Testing the Sphericity of a covariance matrix when the dimension is much larger than the sample size

This paper focuses on the prominent sphericity test when the dimension $p$ is much lager than sample size $n$. The classical likelihood ratio test(LRT) is no longer applicable when $p\gg n$. Therefore a Quasi-LRT is proposed and asymptotic distribution of the test statistic under the null when $p/n\rightarrow\infty, n\rightarrow\infty$ is well established in this paper. Meanwhile, John's test has been found to possess the powerful {\it dimension-proof} property, which keeps exactly the same limiting distribution under the null with any $(n,p)$-asymptotic, i.e. $p/n\rightarrow[0,\infty]$, $n\rightarrow\infty$. All asymptotic results are derived for general population with finite fourth order moment. Numerical experiments are implemented for comparison

Restricted Flows and the Soliton Equation with Self-Consistent Sources

The KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Backlund transformation for the restricted flows (by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations for the KdV equation with self-consistent sources (KdVES). These Darboux transformations are used to get some explicit solutions of the KdVES, which include soliton, rational, positon, and negaton solutions.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On singular value distribution of large dimensional auto-covariance matrices

Let $(\varepsilon_j)_{j\geq 0}$ be a sequence of independent $p-$dimensional random vectors and $\tau\geq1$ a given integer. From a sample $\varepsilon_1,\cdots,\varepsilon_{T+\tau-1},\varepsilon_{T+\tau}$ of the sequence, the so-called lag $-\tau$ auto-covariance matrix is $C_{\tau}=T^{-1}\sum_{j=1}^T\varepsilon_{\tau+j}\varepsilon_{j}^t$. When the dimension $p$ is large compared to the sample size $T$, this paper establishes the limit of the singular value distribution of $C_\tau$ assuming that $p$ and $T$ grow to infinity proportionally and the sequence satisfies a Lindeberg condition on fourth order moments. Compared to existing asymptotic results on sample covariance matrices developed in random matrix theory, the case of an auto-covariance matrix is much more involved due to the fact that the summands are dependent and the matrix $C_\tau$ is not symmetric. Several new techniques are introduced for the derivation of the main theorem

An investigation of decision-making style of Chinese college student online apparel shoppers

Internet users in China increased to 210 million with an annual growth rate of 53.3 percent in 2007 (CNNIC, 2008). This dramatic increase of Internet usage in China provides numerous opportunities for online marketers. Thirty-eight percent of Chinese netizens are 18 to 24 years old, among whom college netizens account for a large proportion in China (CNNIC, 2008). Given the market potential of targeting this group, research is needed to understand Chinese college students’ online shopping behavior. The purpose of this research was to better understand Chinese college student online apparel shoppers by investigating their decision-making style and explore the relationships between their decision-making characteristics and related online apparel shopping behavior and consumption. Consumer Style Inventory (CSI) developed by Sproles and Kendall (1986) was adopted as a theoretical framework to guide this study. CSI has been recognized as a useful tool to understand consumers’ shopping orientation. This market tool has been applied to effectively understand consumers from different countries and cultures (Lysonski, Srini, & Zotos, 1996). However, no research has been done to apply this tool to understanding Chinese college students as online apparel shoppers. This research intends to fill the identified gap. This empirical study employed an online survey for data collection. A questionnaire was developed and administered to students at five universities from different cities in China. This study found that Chinese college students spent more time online on pre-purchase decision-making activities. Most of the respondents spent time looking for interesting apparel products and evaluating different apparel products online, but not on ordering the selected products. The results demonstrated that some of the characteristics of the CSI are related to the frequency of buying apparel online, and the dollar amount spent online for apparel purchasing. The findings show that recreational consciousness, hedonistic consciousness, brand consciousness, habitual consciousness, and brand-loyalty consciousness have significant correlations with the frequency of online apparel purchases. However, only brand conscious and habitual conscious, brand-loyalty conscious are significantly correlated with the amount of money spent online for apparel purchases by Chinese college students

The generalized Kupershmidt deformation for constructing new integrable systems from integrable bi-Hamiltonian systems

Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable bi-Hamiltonian systems, which provides a nonholonomic perturbation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is verified in a few representative cases: KdV equation, Boussinesq equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific cases, we present a general procedure to convert the generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a $t$-type bi-Hamiltonian system. By using this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius deformation of soliton equation with self-consistent sources.Comment: 21 pages, to appear in Journal of Mathematical Physic