterizations of these operator classes from the viewpoint of the polar decomposition.     

Characterization of simple K 4-groups

In this paper, we prove that a group G is isomorphic to M, where M is a simple K ₄-group, if and only if the following hold: (1) |G| = |M|, (2) nse(G) = nse(M).      

Short cycle covers of cubic graphs

Let G be a bridgeless cubic graph. We prove that the edges of G can be covered by circuits whose total length is at most (44/27) |E(G)|, and if Tutte's 3-flow Conjecture is true, at most (92/57) |E(G)|.     

Edge disjoint cycles through specified vertices

We give a sufficient condition for a simple graph G to have k pairwise edge‐disjoint cycles, each of which contains a prescribed set W of vertices. The condition is that the induced subgraph G[W] be 2k‐connected, and that for any two vertices at distance two in G[W], at least one of the two has degree at least |V(G)|/2 + 2(k ? 1) in G. This is a common generalization of special cases previously obtained by Bollobás/Brightwell (where k = 1) and Li (where W = V(G)). A key lemma is of independent interest. Let G be the complement of a bipartite graph with partite sets X, Y. If G is 2k connected, then G contains k Hamilton cycles that are pairwise edge‐disjoint except for edges in G[Y]. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Integral Operators in Sobolev Spaces on Domains with Boundary

Properties of integral operators with weak singularities arc investigated. It is assumed that G ? ?ⁿ is a bounded domain. The boundary δG should be smooth concerning the Sobolev trace theorem. It will be proved that the integral operators $\int {_G \frac{{f\left(\Theta \right)}}{{x - y|^{n - 1} }}u\left(\nu \right)d\partial G_\nu }$ and $ \int {_{\partial G} \frac{{f\left(\Theta \right)}}{{|x - y|^{n - 1} }}u\left(y \right)d\partial G_y }$ maps W_p^k(G) into W_p^k+1(G) and W_p^k?1(G) into W_p^k/p(G), respectively, and are bounded. Here θ ∈ S ? ?ⁿ, where S is the unit sphere. Furthermore, f possesses bounded first order derivatives and is bounded on S. Then applications to first order systems are discussed.     

Improvements of goodness-of-fit statistics for sparse multinomials based on normalizing transformations

We consider multinomial goodness-of-fit tests for a specified simple hypothesis under the assumption of sparseness. It is shown that the asymptotic normality of the PearsonX ² statistic (X _k ² ) and the log-likelihood ratio statistic (G _k ² ) assuming sparseness. In this paper, we improve the asymptotic normality ofX _k ² andG _k ² statistics based on two kinds of normalizing transformation. The performance of the transformed statistics is numerically investigated.     

On packing 3‐vertex paths in a graph

Let G be a connected graph and let eb(G) and λ(G) denote the number of end‐blocks and the maximum number of disjoint 3‐vertex paths Λ in G. We prove the following theorems on claw‐free graphs: (t1) if G is claw‐free and eb(G) ≤ 2 (and in particular, G is 2‐connected) then λ(G) = ⌊| V(G)|/3⌋; (t2) if G is claw‐free and eb(G) ≥ 2 then λ(G) ≥ ⌊(| V(G) | − eb(G) + 2)/3 ⌋; and (t3) if G is claw‐free and Δ^*‐free then λ(G) = ⌊| V(G) |/3⌋ (here Δ^* is a graph obtained from a triangle Δ by attaching to each vertex a new dangling edge). We also give the following sufficient condition for a graph to have a Λ‐factor: Let n and p be integers, 1 ≤ p ≤ n − 2, G a 2‐connected graph, and |V(G)| = 3n. Suppose that G − S has a Λ‐factor for every S ⊆ V(G) such that |S| = 3p and both V(G) − S and S induce connected subgraphs in G. Then G has a Λ‐factor. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 175–197, 2001     

A new control chart in contaminated data of t‐Student distribution for individual observations

Control charts are the most popular tool for monitoring production quality. In traditional control charts, it is usually supposed that the observations follow a multivariate normal distribution. Nevertheless, there are many practical applications where the normality assumption is not fulfilled. Furthermore, the performance of these charts in the presence of measurement errors (outliers) in the historical data has been improved using robust control charts when the observations follow a normal distribution. In this paper, we develop a new control chart for t‐Student data based on the trimmed T² control chart () through the adaptation of the elements of this chart to the case of this distribution. Simulation studies show that a control chart performs better than T² in t‐Student samples for individual observations. Copyright © 2012 John Wiley & Sons, Ltd.     

Principal components in the nonnormal case: The test of equality of Q roots

The limiting distribution of the likelihood ratio statistic W_q for testing the hypothesis of equality of q characteristic roots of a covariance matrix is studied in the case of nonnormal populations. It is shown, both theoretically and empirically, that the limiting distribution of W_q is not robust to departures from normality and that W_q cannot be used for nonnormal populations with long tails. For the class of elliptical populations the limiting distribution of W_q can be expressed in close form. A corrected test statistic, W_q^*, is proposed that can be used in principal component analysis when sampling from elliptical populations.     

A generalization of Hall-Higman's reduction theorem

In this paper, we generalize the Hall-Higman's reduction theorem by dropping the restrictive hypothesis (|G|, |H|)=1 and determine the detailed structure ofG.Supported in part by NSCF and Guangdong STCF. The author deeply thanks Professor Chen Zhongmu for his kind advice and help.     

Weinberg bounds over nonspherical graphs

Let Aut(G) and E(G) denote the automorphism group and the edge set of a graph G, respectively. Weinberg's Theorem states that 4 is a constant sharp upper bound on the ratio |Aut(G)|/|E(G)| over planar (or spherical) 3‐connected graphs G. We have obtained various analogues of this theorem for nonspherical graphs, introducing two Weinberg‐type bounds for an arbitrary closed surface Σ, namely: where supremum is taken over the polyhedral graphs G with respect to Σ for W_P(Σ) and over the graphs G triangulating Σ for W_T(Σ). We have proved that Weinberg bounds are finite for any surface; in particular: W_P = W_T = 48 for the projective plane, and W_T = 240 for the torus. We have also proved that the original Weinberg bound of 4 holds over the graphs G triangulating the projective plane with at least 8 vertices and, in general, for the graphs of sufficiently large order triangulating a fixed closed surface Σ. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 220–236, 2000     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

Precoloring extension for K4‐minor‐free graphs

Let G=(V, E) be a graph where every vertex v∈V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v)={1, …, k} for all v∈V then a corresponding list coloring is nothing other than an ordinary k‐coloring of G. Assume that W?V is a subset of V such that G[W] is bipartite and each component of G[W] is precolored with two colors taken from a set of four. The minimum distance between the components of G[W] is denoted by d(W). We will show that if G is K₄‐minor‐free and d(W)≥7, then such a precoloring of W can be extended to a 4‐coloring of all of V. This result clarifies a question posed in 10. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)|=4 for all v∈V\W and d(W)≥7. In both cases the bound for d(W) is best possible. © 2009 Wiley Periodicals, Inc. J Graph Theory 60: 284‐294, 2009     

Bayesian statistical process control for Phase I count type data

Count data, most often modeled by a Poisson distribution, are common in statistical process control. They are traditionally monitored by frequentist c or u charts, by cumulative sum and by exponentially weighted moving average charts. These charts all assume that the in‐control true mean is known, a common fiction that is addressed by gathering a large Phase I sample and using it to estimate the mean. “Self‐starting” proposals that ameliorate the need for a large Phase I sample have also appeared. All these methods are frequentist, ie, they allow only retrospective inference during Phase I, and they have no coherent way to incorporate less‐than‐perfect prior information about the in‐control mean. In this paper, we introduce a Bayesian procedure that can incorporate prior information, allow online inference, and should be particularly attractive for short‐run settings where large Phase I calibration exercises are impossible or unreasonable.     

Some new characterizations of solvable PST-groups

A subgroup H of a finite group G is said to be ??-semipermutable in G if it permutes with all the Sylow subgroups Q of G such that (|H|, |Q|) = 1 and (|H|, |Q ^G |) ?? 1. A rather remarkable result of Lukyanenko and Skiba (Rend Semin Mat Univ Padova, 124:231?C246, 2010) is: a finite solvable group G is a PST-group if and only if every subgroup of Fit(G) is ??-semipermutable. A local version of this result is established in this paper. A subgroup H of G is said to be ??-seminormal provided that it is normalized by all Sylow subgroups Q such that (|H|, |Q|) =?1 and (|H|, |Q ^G |) ???1. It is shown that a finite solvable group is a PST-group if and only if every subgroup of Fit(G) is ??-seminormal in G.     

When do random subsets decompose a finite group?

Let A, B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group, i.e. [AB ∪ BA = G]. The study of this event gives rise to a group invariant we call Θ(G). Θ(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes √Θ(G)|G| log |G|; i.e. for any ɛ > 0, if the size of A and B is less than (1 − ɛ)√Θ(G)|G| log |G|, then with high probability AB ∪ BA ≠ G. If A and B are larger than (1 + ɛ)√Θ(G)|G| log |G|, then AB ∪ BA = G with high probability.     

Continuity Adjustment for Control Charts for Attributes

A unified approach is proposed for making a continuity adjustment on some control charts for attributes, e.g., np-chart and c-chart. through adding a uniform (0,1) random observation to the conventional sample statistic (e.g., npi and ci). The adjusted sample statistic then has a continuous distribution. Consequently, given any Type I risk a (the probability that the sample statistic is on or beyond the control limits), control charts achieving the exact value of a can be readily constructed. Guidelines are given for when to use the continuity adjustment control chart, the conventional Shewhart control chart (with ±3 standard deviations control limits), and the control chart based on the exact distribution of the sample statistic before adjustment.