相伴于随机自相似分形的随机测度的强测度的收敛性

构造了随机自相似分形及其上的记忆函数,并得出了有关结论,在此基础上,我们可以定义一个随机概率测度dΦn(τ)=Kn(τ)dτ,Φn(τ)弱收敛于Φ,进一步可得到强测度序列Ψn(·)=EΦn(·),则{Ψn}弱收敛于Ψ=EΦ.     

Yates type estimators of a common mean

Letx, y, S, T andW be independent random variables such that,～N(μασ ²),y～N(μ,βη ²), S/σ²～χ²(m), T/η²～χ²(n) andW/(ασ ²+βη²)～x²(q), where μ, σ², η² are unknown. For estimating μ, consider the estimator \(\hat \mu = x + \left( {y - x} \right){{aS} \mathord{\left/ {\vphantom {{aS} {\left[ {S + cT + d\left\{ {\left( {y - x} \right)^2 + W} \right\}} \right],a,c,d > 0}}} \right. \kern-0em} {\left[ {S + cT + d\left\{ {\left( {y - x} \right)^2 + W} \right\}} \right],a,c,d > 0}}\) . Note that the performance of \(\hat \mu \) depends onτ=βη ²/ασ², which is unknown. Assumeq+n≧2 and leta ₀=(n+q?1)/(m+2), c^*=cα/β, d^*=dα. Two main results are:

1. for all τ>0, \(\hat \mu \) has a variance smaller than that ofx ifa≦2 min (1,c ^*a₀, d^*a₀);
2. for all τ≧τ₀, where τ₀>0 is arbitrary, \(\hat \mu \) has a variance smaller than that ofx ifa≦2a ₀ min [c ^*τ₀/(1+τ₀),d^*].

We also obtain some necessary conditions for \(\hat \mu \) to have a variance smaller than that ofx. It can be seen that with the exception of linked block designs for any design belonging to the class calledD ₁-class by Shah [16], Yates-Rao estimator for recovery of interblock information has the same form as that of \(\hat \mu \) . Hence, for such designs the above results can be used to examine if Yates estimator is good i.e., better than the intra-block estimator. Shah [16] resolved this question for linked block designs, which include the symmetrical BIBD's. Here, we consider asymmetrical BIBD's and show that Yates' estimator is good for all such designs listed in Fisher and Yates' table [5], with two exceptions. For one of these two designs, we show that Yates' estimator is not uniformly better than the intra-block estimator.     

Estimating a largest eigenvector by Lanczos and polynomial algorithms with a random start

We study Lanczos and polynomial algorithms with random start for estimating an eigenvector corresponding to the largest eigenvalue of an n × n large symmetric positive definite matrix. We analyze the two error criteria: the randomized error and the randomized residual error. For the randomized error, we prove that it is not possible to get distribution-free bounds, i.e., the bounds must depend on the distribution of eigenvalues of the matrix. We supply such bounds and show that they depend on the ratio of the two largest eigenvalues. For the randomized residual error, distribution-free bounds exist and are provided in the paper. We also provide asymptotic bounds, as well as bounds depending on the ratio of the two largest eigenvalues. The bounds for the Lanczos algorithm may be helpful in a practical implementation and termination of this algorithm. © 1998 John Wiley & Sons, Ltd.     

A random measure model for the emission of pollutants by vehicles on a highway

Consider a collection of vehicles on a highway. The velocity process of the k^th vehicle is a Markov process with continuous paths independent of the velocity processes of the other vehicles but with the same probability law. The discharge of pollutants by the k^th vehicle is described by a nondecreasing right continuous process whose probability law depends on the path of the vehicle in such a way that the discharge process has independent increments given the velocity process of the vehicle. The limiting behavior of the total amount of pollutants discharged in an interval of the highway by time t by all the vehicles is studied as t → ∞.     

Optimal linear estimators of the unknown mean of a random field observed on a cross

Explicit formulas of minimum-variance linear unbiased estimators of the unknown mean are derived for homogeneous random fields with correlation functions observed on the set (a cross).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 116–121, 1988.     

Estimating parameters of a log-linear intensity for a repairable system

The log-linear intensity is often used in survival analysis of technical products with rapid deterioration. It is an extremely important intensity to characterize the probabilistic behavior of a large number of real phenomena. In this paper we develop statistical methods for an alternating repair model using a log-linear intensity. The maximum likelihood estimator is considered for determining the estimations of the model parameters. The distribution of the life times after perfect repairs and imperfect repairs are obtained. The estimation of the Fisher information matrix is given. Simultaneous confidence regions based on the likelihood ratio statistics are developed for the estimators of the parameters. The proposed model is demonstrated using the well known data on airplane air-conditioning failures from Plane 7.     

Sensitivity of GLS estimators in random effects models

This paper studies the sensitivity of random effects estimators in the one-way error component regression model. Maddala and Mount (1973) [6] give simulation evidence that in random effects models the properties of the feasible GLS estimator are not affected by the choice of the first-step estimator used for the covariance matrix. Taylor (1980) [8] gives a theoretical example of this effect. This paper provides a reason for this in terms of sensitivity. The properties of are transferred via an uncorrelated (and independent under normality) link, called sensitivity. The sensitivity statistic counteracts the improvement in . A Monte Carlo experiment illustrates the theoretical findings.     

随机截断下部分线性模型中参数估计的渐近性质

考虑部分线性回归模型Yi＝xiβ g(ti) σiej,i＝1，2，…，n其中σi^2＝／f(ui)．当Yi因受某种随机干扰而被右截断时，就截断分布巳知的情形，利用所获得的截断观察数据构造了β，g，f的估计量β^～n，g^～n，f^～n，并在一定条件下，证明了β^～n的渐近正态性，同时得到了g^～n，f^～n的最优收敛速度。     

Estimating the size of a union of random subsets of fixed cardinality

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Estimating the size of a union of random subsets of fixed cardinality

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Types of laws of random vectors and best estimators

Optimality of estimators of a vector parameter in terms of the probability of the estimators being contained in suitable region(s) around the parameter point is defined. Conditions under which optimal estimators in the usual senses are also optimal in the above sense are investigated.     

Optimisation of linear unbiased intensity estimators for point processes

A general non-stationary point process whose intensity function is given up to unknown numerical factor λ is considered. As an alternative to the conventional estimator of λ based on counting the points, we consider general linear unbiased estimators of λ given by sums of weights associated with individual points. A necessary and sufficient condition for a linear, unbiased estimator for the intensity λ to have the minimum variance is determined. It is shown that there are “nearly” no other processes than Poisson and Cox for which the unweighted estimator of λ, which counts the points only, is optimal. The properties of the optimal estimator are illustrated by simulations for the Matérn cluster and the Matérn hard-core processes. This research was partially supported by Grant Agency of Czech Republic, project No. 201/03/D062.     

Influence measure based on probabilistic behavior of regression estimators

An influence measure for investigating the influence of deleting an observation in linear regression is proposed based on geometric thoughts of the sampling distribution of the distance between two estimators of regression coefficients computed with and without a single specific observation. The covariance matrix of the above sampling distribution plays a key role in deriving the influence measure. It turns out that geometrically, this distance is distributed entirely along the axis associated with the nonnull eigenvalue of the covariance matrix. The deviation of the regression coefficients computed without an observation from the regression coefficients computed with the full data is reflected in the eigenvalue of the covariance matrix which can be used for investigating the influence. The distance is normalized using the associated covariance matrix and this normalized distance turns out to be the square of internally studentized residuals. Illustrative examples for showing the effectiveness of the influence measure proposed here are given. In judging the influence of observations on the least squares estimates of regression coefficients, Cook’s distance does not work well for one example and therefore we should be cautious about a blind use of the Cook’s distance.     

Large deviations for stochastic partial differential equations driven by a Poisson random measure

Stochastic partial differential equations driven by Poisson random measures (PRMs) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equation (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRMs, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway.     

Estimating the observation parameters of a switching Poisson stream at random moments

We investigate the properties of maximum likelihood estimators in observation schemes of random variables that arrive at random time moments on a Poisson trajectory defined on some ergodic process.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 98–105, 1986.     

Improvement of estimators in a linear regression problem with random errors in coefficients

Under consideration is the problem of estimating the linear regression parameter in the case when the variances of observations depend on the unknown parameter of the model, while the coefficients (independent variables) are measured with random errors. We propose a new two-step procedure for constructing estimators which guarantees their consistency, find general necessary and sufficient conditions for the asymptotic normality of these estimators, and discuss the case in which these estimators have the minimal asymptotic variance.     

随机分形的测度性质

本文主要介绍随机过程样本轨道、Ｈａｗｋｅｓ模型、统计自相似集、统计自仿射集的测度性质，同时也将介绍一些离散分形的结果．文中还列出一些尚未解决的问题．