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Register a new userA note on the permutation distribution of generalized correlation coefficients
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Added by
Hao Chen
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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We provide sufficient conditions for the asymptotic normality of the generalized correlation coefficient $sum a_{ij}b_{ij}$ under the permutation null distribution when $a_{ij}$s are symmetric and $b_{ij}$s are symmetric.
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This paper proposes a new statistic to test independence between two high dimensional random vectors ${mathbf{X}}:p_1times1$ and ${mathbf{Y}}:p_2times1$. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of ${mathbf{X}}$ and ${mathbf{Y}}$. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when $p_1$ and $p_2$ are both comparable to the sample size $n$. As applications of the developed independence test, various types of dependent structures, such as factor models, ARCH models and a general uncorrelated but dependent case, etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily stock returns of companies between different sections in the New York Stock Exchange (NYSE) is detected by the proposed test.
We analyse the prediction error of principal component regression (PCR) and prove non-asymptotic upper bounds for the corresponding squared risk. Under mild assumptions, we show that PCR performs as well as the oracle method obtained by replacing empirical principal components by their population counterparts. Our approach relies on upper bounds for the excess risk of principal component analysis.
Consider a Gaussian vector $mathbf{z}=(mathbf{x},mathbf{y})$, consisting of two sub-vectors $mathbf{x}$ and $mathbf{y}$ with dimensions $p$ and $q$ respectively, where both $p$ and $q$ are proportional to the sample size $n$. Denote by $Sigma_{mathbf{u}mathbf{v}}$ the population cross-covariance matrix of random vectors $mathbf{u}$ and $mathbf{v}$, and denote by $S_{mathbf{u}mathbf{v}}$ the sample counterpart. The canonical correlation coefficients between $mathbf{x}$ and $mathbf{y}$ are known as the square roots of the nonzero eigenvalues of the canonical correlation matrix $Sigma_{mathbf{x}mathbf{x}}^{-1}Sigma_{mathbf{x}mathbf{y}}Sigma_{mathbf{y}mathbf{y}}^{-1}Sigma_{mathbf{y}mathbf{x}}$. In this paper, we focus on the case that $Sigma_{mathbf{x}mathbf{y}}$ is of finite rank $k$, i.e. there are $k$ nonzero canonical correlation coefficients, whose squares are denoted by $r_1geqcdotsgeq r_k>0$. We study the sample counterparts of $r_i,i=1,ldots,k$, i.e. the largest $k$ eigenvalues of the sample canonical correlation matrix $S_{mathbf{x}mathbf{x}}^{-1}S_{mathbf{x}mathbf{y}}S_{mathbf{y}mathbf{y}}^{-1}S_{mathbf{y}mathbf{x}}$, denoted by $lambda_1geqcdotsgeq lambda_k$. We show that there exists a threshold $r_cin(0,1)$, such that for each $iin{1,ldots,k}$, when $r_ileq r_c$, $lambda_i$ converges almost surely to the right edge of the limiting spectral distribution of the sample canonical correlation matrix, denoted by $d_{+}$. When $r_i>r_c$, $lambda_i$ possesses an almost sure limit in $(d_{+},1]$. We also obtain the limiting distribution of $lambda_i$s under appropriate normalization. Specifically, $lambda_i$ possesses Gaussian type fluctuation if $r_i>r_c$, and follows Tracy-Widom distribution if $r_i<r_c$. Some applications of our results are also discussed.
In this note we consider the optimal design problem for estimating the slope of a polynomial regression with no intercept at a given point, say z. In contrast to previous work, which considers symmetric design spaces we investigate the model on the interval $[0, a]$ and characterize those values of $z$, where an explicit solution of the optimal design is possible.
Permutation tests are widely used in statistics, providing a finite-sample guarantee on the type I error rate whenever the distribution of the samples under the null hypothesis is invariant to some rearrangement. Despite its increasing popularity and empirical success, theoretical properties of the permutation test, especially its power, have not been fully explored beyond simple cases. In this paper, we attempt to fill this gap by presenting a general non-asymptotic framework for analyzing the power of the permutation test. The utility of our proposed framework is illustrated in the context of two-sample and independence testing under both discrete and continuous settings. In each setting, we introduce permutation tests based on U-statistics and study their minimax performance. We also develop exponential concentration bounds for permuted U-statistics based on a novel coupling idea, which may be of independent interest. Building on these exponential bounds, we introduce permutation tests which are adaptive to unknown smoothness parameters without losing much power. The proposed framework is further illustrated using more sophisticated test statistics including weighted U-statistics for multinomial testing and Gaussian kernel-based statistics for density testing. Finally, we provide some simulation results that further justify the permutation approach.
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