Monotonicity of the Power Functions of Some Tests of the Multivariate Linear Hypothesis
TLDR
In this paper, sufficient conditions on the procedure for the power function to be a monotonically increasing function of each of the parameters are obtained, and the likelihood-ratio test, Lawley-Hotelling trace test, and Roy's maximum root test satisfy these conditions.Abstract:
The test procedures, invariant under certain groups of transformations [4], for testing a set of multivariate linear hypotheses in the linear normal model depend on the characteristic roots of a random matrix. The power function of such a test depends on the characteristic roots of a corresponding population matrix as parameters; these roots may be regarded as measures of deviation from the hypothesis tested. In this paper sufficient conditions on the procedure for the power function to be a monotonically increasing function of each of the parameters are obtained. The likelihood-ratio test [1], Lawley-Hotelling trace test [1], and Roy's maximum root test [6] satisfy these conditions. The monotonicity of the power function of Roy's test has been shown by Roy and Mikhail [5] using a geometrical method.read more
Citations
More filters
Journal ArticleDOI
Minimax risk of matrix denoising by singular value thresholding
David L. Donoho,Matan Gavish +1 more
Abstract: An unknown $m$ by $n$ matrix $X_0$ is to be estimated from noisy measurements $Y=X_0+Z$, where the noise matrix $Z$ has i.i.d. Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem $\operatorname {min}_X\|Y-X\|_F^2/2+\lambda\|X\|_*$, where $\|X\|_*$ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of $\ell_1$ penalization in the vector case. It has been empirically observed that if $X_0$ has low rank, it may be recovered quite accurately from the noisy measurement $Y$. In a proportional growth framework where the rank $r_n$, number of rows $m_n$ and number of columns $n$ all tend to $\infty$ proportionally to each other ($r_n/m_n\rightarrow \rho$, $m_n/n\rightarrow \beta$), we evaluate the asymptotic minimax MSE $\mathcal {M}(\rho,\beta)=\lim_{m_n,n\rightarrow \infty}\inf_{\lambda}\sup_{\operatorname {rank}(X)\leq r_n}\operatorname {MSE}(X_0,\hat{X}_{\lambda})$. Our formulas involve incomplete moments of the quarter- and semi-circle laws ($\beta=1$, square case) and the Marcenko-Pastur law ($\beta<1$, nonsquare case). For finite $m$ and $n$, we show that MSE increases as the nonzero singular values of $X_0$ grow larger. As a result, the finite-$n$ worst-case MSE, a quantity which can be evaluated numerically, is achieved when the signal $X_0$ is "infinitely strong." The nuclear norm penalization problem is solved by applying soft thresholding to the singular values of $Y$. We also derive the minimax threshold, namely the value $\lambda^*(\rho)$, which is the optimal place to threshold the singular values. All these results are obtained for general (nonsquare, nonsymmetric) real matrices. Comparable results are obtained for square symmetric nonnegative-definite matrices.
Journal ArticleDOI
Power comparisons of tests of two multivariate hypotheses based on four criteria
Journal ArticleDOI
Reflection Groups, Generalized Schur Functions, and the Geometry of Majorization
TL;DR: In this article, a geometrical study of convex polytopes and the ordering of the majorization and Schur-concave functions in reflection groups is presented.
Journal ArticleDOI
Minimax risk of matrix denoising by singular value thresholding
David L. Donoho,Matan Gavish +1 more
TL;DR: In this article, the problem of matrix denoising is solved by applying soft thresholding to the singular values of the noisy measurement, where the noise matrix has i.i.d. Gaussian entries.
Journal ArticleDOI
Admissible Bayes Character of $T^2-, R^2-$, and Other Fully Invariant Tests for Classical Multivariate Normal Problems
J. Kiefer,R. Schwartz +1 more
TL;DR: In this paper, it was shown that certain procedures, often fully invariant, similar, and/or likelihood ratio, are admissible Bayes procedures for a variety of standard multivariate normal testing problems.
References
More filters
Book
An Introduction to Multivariate Statistical Analysis
TL;DR: In this article, the distribution of the Mean Vector and the Covariance Matrix and the Generalized T2-Statistic is analyzed. But the distribution is not shown to be independent of sets of Variates.
Journal ArticleDOI
Introduction to Multivariate Statistical Analysis.
William G. Madow,T. W. Anderson +1 more
Book
Testing statistical hypotheses
TL;DR: The general decision problem, the Probability Background, Uniformly Most Powerful Tests, Unbiasedness, Theory and First Applications, and UNbiasedness: Applications to Normal Distributions, Invariance, Linear Hypotheses as discussed by the authors.
Journal ArticleDOI