Sample Correlation

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Yukihiro Ozaki - One of the best experts on this subject based on the ideXlab platform.

Ran Guo - One of the best experts on this subject based on the ideXlab platform.

Wang Zhou - One of the best experts on this subject based on the ideXlab platform.

  • Tracy-Widom law for the extreme eigenvalues of Sample Correlation matrices
    Electronic Journal of Probability, 2012
    Co-Authors: Zhigang Bao, Guangming Pan, Wang Zhou
    Abstract:

    Let the Sample Correlation matrix be $W=YY^T$ , where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetrically distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0

  • tracy widom law for the extreme eigenvalues of Sample Correlation matrices
    arXiv: Statistics Theory, 2011
    Co-Authors: Zhigang Bao, Guangming Pan, Wang Zhou
    Abstract:

    Let the Sample Correlation matrix be $W=YY^T$, where $Y=(y_{ij})_{p,n}$ with $y_{ij}=x_{ij}/\sqrt{\sum_{j=1}^nx_{ij}^2}$. We assume $\{x_{ij}: 1\leq i\leq p, 1\leq j\leq n\}$ to be a collection of independent symmetric distributed random variables with sub-exponential tails. Moreover, for any $i$, we assume $x_{ij}, 1\leq j\leq n$ to be identically distributed. We assume $0eigenvalues of $W$. If $x_{ij}$ are i.i.d. standard normal, we can derive the $TW_1$ for both the largest and smallest eigenvalues of the matrix $\mathcal{R}=RR^T$, where $R=(r_{ij})_{p,n}$ with $r_{ij}=(x_{ij}-\bar x_i)/\sqrt{\sum_{j=1}^n(x_{ij}-\bar x_i)^2}$, $\bar x_i=n^{-1}\sum_{j=1}^nx_{ij}$.

  • Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices
    Journal of Theoretical Probability, 2010
    Co-Authors: Han Xiao, Wang Zhou
    Abstract:

    Let X ^( n )=( X _ ij ) be a p × n data matrix, where the n columns form a random Sample of size n from a certain p -dimensional distribution. Let R ^( n )=( ρ _ ij ) be the p × p Sample Correlation coefficient matrix of X ^( n ), and $S^{(n)}=(1/n)X^{(n)}(X^{(n)})^{\ast}-\bar{X}\bar{X}^{\ast}$ be the Sample covariance matrix of X ^( n ), where $\bar{X}$ is the mean vector of the n observations. Assuming that X _ ij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R ^( n ) converges almost surely to the limit $(1-\sqrt{c}\,)^{2}$ as n →∞ and p / n → c ∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S ^( n ) converges almost surely to $(1-\sqrt{c}\,)^{2}$ .

  • Almost Sure Limit of the Smallest Eigenvalue of Some Sample Correlation Matrices
    Journal of Theoretical Probability, 2010
    Co-Authors: Han Xiao, Wang Zhou
    Abstract:

    Let X(n)=(Xij) be a p×n data matrix, where the n columns form a random Sample of size n from a certain p-dimensional distribution. Let R(n)=(ρij) be the p×p Sample Correlation coefficient matrix of X(n), and \(S^{(n)}=(1/n)X^{(n)}(X^{(n)})^{\ast}-\bar{X}\bar{X}^{\ast}\) be the Sample covariance matrix of X(n), where \(\bar{X}\) is the mean vector of the n observations. Assuming that Xij are independent and identically distributed with finite fourth moment, we show that the smallest eigenvalue of R(n) converges almost surely to the limit \((1-\sqrt{c}\,)^{2}\) as n→∞ and p/n→c∈(0,∞). We accomplish this by showing that the smallest eigenvalue of S(n) converges almost surely to \((1-\sqrt{c}\,)^{2}\) .

  • Asymptotic distribution of the largest off-diagonal entry of Correlation matrices
    Transactions of the American Mathematical Society, 2007
    Co-Authors: Wang Zhou
    Abstract:

    Suppose that we have observations from a -dimensional population. We are interested in testing that the variates of the population are independent under the situation where goes to infinity as . A test statistic is chosen to be , where is the Sample Correlation coefficient between the -th coordinate and the -th coordinate of the population. Under an independent hypothesis, we prove that the asymptotic distribution of is an extreme distribution of type , by using the Chen-Stein Poisson approximation method and the moderate deviations for Sample Correlation coefficients. As a statistically more relevant result, a limit distribution for , where is Spearman's rank Correlation coefficient between the -th coordinate and the -th coordinate of the population, is derived

Isao Noda - One of the best experts on this subject based on the ideXlab platform.

Xin Zhang - One of the best experts on this subject based on the ideXlab platform.

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