Statistical and machine learning theory has developed several conditions ensuring that popular estimators such as the Lasso or the Dantzig selector perform well in high-dimensional sparse regression, including the restricted eigenvalue, compatibility, and $\ell _q$ sensitivity properties. However, some of the central aspects of these conditions are not well understood. For instance, it is unknown if these conditions can be checked efficiently on any given dataset. This is problematic, because they are at the core of the theory of sparse regression. Here we provide a rigorous proof that these conditions are NP-hard to check. This shows that the conditions are computationally infeasible to verify, and raises some questions about their practical applications. However, by taking an average-case perspective instead of the worst-case view of NP-hardness, we show that a particular condition, $\ell _q$ sensitivity, has certain desirable properties. This condition is weaker and more general than the others. We show that it holds with high probability in models where the parent population is well behaved, and that it is robust to certain data processing steps. These results are desirable, as they provide guidance about when the condition, and more generally the theory of sparse regression, may be relevant in the analysis of high-dimensional correlated observational data.

There has been a rapid progress in designing valid and effective statistical hypothesis tests for the order of a finite mixture model. In particular, EM-test for the order of the mixture model has been developed and found effective when the component distribution contains a single parameter. EM-test is found to be particularly effective and elegant for the order of normal mixture in both mean and variance. The idea behind EM-test has been found widely applicable. In this paper, we investigate the use of EM-test for the order of a finite normal mixture in the mean parameter with equal but unknown component variances. We show that for any positive integer $m_0 \ge 2,$ the limiting distribution of the EM-test for the order of $m_0$ against the higher order alternative is $\chi ^2_{m_0-1}.$ A genetic example is used to illustrate the application of the EM-test.

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F and G, the two nonzero generalized derivations of R, I an ideal of R and $f(x_1,\ldots ,x_n)$ a multilinear polynomial over C which is not central valued on R. If

for all $x_1,\ldots ,x_n \in I$, then one of the followings holds: (1) there exist $a,b\in U$ such that $F(x)=ax$ and $G(x)=bx$ for all $x\in R$ with $ab=0$; (2) there exist $a,b,p\in U$ such that $F(x)=ax+xb$ and $G(x)=px$ for all $x\in R$ with $F(p)=0$ and $f(x_1,\ldots ,x_n)^2$ is central valued on R. We also obtain some related results in cases where R is a semiprime ring and Banach algebra.

Let R be a commutative ring with nonzero identity. In this article, we introduce the notion of 2-absorbing quasi-primary ideal which is a generalization of quasi-primary ideal. We define a proper ideal I of R to be 2-absorbing quasi primary if $\sqrt{I}$ is a 2-absorbing ideal of R. A number of results concerning 2-absorbing quasi-primary ideals and examples of 2-absorbing quasi-primary ideals are given.

In this paper the compositions $(x_+^\mu )_-^{{-}s},\, (x_+^\mu )_+^{{-}s},\, (|x|^\mu )_-^{{-}s}$ and $(|x|^\mu )_+^{{-}s}$ of distributions $x_+^\mu ,\,|x|^\mu $ and $x^{{-}s}$ are considered. They are defined via neutrix calculus for $\mu >0, \, s=1,\,2,\ldots $ and $\mu s\in {\mathbb {Z}}^+.$ In addition, the composition of $x^{{-}s}\ln |x|$ and $x_+^r$ is also defined for $r,\,s\in {\mathbb {Z}}^+.$

In this paper, we prove that every $*$-Lie derivable mapping on a von Neumann algebra with no central abelian projections can be expressed as the sum of an additive $*$-derivation and a mapping with image in the center vanishing at commutators.