密度的求导插值估计

密度估计在应用上有重要意义.密度估计方法有直方图估计、核估计、最近邻估计等.本文提出一种新的密度估计方法,对连续经验分布函数求导插值而求密度估计.经验分布函数就是     

核密度估计在预测风险价值中的应用

通过研究核密度估计理论,提出了一种适应估计金融时间序列分布的L ap lace核密度函数.在单变量核密度估计的基础上建立了风险价值(V a lua at R isk,简记为VaR)预测的预测模型.通过对核密度估计变异系数的加权处理建立了两种加权VaR预测模型.最后,通过上证指数收益率对建立的VaR预测模型进行了实证分析,结果显示两种加权方法对上证指数收益率的VaR预测具有较高的效率.     

截尾情形下随机窗宽核密度估计

本文研究了截尾情形下随机窗宽核密度估计.在关于随机窗宽的较弱的条件下,我们得到了精确的收敛速度及渐近分布.这些结果与 Diehl 和 Stute(1988)的关于非随机窗宽核密度估计的结果相一致.     

商业银行操作风险的度量——基于非参数方法

在损失分布方法的基础上,本文基于非参数方法对商业银行操作风险的度量进行了研究。非参数方法对损失额的分布不作过多的设定,避免了由于分布误设可能出现的偏差。古典的核密度估计对损失额拟合的效果不太好,特别是尾部的拟合效果更差。变换后的核密度估计的拟合效果比古典的核密度估计改善很多.基于变换后的核密度估计对商业银行操作风险损失度量可以得到不同置信水平的VaR与ES,并且不同置信水平的差距比较大。基于非参数与基于参数方法得到的各个置信水平的VaR与ES有一定差距。     

具有核的态射的 w -加权Drazin逆

该文中, a: X→Y, w: Y→ X为加法范畴 £ 中的态射, k₁: K ₁→X是(aw)ⁱ 的核, k₂: K₂ →Y是(wa)^j 的核. 那么下列命题等价: (1) a 在 £ 中有w -加权Drazin逆a _d,w; (2) ₁:X→ L₁是(aw)ⁱ 的上核,k_{1 1}(aw)ⁱ⁺¹}+ ₁(k_{1 1})^-1k₁是可逆的; (3) ₂: Y→ L₂是(wa)^j 的上核, k_{2 2}和(wa)^j+1+ ₂(k_{2 2})^-1k₂是可逆的. 作者又研究了具有{1} -逆的正合加法范畴中态射的w -加权Drazin逆的柱心幂零分解, 证明了其存在性. 作者把具有核的态射的Drazin逆及其柱心幂零分解推广到具有核的态射的w -加权 Drazin逆及其柱心幂零分解, 并给出了表达式.     

非晶态合金晶化过程中的形核及核长大激活能——Ⅰ.计算方法

本文从非晶态合金晶化动力学理论出发,结合非晶态Ni-P合金晶化过程中的阶段激活能 E_c(x)及阶段Avrami指数n(x)随晶化转变分数的变化关系等实验结果,提出了一种计算形核激活能(E_n)及核长大激活能(E_g)的新方法.对非晶态Ni-P合金晶化过程中E_n和E_g的计算结果表明这种方法实验过程简单,所得结果精确.     

独立样本核密度估计的r阶均方相合性及收敛速度

本文旨在讨论并改进独立样本核密度估计的大样本性质．本文在改进Parzen(1962)给出的一个基本引理的基础上得出了独立样本核密度估计的r阶均方及一致均方相合性和收敛速度．此结果是文献[1]中独立样本核密度估计均方相合性的推广．     

非晶态合金晶化过程中的形核及核长大激活能——Ⅱ.与非晶态结构和温度的关系

本文利用第一部分提出的计算形核及核长大激活能(E_n和E_g)的方法分别测算了预退火热处理后非晶态Ni-P合金样品以及在不同温度区间内激冷非晶态Ni-P合金样品的E_n和E_g值.结果表明E_n和E_g值与非晶态合金微观结构和温度有密切的关系,非晶态合金有序程度越高,E_n和E_g值越小;晶化时的温度越高,E_n和E_g值越大.运用新近提出的晶化微观机制对这种用经典晶化机制难以解释的实验现象进行了合理的解释.     

太阳耀斑核块中的电流

本文利用1980年4月6日AR2372活动区中1B/M1级耀斑的高分辨H_α偏带单色像和真正同时性观测的向量磁场资料,研究了耀斑核块在闪相期间的发展及其与磁图和纵向电流图中特征的关系,得到了如下主要结论: 1.发生于20:53UT的耀斑开始由5个核块组成,这些核块不断增亮,于21:00UT同时达到极大,然而它们的寿命并不相同(图2). 2.耀斑起始核块处在磁场中性线附近,对应于Hagyard等所定义和计算的最大剪切区域(图3). 3.耀斑起始核块的位置与Krall等所计算的纵向电流密度极大位置一致(图4). 4.耀斑前夕和耀斑期间的活动区磁场是高度剪切的.自开始至耀斑极大期间,纵向磁场B_∥和横向磁场B_⊥均未发生实质性变化. 5.1980年4月6—7日,AR2372活动区中至少存在10个联结不同磁极性的H_α环系(图5).其中4个位于比较活动的南部区域.这4个H_α环系的位形与Machado等从X光观测推测出的环系一致.     

表面活性剂在水溶液中自聚集的NMR研究

用质子化学移位 ,自扩散系数和核磁弛豫方法研究了正十二烷基磺酸钠(SDSN) ,正十六烷基三甲基溴化铵 (CTAB)和TritonX 1 0 0等 3种类型表面活性剂及其在溶液中生成的胶团的动态行为 .3种体系的化学移位 ,自扩散系数和自旋 晶格弛豫时间 (T₁) ,自旋 自旋弛豫时间 (T₂ )均在临界胶团浓度 (cmc)附近有明显的转折点 ,并在 5～ 1 0倍cmc左右时趋向平衡 .T₁和T₂值的测量表明TritonX 1 0 0分子在溶液中的浓度低于cmc时进行着快速的各向同性运动 .生成胶团后 ,胶团核中的疏水碳氢链紧密堆积使其CH₂ 和CH₃ 基团进行着偏离极限窄化条件的运动 .而胶团核外的亲水醚氧长链则进行着相对自由的运动 .但是由于胶团中的分子密集 ,亲水长链产生局部缠结引起T1和T2 值变短 .SDSN和CTAB生成的胶团核内的碳氢链的状态和TritonX 1 0 0的相同 .     

Application of fast spherical Fourier transform to density estimation

This paper is on density estimation on the 2-sphere, S², using the orthogonal series estimator corresponding to spherical harmonics. In the standard approach of truncating the Fourier series of the empirical density, the Fourier transform is replaced with a version of the discrete fast spherical Fourier transform, as developed by Driscoll and Healy. The fast transform only applies to quantitative data on a regular grid. We will apply a kernel operator to the empirical density, to produce a function whose values at the vertices of such a grid will be the basis for the density estimation. The proposed estimation procedure also contains a deconvolution step, in order to reduce the bias introduced by the initial kernel operator. The main issue is to find necessary conditions on the involved discretization and the bandwidth of the kernel operator, to preserve the rate of convergence that can be achieved by the usual computationally intensive Fourier transform. Density estimation is considered in L²(S²) and more generally in Sobolev spaces H_v(S²), any v?0, with the regularity assumption that the probability density to be estimated belongs to H_s(S²) for some s>v. The proposed technique to estimate the Fourier transform of an unknown density keeps computing cost down to order O(n), where n denotes the sample size.     

Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous

Summary Asymptotic properties of the mean integrated squared error (MISE) of kernel estimators of a density function, based on a sampleX ₁, …,X _n, were obtained by Rosenblatt [4] and Epanechnikov [1] for the case when the densityf and its derivativef′ are continuous. They found, under certain additional regularity conditions, that the optimal choiceh _n0 for the scale factorh _n=Kn^−α is given byh _n0=K₀n^−1/5 withK ₀ depending onf and the kernel; they also showed that MISE(h _n0)=O(n^−4/5) and Epanechnikov [1] found the optimal kernel. In this paper we investigate the robustness of these results to departures from the assumptions concerning the smoothness of the density function. In particular it is shown, under certain regularity conditions, that whenf is continuous but its derivativef′ is not, the optimal value of α in the scale factor becomes 1/4 and MISE(h _n0)=O(n^−3/4); for the case whenf is not continuous the optimal value of α becomes 1/2 and MISE(h _n0)=O(n^−1/2). For this last case the optimal kernel is shown to be the double exponential density. Supported by the Natural Sciences and Engineering Research Council of Canada under Grant Nr. A 3114 and by the Gouvernement du Québec, Programme de formation de chercheurs et d'action concertée.     

Density estimation in the simple proportional hazards model

In the simple proportional hazards model of random right censorship the limiting variance v_ACL²(x) at x of the kernel density estimator based on the Abdushukurov-Cheng-Lin estimator is shown to be equal to the corresponding variance pertaining to the Kaplan-Meier estimator times the expected proportion p of uncensored observations. More surprisingly, for appropriate p, v_ACL²(x) is smaller than the asymptotic variance of the classical kernel estimator based on a complete sample, for any x below the (1 − e⁻¹)-quantile.     

Large deviations and moderate deviations for kernel density estimators of directional data

Let fn be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere Sd-1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {sup x∈sd-1 ｜fn（x） - E（fn（x））｜, n ≥ 1} hold.     

Density Estimation on the Stiefel Manifold

This paper develops the theory of density estimation on the Stiefel manifoldV_{k, m}, whereV_{k, m}is represented by the set ofm×kmatricesXsuch thatX′X=I_k, thek×kidentity matrix. The density estimation by the method of kernels is considered, proposing two classes of kernel density estimators with small smoothing parameter matrices and for kernel functions of matrix argument. Asymptotic behavior of various statistical measures of the kernel density estimators is investigated for small smoothing parameter matrix and/or for large sample size. Some decompositions of the Stiefel manifoldV_{k, m}play useful roles in the investigation, and the general discussion is applied and examined for a special kernel function. Alternative methods of density estimation are suggested, using decompositions ofV_{k, m}.     

Probability Density Function Estimation Using Gamma Kernels

We consider estimating density functions which have support on [0, ) using some gamma probability densities as kernels to replace the fixed and symmetric kernel used in the standard kernel density estimator. The gamma kernels are non-negative and have naturally varying shape. The gamma kernel estimators are free of boundary bias, non-negative and achieve the optimal rate of convergence for the mean integrated squared error. The variance of the gamma kernel estimators at a distance x away from the origin is O(n ^–4/5 x ^–1/2) indicating a smaller variance as x increases. Finite sample comparisons with other boundary bias free kernel estimators are made via simulation to evaluate the performance of the gamma kernel estimators.     

On the risk of estimates for block decreasing densities

A density f=f(x₁,…,x_d) on [0,∞)^d is block decreasing if for each j∈{1,…,d}, it is a decreasing function of x_j, when all other components are held fixed. Let us consider the class of all block decreasing densities on [0,1]^d bounded by B. We shall study the minimax risk over this class using n i.i.d. observations, the loss being measured by the L₁ distance between the estimate and the true density. We prove that if S=log(1+B), lower bounds for the risk are of the form C(S^d/n)^1/(d+2), where C is a function of d only. We also prove that a suitable histogram with unequal bin widths as well as a variable kernel estimate achieve the optimal multivariate rate. We present a procedure for choosing all parameters in the kernel estimate automatically without loosing the minimax optimality, even if B and the support of f are unknown.     

Nonparametric density estimation for linear processes with infinite variance

We consider nonparametric estimation of marginal density functions of linear processes by using kernel density estimators. We assume that the innovation processes are i.i.d. and have infinite-variance. We present the asymptotic distributions of the kernel density estimators with the order of bandwidths fixed as h =  cn ^−1/5, where n is the sample size. The asymptotic distributions depend on both the coefficients of linear processes and the tail behavior of the innovations. In some cases, the kernel estimators have the same asymptotic distributions as for i.i.d. observations. In other cases, the normalized kernel density estimators converge in distribution to stable distributions. A simulation study is also carried out to examine small sample properties.     

Estimating and Visualizing Conditional Densities

Abstract

We consider the kernel estimator of conditional density and derive its asymptotic bias, variance, and mean-square error. Optimal bandwidths (with respect to integrated mean-square error) are found and it is shown that the convergence rate of the density estimator is order n ^–2/3. We also note that the conditional mean function obtained from the estimator is equivalent to a kernel smoother. Given the undesirable bias properties of kernel smoothers, we seek a modified conditional density estimator that has mean equivalent to some other nonparametric regression smoother with better bias properties. It is also shown that our modified estimator has smaller mean square error than the standard estimator in some commonly occurring situations. Finally, three graphical methods for visualizing conditional density estimators are discussed and applied to a data set consisting of maximum daily temperatures in Melbourne, Australia.     

One Initial Boundary-Value Problem for Integro-Differential Equation of the Second Order With Power Nonlinearity

In the Sobolev space W _∞ ² (?⁺) we investigate one initial boundary-value problem for integro-differential equation of the second order with power nonlinearity on a semi-axis. Assuming that summary-difference even kernel serves for the considered kernel as minorant in the sense of M. A. Krasnosel’skii, we prove the existence of a nonnegative (nontrivial) solution in the Sobolev spaceW _∞ ² (?⁺). We also calculate the limits of constructed solution at infinity.