اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn testing for high-dimensional white noise
92
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Testing for white noise is a classical yet important problem in statistics, especially for diagnostic checks in time series modeling and linear regression. For high-dimensional time series in the sense that the dimension $p$ is large in relation to the sample size $T$, the popular omnibus tests including the multivariate Hosking and Li-McLeod tests are extremely conservative, leading to substantial power loss. To develop more relevant tests for high-dimensional cases, we propose a portmanteau-type test statistic which is the sum of squared singular values of the first $q$ lagged sample autocovariance matrices. It, therefore, encapsulates all the serial correlations (upto the time lag $q$) within and across all component series. Using the tools from random matrix theory and assuming both $p$ and $T$ diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. As the actual implementation of the test requires the knowledge of three characteristic constants of the population cross-sectional covariance matrix and the value of the fourth moment of the standardized innovations, non trivial estimations are proposed for these parameters and their integration leads to a practically usable test. Extensive simulation confirms the excellent finite-sample performance of the new test with accurate size and satisfactory power for a large range of finite $(p,T)$ combinations, therefore ensuring wide applicability in practice. In particular, the new tests are consistently superior to the traditional Hosking and Li-McLeod tests.
قيم البحث
اقرأ أيضاً
Covariance matrix testing for high dimensional data is a fundamental problem. A large class of covariance test statistics based on certain averaged spectral statistics of the sample covariance matrix are known to obey central limit theorems under the
null. However, precise understanding for the power behavior of the corresponding tests under general alternatives remains largely unknown. This paper develops a general method for analyzing the power behavior of covariance test statistics via accurate non-asymptotic power expansions. We specialize our general method to two prototypical settings of testing identity and sphericity, and derive sharp power expansion for a number of widely used tests, including the likelihood ratio tests, Ledoit-Nagao-Wolfs test, Cai-Mas test and Johns test. The power expansion for each of those tests holds uniformly over all possible alternatives under mild growth conditions on the dimension-to-sample ratio. Interestingly, although some of those tests are previously known to share the same limiting power behavior under spiked covariance alternatives with a fixed number of spikes, our new power characterizations indicate that such equivalence fails when many spikes exist. The proofs of our results combine techniques from Poincare-type inequalities, random matrices and zonal polynomials.
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 coefficie
nts 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 prove a Bernstein-von Mises theorem for a general class of high dimensional nonlinear Bayesian inverse problems in the vanishing noise limit. We propose a sufficient condition on the growth rate of the number of unknown parameters under which the
posterior distribution is asymptotically normal. This growth condition is expressed explicitly in terms of the model dimension, the degree of ill-posedness of the inverse problem and the noise parameter. The theoretical results are applied to a Bayesian estimation of the medium parameter in an elliptic problem.
Let $bX=(X_1, hdots, X_d)$ be a $mathbb R^d$-valued random vector with i.i.d. components, and let $VertbXVert_p= (sum_{j=1}^d|X_j|^p)^{1/p}$ be its $p$-norm, for $p>0$. The impact of letting $d$ go to infinity on $VertbXVert_p$ has surprising consequ
ences, which may dramatically affect high-dimensional data processing. This effect is usually referred to as the {it distance concentration phenomenon} in the computational learning literature. Despite a growing interest in this important question, previous work has essentially characterized the problem in terms of numerical experiments and incomplete mathematical statements. In the present paper, we solidify some of the arguments which previously appeared in the literature and offer new insights into the phenomenon.
We focus on the high dimensional linear regression $Ysimmathcal{N}(Xbeta^{*},sigma^{2}I_{n})$, where $beta^{*}inmathds{R}^{p}$ is the parameter of interest. In this setting, several estimators such as the LASSO and the Dantzig Selector are known to s
atisfy interesting properties whenever the vector $beta^{*}$ is sparse. Interestingly both of the LASSO and the Dantzig Selector can be seen as orthogonal projections of 0 into $mathcal{DC}(s)={betainmathds{R}^{p},|X(Y-Xbeta)|_{infty}leq s}$ - using an $ell_{1}$ distance for the Dantzig Selector and $ell_{2}$ for the LASSO. For a well chosen $s>0$, this set is actually a confidence region for $beta^{*}$. In this paper, we investigate the properties of estimators defined as projections on $mathcal{DC}(s)$ using general distances. We prove that the obtained estimators satisfy oracle properties close to the one of the LASSO and Dantzig Selector. On top of that, it turns out that these estimators can be tuned to exploit a different sparsity or/and slightly different estimation objectives.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
