A family of minimax estimators of a multivariate normal mean

Let X be a p-dimensional normal random vector with unknown mean vector θ and covariance σ²I. Let S/σ², independent of X, be chi-square with n degrees of freedom. Relative to the squared error loss, James and Stein (1961) have obtained an estimator which dominates the usual estimator X. Baranchik (1970) has extended James and Stein's results. We obtain a theorem which can provide a different family of minimax estimators containing James-Stein's estimator. Two interesting minimax estimators are presented in this paper.     

Stochastic integrals of empirical-type processes with applications to censored regression

Motivated by the analysis of linear rank estimators and the Buckley-James nonparametric EM estimator in censored regression models, we study herein the asymptotic properties of stochastic integrals of certain two-parameter empirical processes. Applications of these results on empirical processes and their stochastic integrals to the asymptotic analysis of censored regression estimators are also given.     

Quaternionic multi-S4 instantons in general covariant SU(2) Yang-Mills and HP(n) σ-models

It is shown that the quaternionic maps xⁿ = S⁴ → S⁴ are realized by the n-instanton solutions of two related types of general covariant field theories in four dimensions. They are, respectively, the Einstein-SU(2) Yang-Mills theory with a cosmological term and the new HP(m) σ-models     

Theory and applications of multivariate self-normalized processes

Multivariate self-normalized processes, for which self-normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case), are ubiquitous in statistical applications. In this paper we make use of a technique called “pseudo-maximization” to derive exponential and moment inequalities, and bounds for boundary crossing probabilities, for these processes. In addition, Strassen-type laws of the iterated logarithm are developed for multivariate martingales, self-normalized by their quadratic or predictable variations.     

Complex and quaternionic analyticity in chiral and gauge theories,I

A comparative and systematic study is made of 2-dimensional CP(n) σ-models and new 4-dimensional HP(n) σ-models and their respective embedded U(1) and Sp(1) holonomic gauge field structures. The central theme is complex versus quaternionic analyticity. A unified formulation is achieved by way of Cartan's method of moving frames adapted to the hypercomplex geometries of the harmonic symmetric spaces $\text{CP(n) ≈ SU(n + 1)}\text{SU(n) × U(1)}$ and $\text{HP(n) ≈ Sp(n + 1)}\text{Sp(n) × Sp(1)}$ respectively. Elements of complex Kähler manifolds are applied to a detailed analysis of the CP(n) σ-model and its instanton sector. Generalization to any Kählerian σ-model is manifest. On the basis of Cauchy-Riemann analyticity, Kählerian models are shown to have an infinite number of local continuity equations. In a parallel manner, new 4-dimensional conformally invariant HP(n) σ-models are constructed. Focus is on the latter's hidden local gauge invariance in their holonomy group Sp(n) × Sp(1) which allows a natural embedding of the Sp(1) ≈ SU(2) pure Yang-Mills theory. The associated quaternionic structure is discussed in light of both quaternionic quantum mechanics and Kählerian geometry. In this chiral setting, the SU(2) Yang-Mills duality equations are cast into quaternionic Cauchy-Riemann equations over S⁴ ≈ HP(1), the conformal spacetime. In analogy to the CP(n) case, their rational solutions are the most general (8n ? 3) parameter instantons where the associated algebraic nonlinear equations of the type of Atiyah, Drinfeld, Hitchin, and Manin are now expressed in a new conformally invariant form. Geometrically, the SU(2) instantons solve the Frenet-Serret equations for quaternionic holomorphic curves; they are conformal maps from HP(1) into HP(n) with n their second Chern index. Fueter's quaternionic analysis is presented, then applied: Fueter functions are particularly suited for the solutions of 't Hooft, of Jackiw, Nohl and Rebbi, and of Witten and Peng, as well as the self-dual finite action per unit time solution of Bogomol'nyi, Prasad and Sommerfield. Generalizing the latter, a new solution with unit Chern index and finite action per unit spacetime cell is found. It is expressed in terms of the quaternionic fourfold quasi-periodic Weierstrass Zeta function. Finally the essence of our method is revealed in terms of universal connections over Stiefel bundles; generalization to real, complex and quaternionic classifying Grassmanian σ-models with their embedded SO(m), SU(m) and Sp(m) gauge fields is outlined in terms of gauge invariant projector valued chiral fields. Other outstanding problems are briefly discussed.     

Study of a molybdenum dithiocarboxylato complex. Chemical characterization,crystal and molecular structure of tetrakis(α-dithionaphthoato)molybdenum(IV)

Summary The complex Mo(-C₁₀H₇CS₂)₄ has been prepared from several starting materials: [Mo(NNHPh)₂(butane-2,3-diolate)₂] · H₂NNHPh, [MoCl(NNMe₂)₂(PPh₃)₂]⁺Cl^–, [MoO₂ (MeCHOCHOHMe)₂] · 2MeCHOHCHOHMe or [(C₄H₉)₄N]₆Mo₇O₂₄, reacted with-C₁₀H₇CS₂H. The complex Mo(-C₁₀H₇CS₂)₄ crystallizes from CH₂Cl₂-Et₂O in the orthorhombic space group Pbca,a=13.655(3)Å,b=21.707(5)Å,c= 26.365(2)Å, V=7814.81ų, Z=8; 5075 reflections collected, 2723 used in solution to give R=0.0517, R_w=0.0564. The molybdenum is eight coordinate with geometry approximating to aD _2d dodecahedron. The average Mo-S bond lengths are 2.526(3)Å and 2.479(3)Å for the dodecahedrala andb sites, respectively.     

Role of Nitrite on Nitration of 2′‐Deoxyguanosine by Nitryl Chloride

Nitryl chloride and peroxynitrite are reactive nitrogen species generated by activated phagocytes against invading pathogens during infections and inflammation. In our previous report, formation of 8‐nitroxanthine and 8‐nitroguanine was observed in reaction of 2′‐deoxyguanosine or calf thymus DNA with nitryl chloride generated by mixing hypochlorous acid (HOCl) with nitrite (NC₂^?). The present study investigates factors control ling the yields of 8‐nitroxanthine and 8‐nitroguanine formation in nitration of 2′‐deoxyguanosine by nitryl chloride. We found that the yields of 8‐nitroxanthine and 8‐nitroguanine in reaction of 2′‐deoxyguanosine with nitryl chloride were highly dependent on the ratio of NO₂^? versus HOCl concentration. The yields of 8‐nitroxanthine and 8‐nitroguanine reached a plateau when the ratio of NC₂^? versus HOCl concentration was higher than 2. A possible mechanism was postulated to explain this observation. While 8‐nitroguanine is not stable in the presence of peroxynitrite, 8‐nitroxanthine is sensitive to HOCl. The stability of these two nitrated ad ducts might be a factor on their final yields in this reaction. Since HOCl is produced by neutrophils at sites of inflammation where the level of NC₂^? is elevated, it is conceivable that nitryl chloride contributes to DNA base nitration in vivo, forming 8‐nitroxanthine and 8‐nitroguanine.     

Synthesis and characterization of nonaqueous dispersion particles with photolabile α‐heptadecylphenacyl ester stabilizer chains

We describe a new method for the synthesis of core–shell photolabile nanoparticles. The synthesis begins with the batch emulsion copolymerization of n‐butyl methacrylate (BMA) and ethylene glycol dimethacrylate to form small (20‐nm‐diameter) crosslinked particles with a narrow size distribution. These seeds are then used for a second‐stage emulsion copolymerizations in which BMA and various polar monomers, including methacrylic acid, are added under monomer‐starved conditions. After characterization of the particles, they are transferred to an N,N‐dimethylformamide solution. The cesium salt of the carboxylic acid groups is reacted with 2‐bromo‐1‐phenyl‐octadecan‐1‐one to convert various fractions of the ? COOH groups to the corresponding 2‐benzoylheptadecyl ester groups. These aliphatic ester groups render the surface sufficiently hydrophobic that the particles can be dispersed in common aliphatic hydrocarbons solvents to yield colloidal dispersions, sterically stabilized by the dangling aliphatic chains. Ester groups with a phenyl ketone attached to the β‐carbon are photolabile. Irradiation of the particles with UV light detaches the sterically stabilizing chains from the particle and transforms the surface groups back to COOH groups. This leads to flocculation of the particles. The emphasis in this article is on the optimization of the particle synthesis and the characterization of the particles obtained. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 2642–2657, 2001