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Nitis Mukhopadhyay - One of the best experts on this subject based on the ideXlab platform.
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multiple crossing sequential fixed size Confidence Region methodologies for a multivariate normal mean vector
Statistical Methodology, 2014Co-Authors: Nitis Mukhopadhyay, Sankha Muthu PoruthotageAbstract:Abstract The asymptotically efficient and asymptotically consistent purely sequential procedure of Mukhopadhyay and Al-Mousawi (1986) is customarily used to construct a Confidence Region R for the mean vector μ of N p ( μ , σ 2 H ) . This procedure does not have the exact consistency property. H p × p is assumed known and positive definite with σ 2 unknown. The maximum diameter of R and the Confidence coefficient are prefixed. A purely sequential sampling strategy is proposed allowing sampling until sample size crosses the boundary multiple times. We ascertain asymptotic efficiency and asymptotic consistency properties ( Theorem 3.1 ). Its ability to nearly achieve required coverage probability without significant over-sampling is demonstrated with simulations. A truncation technique plus fine-tuning of the multiple crossing rule are proposed to increase practicality. Two real data illustrations are highlighted.
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second order properties of a two stage fixed size Confidence Region for the mean vector of a multivariate normal distribution
Journal of Multivariate Analysis, 1999Co-Authors: Nitis MukhopadhyayAbstract:We consider the classical fixed-size Confidence Region estimation problem for the mean vector?in theNp(?,?) population where ? is unknown but positive definite. We write?1for the largest characteristic root of ? and assume that?1is simple. Moreover, we suppose that, in many practical applications, we will often have available a number?*(>0) and that we can assume?1>?*. Given this addi- tional, and yet very minimal, knowledge regarding?1, the two-stage procedure of Chatterjee (Calcutta Statist. Assoc. Bull.8(1959a), 121?148;9(1959b), 20?28;11(1962), 144?159) is revised appropriately. The highlight in this paper involves the verification ofsecond-order propertiesassociated with such revised two-stage estimation techniques, along with the maintenance of the nominal Confidence coefficient.
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Second-order properties of a two-stage fixed-size Confidence Region when the covariance matrix has a structure
Communications in Statistics - Theory and Methods, 1999Co-Authors: Makoto Aoshima, Nitis MukhopadhyayAbstract:We study second-order properties of a two-stage fixed -size Confidence Region for estimating the mean vector μ in the Np(μ∑) population when some auxiliary information about the structure of ∑ is available. In the case when we do not have such nrior information regarding ∑. Mukhopauiiyay (199/ ) de rived second-order properties of the classical two stage fixed-size Confidence Region, when properly modified. It was assumed that the maximum latent root of ∑ was simple and bounded below by a known positive number. In this paper we allow the maximum latent root to have general multiplicity for the verification of the second order properties of the two-stage procedure incorpo rating ∑'s structural information. We also maintain the nominal Confidence coefficient.
Dong Xia - One of the best experts on this subject based on the ideXlab platform.
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Confidence Region of singular subspaces for low rank matrix regression
IEEE Transactions on Information Theory, 2019Co-Authors: Dong XiaAbstract:Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. Numerous statistical methods have been developed over the recent decade for efficiently reconstructing the unknown low-rank matrices. It is often interesting, in certain applications, to estimate the unknown singular subspaces. In this paper, we revisit the low-rank matrix regression model and introduce a two-step procedure to construct Confidence Regions of the singular subspaces. We investigate distributions of the joint projection distance between the empirical singular subspaces and the unknown true singular subspaces. We prove asymptotical normality of the joint projection distance with data-dependent centering and normalization when $r^{3/2}(m_{1}+m_{2})^{3/2}=o(n/\log n)$ where $m_{1}, m_{2}$ denote the matrix row and column sizes, $r$ is the rank and $n$ is the number of independent random measurements. Consequently, data-dependent Confidence Regions of the true singular subspaces are established which attain pre-determined Confidence levels asymptotically. Additionally, non-asymptotic convergence rates are also established. Numerical results are presented to show the merits of our methods.
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normal approximation and Confidence Region of singular subspaces
arXiv: Statistics Theory, 2019Co-Authors: Dong XiaAbstract:This paper is on the normal approximation of singular subspaces when the noise matrix has i.i.d. entries. Our contributions are three-fold. First, we derive an explicit representation formula of the empirical spectral projectors. The formula is neat and holds for deterministic matrix perturbations. Second, we calculate the expected projection distance between the empirical singular subspaces and true singular subspaces. Our method allows obtaining arbitrary $k$-th order approximation of the expected projection distance. Third, we prove the non-asymptotical normal approximation of the projection distance with different levels of bias corrections. By the $\lceil \log(d_1+d_2)\rceil$-th order bias corrections, the asymptotical normality holds under optimal signal-to-noise ration (SNR) condition where $d_1$ and $d_2$ denote the matrix sizes. In addition, it shows that higher order approximations are unnecessary when $|d_1-d_2|=O((d_1+d_2)^{1/2})$. Finally, we provide comprehensive simulation results to merit our theoretic discoveries. Unlike the existing results, our approach is non-asymptotical and the convergence rates are established. Our method allows the rank $r$ to diverge as fast as $o((d_1+d_2)^{1/3})$. Moreover, our method requires no eigen-gap condition (except the SNR) and no constraints between $d_1$ and $d_2$.
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Confidence Region of singular subspaces for low rank matrix regression
arXiv: Statistics Theory, 2018Co-Authors: Dong XiaAbstract:Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. In the last decade, numerous statistical methodologies have been developed for efficiently recovering the unknown low-rank matrices. However, in some applications, the unknown singular subspace is scientifically more important than the low-rank matrix itself. In this article, we revisit the low-rank matrix regression model and introduce a two-step procedure to construct Confidence Regions of the singular subspace. The procedure involves the de-biasing for the typical low-rank estimators after which we calculate the empirical singular vectors. We investigate the distribution of the joint projection distance between the empirical singular subspace and the unknown true singular subspace. We specifically prove the asymptotical normality of the joint projection distance with data-dependent centering and normalization when $r^{3/2}(m_1+m_2)^{3/2}=o(n/\log n)$ where $m_1, m_2$ denote the matrix row and column sizes, $r$ is the rank and $n$ is the number of independent random measurements. Consequently, we propose data-dependent Confidence Regions of the true singular subspace which attains any pre-determined Confidence level asymptotically. In addition, non-asymptotical convergence rates are also established. Numerical results are presented to demonstrate the merits of our methods.
Sankha Muthu Poruthotage - One of the best experts on this subject based on the ideXlab platform.
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multiple crossing sequential fixed size Confidence Region methodologies for a multivariate normal mean vector
Statistical Methodology, 2014Co-Authors: Nitis Mukhopadhyay, Sankha Muthu PoruthotageAbstract:Abstract The asymptotically efficient and asymptotically consistent purely sequential procedure of Mukhopadhyay and Al-Mousawi (1986) is customarily used to construct a Confidence Region R for the mean vector μ of N p ( μ , σ 2 H ) . This procedure does not have the exact consistency property. H p × p is assumed known and positive definite with σ 2 unknown. The maximum diameter of R and the Confidence coefficient are prefixed. A purely sequential sampling strategy is proposed allowing sampling until sample size crosses the boundary multiple times. We ascertain asymptotic efficiency and asymptotic consistency properties ( Theorem 3.1 ). Its ability to nearly achieve required coverage probability without significant over-sampling is demonstrated with simulations. A truncation technique plus fine-tuning of the multiple crossing rule are proposed to increase practicality. Two real data illustrations are highlighted.
John J Peterson - One of the best experts on this subject based on the ideXlab platform.
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a Confidence Region for the ridge path in multiple response surface optimization
European Journal of Operational Research, 2016Co-Authors: Liangxing Shi, Dennis K J Lin, John J PetersonAbstract:Abstract Ridge analysis allows the analyst to explore the optimal operating conditions of the experimental factors. A Confidence Region is desirable for the estimated ridge path. Most literature concentrates on the univariate response situation. Little is known for the Confidence Region of the ridge path for the multivariate response; only a large-sample Confidence interval for the ridge path is available. The simultaneous coverage rate for the existing interval is typically too conservative in practice, especially for small sample sizes. In this paper, the ridge path (via desirability function) is estimated based on the seemingly unrelated regression (SUR) model as well as standard multivariate regression (SMR) model, and a conservative Confidence interval suitable for small sample sizes is proposed. It is shown that the proposed method outperforms the existing methods. Real-life examples and simulative study are given for illustration.
Arturo J Fernandez - One of the best experts on this subject based on the ideXlab platform.
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minimizing the area of a pareto Confidence Region
European Journal of Operational Research, 2012Co-Authors: Arturo J FernandezAbstract:Abstract A constrained optimization problem is formulated and solved in order to determine the smallest Confidence Region for the parameters of the Pareto distribution in a proposed family of sets. The objective function is the area of the Region, whereas the constraints are related to the required Confidence level. Explicit expressions for the area and Confidence level of a given Region are first deduced. An efficient procedure based on minimizing the corresponding Lagrangian function is then presented to solve the nonlinear programming problem. The process is valid when some of the smallest and largest observations have been discarded or censored, i.e., both single (right or left) and double censoring are allowed. The optimal Pareto Confidence Region is derived by simultaneously solving three (four) nonlinear equations in the right (double) censoring case. In most practical situations, Newton’s method with the balanced set as the starting point only needs a few iterations to find the global solution. In general, the reduction in area of the optimal Pareto Region with respect to the balanced set is considerable if the sample size, n , is small or moderately large, which is usual in practice. This reduction is sometimes impressive when n is quite small and the censoring degree is fairly high. Two numerical examples regarding component lifetimes and fire claims are included for illustrative and comparative purposes.