On the modal resolution of kernel density estimators

The ability of a kernel density estimator to resolve modes of the underlying density is investigated. For various bimodal densities and three different kernels, the smallest sample size required for the expectation of an optimally smoothed kernel estimator to be bimodal is determined. The optimality criterion employed is equivalent to asymptotic mean integrated squared error for sufficiently smooth densities.     

Estimation of parameters in a two-parameter exponential distribution using ranked set sample

In situations where the experimental or sampling units in a study can be easily ranked than quantified, McIntyre (1952,Aust. J. Agric. Res.,3, 385–390) proposed that the mean ofn units based on aranked set sample (RSS) be used to estimate the population mean, and observed that it provides an unbiased estimator with a smaller variance compared to a simple random sample (SRS) of the same sizen. McIntyre's concept ofRSS is essentially nonparametric in nature in that the underlying population distribution is assumed to be completely unknown. In this paper we further explore the concept ofRSS when the population is partially known and the parameter of interest is not necessarily the mean. To be specific, we address the problem of estimation of the parameters of a two-parameter exponential distribution. It turns out that the use ofRSS and its suitable modifications results in much improved estimators compared to the use of aSRS.     

有限总体中基于广义Liu 估计的预测及其小样本性质

在有限总体中提出了一类基于广义Liu估计的新的预测,得到了基于广义Liu估计的预测在预测均方误差意义下优于最优线性无偏预测的充要条件,并通过实例对理论成果进行了进一步的说明.     

Asymptotic risk behavior of mean vector and variance estimators and the problem of positive normal mean

Asymptotic risk behavior of estimators of the unknow variance and of the unknown mean vector in a multivariate normal distribution is considered for a general loss. It is shown that in both problems this characteristic is related to the risk in an estimation problem of a positive normal mean under quadratic loss function. A curious property of the Brewster-Zidek variance estimator of the normal variance is also noticed.Research supported by NSF Grant DMS 9000999 and by Alexander von Humboldt Foundation Senior Distinguished Scientist Award.University of Münster     

认购证发行总数的估计

利用发行的认购证有编号１，２，……，Ｎ，，本文通过随机抽取的几个认购证号码，给出了认购证发行总数的最大似然估计，一致最小方差无偏估计和Ｂａｙｅｓ估计。     

Existence of the uniformly minimum risk equivariant estimators of parameters in a class of normal linear models

In this paper, we study the existence of the uniformly minimum risk equivariant (UMRE) estimators of parameters in a class of normal linear models, which include the normal variance components model, the growth curve model, the extended growth curve model, and the seemingly unrelated regression equations model, and so on. The necessary and sufficient conditions are given for the existence of UMRE estimators of the estimable linear functions of regression coefficients, the covariance matrixV and (trV)^α, where α > 0 is known, in the models under an affine group of transformations for quadratic losses and matrix losses, respectively. Under the (extended) growth curve model and the seemingly unrelated regression equations model, the conclusions given in literature for estimating regression coefficients can be derived by applying the general results in this paper, and the sufficient conditions for non-existence of UMRE estimators ofV and tr(V) are expanded to be necessary and sufficient conditions. In addition, the necessary and sufficient conditions that there exist UMRE estimators of parameters in the variance components model are obtained for the first time.     

Simultaneous optimal estimates of fixed effects and variance components in the mixed model

For a general linear mixed model with two variance components, a set of simple conditions is obtained, under which, (i) the least squares estimate of the fixed effects and the analysis of variance (ANOVA) estimates of variance components are proved to be uniformly minimum variance unbiased estimates simultaneously; (ii) the exact confidence intervals of the fixed effects and uniformly optimal unbiased tests on variance components are given; (iii) the exact probability expression of ANOVA estimates of variance components taking negative value is obtained.     

指数分布抽样基本定理及在三参数一般指数分布参数估计中的应用

讨论三参数一般指数分布的参数估计,首先讨论了三参数一般指数分布参数的最大似然估计的求解问题,当其中参数α=1时,应用指数分布抽样基本定理,得到了三参数一般指数分布其它参数的一致最小方差无偏估计;并且由此给出求解三参数一般指数分布参数最大似然估计的迭代方法,得到了三参数一般指数分布参数最大似然估计的近似值,给出了模拟结果以说明迭代方法的收敛性;并以相关文献的观察数据作为样本,得到了三参数一般指数分布的参数估计,从而说明了迭代方法的有效性.     

秩亏线性模型中的一个新的线性有偏估计类

本文基于最优线性最小偏差估计的谱分解,定义了秩亏线性模型未知参数的一个新的线性有偏估计类,并讨论了它的许多重要性质,通过选取偏参数的适当形式,构造了许多很有意义的线性有偏估计,最后,给出了一个算例。     

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.     

A class of estimators with estimated optimum values in sample surveys

Srivastava and Jhajj (1981) proposed a class of estimators for population mean of a character using auxiliary information and optimum values involving unknown parameters. From the practical point of view, their results have very little utility. In view of practical utility, we propose a class of estimators with estimated optimum values. Further, it is shown that the proposed class with estimated optimum values attains the same minimum mean square error of the class of estimators based on optimum values.     

最小二乘估计精度及其界限

Puntanen[1]提出用均方误差来度量最小二乘估计的精度,以后Styan[2],Rao[3]等相继讨论了这种精度及其界限.本文考虑采用广义方差,从而引进了一种新的最小二乘估计精度的度量并讨论了它的界.     

Estimation of a common mean of several univariate inverse Gaussian populations

The problem of estimating the common mean of k independent and univariate inverse Gaussian populations IG(, _i ), i=1,..., k with unknown and unequal 's is considered. The difficulty with the maximum likelihood estimator of is pointed out, and a natural estimator % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] of along the lines of Graybill and Deal is proposed. Various finite sample properties and some decision-theoretic properties of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acciGaf8hVd0MbaGaaaaa!3D38!\[\tilde \mu \] are discussed.This research was partially supported by research grants #A3661 and #A3450 from NSERC of Canada.     

带有不等式约束的生长曲线模型中线性估计的容许性

For the growth curve model with respect to inequality restriction: Y =XBZ ε,→ε～ (0,σ2V (×) I),trNB ≥ 0,this paper gives some necessary and sufficient conditions for the linear estimator of KBL to be admissible in the class of homogeneous linear estimators LH and nonhomogeneous linear estimators LI,respectively,under the quadratic loss function tr(d(Y) - KBL)'(d(Y) - KBL).     

A Class of Linear Biased Estimators of Regression ParameterMatrix in the Growth Curve Model

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Operational Variants of the Minimum Mean Squared Error Estimator in Linear Regression Models with Non-Spherical Disturbances

There is a good deal of literature that investigates the properties of various operational variants of Theil's (1971, Principles of Econometrics, Wiley, New York) minimum mean squared error estimator. It is interesting that virtually all of the existing analysis to date is based on the premise that the model's disturbances are i.i.d., an assumption which is not satisfied in many practical situations. In this paper, we consider a model with non-spherical errors and derive the asymptotic distribution, bias and mean squared error of a general class of feasible minimum mean squared error estimators. A Monte-Carlo experiment is conducted to examine the performance of this class of estimators in finite samples.     

Admissibility of the natural estimator of the mean of a Gaussian process

Let {X(t): t [a, b]} be a Gaussian process with mean μ L₂[a, b] and continuous covariance K(s, t). When estimating μ under the loss ∫_a^b ( (t)−μ(t))² dt the natural estimator X is admissible if K is unknown. If K is known, X is minimax with risk ∫_a^b K(t, t) dt and admissible if and only if the three by three matrix whose entries are K(t_i, t_j) has a determinant which vanishes identically in t_i [a, b], i = 1, 2, 3.     

包含多余回归变量的错误指定模型的随机约束Liu估计

对由于包含多余回归自变量而导致的错误指定线性回归模型,本文导出了回归系数的最小二乘估计,普通混合估计以及随机约束Liu估计,并在均方误差矩阵准则下对这三个估计的优良性进行了比较,给出了随机约束Liu估计优于最小二乘估计和普通混合估计的充要条件.此外,对它们所对应的经典预测值的优良性也进行了讨论.     

敏感性问题调查方法的探讨

本文探讨了敏感性问题的四种调查方法，并结合实例进行比较，最后，对各种方法的优劣和应用范围提出作者的见解     

生长曲线模型中参数的Bayes线性无偏估计

本文在平方损失下导出了生长曲线模型中参数的Bayes线性无偏估计(LUE), 并在均方误差矩阵(MSEM)准则下研究了Bayes LUE相对于广义最小二乘估计(GLSE)的优良性. 对于非满秩情形,获得了可估函数的Bayes LUE并讨论了其优良性问题.