Corpus ID: 222177369

High dimensional asymptotics of likelihood ratio tests in Gaussian sequence model under convex constraint

@article{Han2020HighDA,
  title={High dimensional asymptotics of likelihood ratio tests in Gaussian sequence model under convex constraint},
  author={Qiyang Han and Bodhisattva Sen and Yandi Shen},
  journal={arXiv: Statistics Theory},
  year={2020}
}
In the Gaussian sequence model $Y=\mu+\xi$, we study the likelihood ratio test (LRT) for testing $H_0: \mu=\mu_0$ versus $H_1: \mu \in K$, where $\mu_0 \in K$, and $K$ is a closed convex set in $\mathbb{R}^n$. In particular, we show that under the null hypothesis, normal approximation holds for the log-likelihood ratio statistic for a general pair $(\mu_0,K)$, in the high dimensional regime where the estimation error of the associated least squares estimator diverges in an appropriate sense… Expand

Figures from this paper

References

SHOWING 1-10 OF 85 REFERENCES
A general method for power analysis in testing high dimensional covariance matrices
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 matrixExpand
Second-order Stein: SURE for SURE and other applications in high-dimensional inference
Stein's formula states that a random variable of the form $z^\top f(z) - {\rm{div}} f(z)$ is mean-zero for all functions $f$ with integrable gradient. Here, ${\rm{div}} f$ is the divergence of theExpand
Convex Regression in Multidimensions: Suboptimality of Least Squares Estimators
The least squares estimator (LSE) is shown to be suboptimal in squared error loss in the usual nonparametric regression model with Gaussian errors for $d \geq 5$ for each of the following families ofExpand
Estimation of the $l_2$-norm and testing in sparse linear regression with unknown variance.
We consider the related problems of estimating the $l_2$-norm and the squared $l_2$-norm in sparse linear regression with unknown variance, as well as the problem of testing the hypothesis that theExpand
On a phase transition in general order spline regression
In the Gaussian sequence model $Y= \theta_0 + \varepsilon$ in $\mathbb{R}^n$, we study the fundamental limit of approximating the signal $\theta_0$ by a class $\Theta(d,d_0,k)$ of (generalized)Expand
Isotonic regression in general dimensions
We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^d$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed,Expand
Limit distribution theory for block estimators in multiple isotonic regression
We study limit distributions for the tuning-free max-min block estimator originally proposed in [FLN17] in the problem of multiple isotonic regression, under both fixed lattice design and randomExpand
Minimax Rate of Testing in Sparse Linear Regression
TLDR
It is shown that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form sqrt((s/n) log(1 + sqrt(p)/s)). Expand
Samworth, Isotonic regression in general dimensions
  • Ann. Statist
  • 2019
The geometry of hypothesis testing over convex cones: Generalized likelihood tests and minimax radii
TLDR
This work provides a sharp characterization of the GLRT testing radius up to a universal multiplicative constant in terms of the geometric structure of the underlying convex cones, and proves information-theoretic lower bounds for minimax testing radius again in Terms of geometric quantities. Expand
...
1
2
3
4
5
...