On Jiang's asymptotic distribution of the largest entry of a sample correlation matrix

Let $\{X, X_{k,i}; i \geq 1, k \geq 1 \}$ be a double array of nondegenerate i.i.d. random variables and let $\{p_{n}; n \geq 1 \}$ be a sequence of positive integers such that $n/p_{n}$ is bounded away from $0$ and $\infty$. This paper is devoted to the solution to an open problem posed in Li, Liu, and Rosalsky (2010) on the asymptotic distribution of the largest entry $L_{n} = \max_{1 \leq i < j \leq p_{n}} \left | \hat{\rho}^{(n)}_{i,j} \right |$ of the sample correlation matrix ${\bf \Gamma}_{n} = \left ( \hat{\rho}_{i,j}^{(n)} \right )_{1 \leq i, j \leq p_{n}}$ where $\hat{\rho}^{(n)}_{i,j}$ denotes the Pearson correlation coefficient between $(X_{1, i},..., X_{n,i})'$ and $(X_{1, j},..., X_{n,j})'$. We show under the assumption $\mathbb{E}X^{2} < \infty$ that the following three statements are equivalent: \begin{align*} & {\bf (1)} \quad \lim_{n \to \infty} n^{2} \int_{(n \log n)^{1/4}}^{\infty} \left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x} \right) \right) dF(x) = 0, \\ & {\bf (2)} \quad \left ( \frac{n}{\log n} \right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2, \\ & {\bf (3)} \quad \lim_{n \rightarrow \infty} \mathbb{P} \left (n L_{n}^{2} - a_{n} \leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2} \right \}, - \infty < t < \infty \end{align*} where $F(x) = \mathbb{P}(|X| \leq x), x \geq 0$ and $a_{n} = 4 \log p_{n} - \log \log p_{n}$, $n \geq 2$. To establish this result, we present six interesting new lemmas which may be beneficial to the further study of the sample correlation matrix.Comment: 16 page

Reduce computation in profile empirical likelihood method

Since its introduction by Owen in [29, 30], the empirical likelihood method has been extensively investigated and widely used to construct confidence regions and to test hypotheses in the literature. For a large class of statistics that can be obtained via solving estimating equations, the empirical likelihood function can be formulated from these estimating equations as proposed by [35]. If only a small part of parameters is of interest, a profile empirical likelihood method has to be employed to construct confidence regions, which could be computationally costly. In this paper we propose a jackknife empirical likelihood method to overcome this computational burden. This proposed method is easy to implement and works well in practice.profile empirical likelihood; estimating equation; Jackknife

STUDY ON PREFABRICATED CONNECTOR OF DOUBLE-LAYER RECIPROCAL FRAME

A kind of lattice-type steel member is presented and a kind of prefabricated connector suitable for the connection between lattice-type steel members are proposed. The mechanical properties of the connectors are analyzed by using the finite element numerical simulation software ABAQUS. The connectors meet the design goals of the stiffness of connection stronger than members. Parameterized analysis is carried out on the prefabricated connector, and the flexural stiffness expression of the connector is obtained. The suggested values of each component of the prefabricated connectors are given based on the size of connected lattice-type steel members