Statistical Limits of Adaptive Linear Models: Low-Dimensional Estimation and Inference

Estimation and inference in statistics pose significant challenges when data are collected adaptively. Even in linear models, the Ordinary Least Squares (OLS) estimator may fail to exhibit asymptotic normality for single coordinate estimation and have inflated error. This issue is highlighted by a recent minimax lower bound, which shows that the error of estimating a single coordinate can be enlarged by a multiple of $\sqrt{d}$ when data are allowed to be arbitrarily adaptive, compared with the case when they are i.i.d. Our work explores this striking difference in estimation performance between utilizing i.i.d. and adaptive data. We investigate how the degree of adaptivity in data collection impacts the performance of estimating a low-dimensional parameter component in high-dimensional linear models. We identify conditions on the data collection mechanism under which the estimation error for a low-dimensional parameter component matches its counterpart in the i.i.d. setting, up to a factor that depends on the degree of adaptivity. We show that OLS or OLS on centered data can achieve this matching error. In addition, we propose a novel estimator for single coordinate inference via solving a Two-stage Adaptive Linear Estimating equation (TALE). Under a weaker form of adaptivity in data collection, we establish an asymptotic normality property of the proposed estimator.Comment: This paper is accepted at NeurIPS 202

(1,10-Phenanthroline-κ2 N,N′)(triphenyl­phosphine-κP)silver(I) trifluoro­methane­sulfonate

The structure of the title complex, [Ag(C12H8N2)(C18H15P)]CF3SO3, is based on a distorted trigonal–planar N2P coordination of the AgI ion, provided by two N atoms of the bidentate phenanthroline ligand and one P atom of the triphenyl­phosphine ligand. The phenanthroline ligand and one phenyl ring of the triphenyl­phosphine ligand almost lie in one plane (maximum deviation = 0.014 Å from the best planes). The crystal structure may be stabilized by an inter­molecular C—H⋯O hydrogen bond between the phenanthroline ligand and the O atom of the trifluoro­methane­sulfonate anion

Tetra­aqua­(1,10-phenanthroline-κ2 N,N′)cadmium(II) sulfate dihydrate

In the title mononuclear complex, [Cd(C12H8N2)(H2O)4]SO4·2H2O, the coordination geometry around the CdII atom is a distorted octa­hedron, with two aqua ligands occupying the axial positions. Inter­molecular O—H⋯O hydrogen bonds lead to the formation of a two-dimensional layer structure parallel to (001). The layers are connected by π–π inter­actions between the pyridyl and benzene rings of the phenanthroline ligands [centroid–centroid distances = 3.591 (1) and 3.610 (1) Å]

5,6-Dimethyl-1,2,4-triazin-3-amine

In the crystal structure of the title compound, C5H8N4, adjacent mol­ecules are connected through N—H⋯N hydrogen bonds, resulting in a zigzag chain along [100]. The amino groups and heterocyclic N atoms are involved in further N—H⋯N hydrogen bonds, forming R 2 2(8) motifs

A tetra­silver(I)ditungsten(VI) cluster with sulfide and bis­(diphenyl­phosphino)methane ligands

The asymmetric unit of the title complex, [Ag4W2S8(C25H22P2)3]·2C3H7NO, tris­[μ2-bis­(diphenyl­phosphino)meth­ane]-3:6κ2 P:P′;4:5κ2 P:P′;5:6κ2 P:P′-μ5-sulfido-2:3:4:5:6κ5 S-μ3-sulfido-1:3:4κ3 S-tetra-μ2-sulfido-1:3κ2 S;1:4κ2 S;2:5κ2 S;2:6κ2 S-disulfido-1κS,2κS-tetra­silver(I)ditungsten(VI) N,N-dimethyl­formamide disolvate, contains two [WS4]2− anions, four silver cations, three bidentate–bridging bis­(diphenyl­phosphino)methane (dppm) ligands and two N,N-dimethyl­formamide (DMF) solvent mol­ecules. The coordination geometry of each Ag atom is distorted tetra­hedral. Two Ag ions are coordinated by μ2-S and μ5-S atoms, and by two P atoms from two dppm ligands, while the other two Ag atoms are coordinated by μ2-S, μ3-S and μ5-S atoms, and by one P atom from a dppm ligand

catena-Poly[[(isoquinoline-κN)(triphenylphospane-κP)copper(I)]-μ-thio­cyanato-κ2 N:S]

In the title coordination compound, [Cu(NCS)(C9H7N)(C18H15P)]n, the CuI atom is tetra­hedrally coordinated by one N atom from an isoquinoline ligand, one P atom from a triphenyl­phospane ligand, and one N and one S atom from two thio­cyanate anions. The thio­cyanide anions bridge the CuI atoms into a chain along [100]. π–π inter­actions between the pyridine and benzene rings of the isoquinoline ligands connect the chains [centroid-to-centroid distance = 3.722 (3) Å]

The environments of Type Ia supernovae with different relative equivalent width of Si II feature in their spectra

Although type Ia supernovae are so important in many astrophysical field, e.g. in cosmology, their explosion mechanism and progenitor system are still unclear. In physics, the relative equivalent width (REW) of the Si II 635.5 nm absorption feature reflects the velocity interval of silicon in the supernova ejecta and then may provide constraints on the explosion mechanism of SNe Ia. In this paper, we divide the SNe Ia into broad line (BL) and normal line (NL) subsamples based on their REW of Si II 635.5 nm absorption lines around maximum light, and find that the BL SNe Ia have a dimmer mean brightness than NL ones, which possibly results from their different metallicities. However, based on the pixel statistics study on the environments of two subsamples, we do not find any significant potential difference on the environments between BL and NL SNe Ia, which implies that the explosion mechanism of SNe Ia could be independent of their progenitor populations.Comment: 17 pages, 13 figures, accepted for publication in Ap