Minimax kernels for density estimation with biased data

This paper considers the asymptotic properties of two kernel estimates % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\], which have been proposed by Bhattacharyya et al. (1988, Comm. Statist. Theory Methods, A17, 3629–3644) and Jones (1991, Biometrika, 78, 511–519), respectively, for estimating the underlying density f at a point under a general selection biased model. The asymptotic optimality of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]is measured by the corresponding asymptotic minimax mean squared errors under a compactly supported Lipschitz continuous family of the underlying densities. It is shown that, in general, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is a superior local estimate than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]in the sense that the asymptotic minimax risk of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]is lower than that of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaacamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E82!\[\tilde f_{_n }\]. The minimax kernels and bandwidths of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciqa-zgagaqcamaaBaaaleaadaWgaaadbaGaa8NBaaqabaaa% leqaaaaa!3E83!\[\hat f_{_n }\]are computed explicity and shown to have simple forms and depend on the weight functions of the model.     

Zero–One Law for some Brownian Functionals

We prove that for a>0, (B _t) one-dimensional standard Brownian motion and ₀=inf{t>0 : B _t=0} the following zero–one law is valid

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

Green's function study of a mixed spin-1 and spin-3/2 Heisenberg ferrimagnetic system

The magnetic properties of a mixed spin-1 and spin-3/2 Heisenberg ferrimagnetic system on a square lattice are investigated by using the double-time temperature-dependent Green's function technique. In order to decouple the higher order Green's functions, Anderson and Callen's decoupling and random phase approximations have been used. The nearest- and next-nearest-neighbor interactions and the single-ion anisotropies are considered and their effects on compensation and critical temperature are studied.     

Γ分布密度函数的性质

分析了Γ分布密度函数的性质,指出了该密度函数与相应参数之间的关系.主要研究第二个参数对密度的影响,证明了β增大时Γ(α,β)分布密度极大值也增大,还指出了β变化时Γ(α,β)分布密度与另一特定密度曲线交点的变化规律.     

范德瓦耳斯气液状态方程纵横谈

首先强调了范氏方程是气液系统的状态方程,范氏方程可以很好地说明物质气态和液态的相互转变．分析了焦-汤效应,简述了气体的液化与低温的获得．     

On the convergence toward nonequilibrium stationary states in thermostatted kinetic models

A differential equation‐based framework is suitable for the modeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments.     

BPR氧化减色法测定过氧化氢亨利常数

利用冷阱收集气-液平衡管出来的气相过氧化氢,以及氯化血红素(Hemin)催化下溴邻苯三酚红(bromopyrogallolred,BPR)被过氧化氢氧化减色的分光光度法检测,测定了温度依赖的过氧化氢亨利常数。在10~35℃的范围内,过氧化氢亨利常数KH(mol·L-1·atm-1)表达为lnKH=a/T-b,其中a=7269±22,b=13·26±0·08,T为温度(K)。过氧化氢溶解焓ΔH为60·43±0·18kJ·K-1·mol-1。     

Can a Circulating Light Beam Produce a Time Machine

In a recent paper, Mallett found a solution of the Einstein equations in which closed timelike curves (CTC’s) are present in the empty space outside an infinitely long cylinder of light moving in circular paths around an axis. Here we show that, for physically realistic energy densities, the CTC’s occur at distances from the axis greater than the radius of the visible universe by an immense factor. We then show that Mallett’s solution has a curvature singularity on the axis, even in the case where the intensity of the light vanishes. Thus it is not the solution one would get by starting with Minkowski space and establishing a cylinder of light.