Robust Liu estimator for regression based on an M-estimator

Consider the regression model y = beta 0 1 + Xbeta + epsilon. Recently, the Liu estimator, which is an alternative biased estimator beta L (d) = (X'X + I) -1 (X'X + dI)beta OLS , where 0<d<1 is a parameter, has been proposed to overcome multicollinearity . The advantage of beta L (d) over the ridge estimator beta R (k) is that beta L (d) is a linear function of d. Therefore, it is easier to choose d than to choose k in the ridge estimator. However, beta L (d) is obtained by shrinking the ordinary least squares (OLS) estimator using the matrix (X'X + I) -1 (X'X + dI) so that the presence of outliers in the y direction may affect the beta L (d) estimator. To cope with this combined problem of multicollinearity and outliers, we propose an alternative class of Liu-type M-estimators (LM-estimators) obtained by shrinking an M-estimator beta M , instead of the OLS estimator using the matrix (X'X + I) -1 (X'X + dI).     

Differential equation models for statistical functions

Differential equations have been used in statistics to define functions such as probability densities. But the idea of using differential equation formulations of stochastic models has a much wider scope. The author gives several examples, including simultaneous estimation of a regression model and residual density, monotone smoothing, specification of a link function, differential equation models of data, and smoothing over complicated multidimensional domains. This paper aims to stimulate interest in this approach to functional estimation problems, rather than provide carefully worked out methods.     

Convergence of Heavy-tailed Monte Carlo Markov Chain Algorithms

Abstract.  In this paper, we use recent results of Jarner & Roberts ( Ann. Appl. Probab., 12, 2002, 224) to show polynomial convergence rates of Monte Carlo Markov Chain algorithms with polynomial target distributions, in particular random-walk Metropolis algorithms, Langevin algorithms and independence samplers. We also use similar methodology to consider polynomial convergence of the Gibbs sampler on a constrained state space. The main result for the random-walk Metropolis algorithm is that heavy-tailed proposal distributions lead to higher rates of convergence and thus to qualitatively better algorithms as measured, for instance, by the existence of central limit theorems for higher moments. Thus, the paper gives for the first time a theoretical justification for the common belief that heavy-tailed proposal distributions improve convergence in the context of random-walk Metropolis algorithms. Similar results are shown to hold for Langevin algorithms and the independence sampler, while results for the mixing of Gibbs samplers on uniform distributions on constrained spaces are rather different in character.     

Convergence of Slice Sampler Markov Chains

We analyse theoretical properties of the slice sampler. We find that the algorithm has extremely robust geometric ergodicity properties. For the case of just one auxiliary variable, we demonstrate that the algorithm is stochastically monotone, and we deduce analytic bounds on the total variation distance from stationarity of the method by using Foster–Lyapunov drift condition methodology.     

Analytical approach for determining the cost optimum raw material quality level

In this paper, the expected cost of a raw

material quality characteristic is determined and

the cost optimum quality level is found. A series

of piecewise linear functions is used torepresent a general cost function. Examples are given in

which the distributions of quality characteristics are treated as being either

uniform or normal. The relationship between a raw

material characteristic and manufacturing cost is

assumed to be known

Determining cost optimum quality level Manitoba     

论现代逻辑的应用

现代逻辑的价值和意义,在于其在理论上的应用。对于纯数学的基础和结构来说,现代逻辑是一种非常有用的工具,包括哥德尔不完全性定理等,都利用了现代逻辑的技术和成果。某些现代逻辑技术已经广泛应用到了工程学之中,例如,计算机工程就基于现代逻辑的高级成果。反过来,这些应用又刺激了逻辑理论本身取得更多的进步。     

从葛逻禄到喀喇汗王朝

葛逻禄是由三个部落组成的联盟,唐代时分属唐朝4个都督府管辖。从766年起葛逻禄成为伊斯兰世界的紧邻,并在中亚形成一种势力。840年蒙古高原的回纥汗国灭亡,回纥散往四方,其一支逃往葛逻禄避难。喀喇汗王朝是葛逻禄统治者接受了回纥汗国阿史那汗族（或“阿夫拉西亚勃家族”的一支）归这一家族所有的可汗职位。在伊斯兰历史著作中有三条关于萨曼王朝的伊斯玛因于893年对西部喀喇汗国战争的报道,它们可以相互补充。此外,这三个报道中还有关于喀喇汗王朝与葛逻禄为同一部族的重要情报。     

A Comparative Pharmacokinetic Estimate of Mercury in U.S. Infants Following Yearly Exposures to Inactivated Influenza Vaccines Containing Thimerosal

The use of thimerosal preservative in childhood vaccines has been largely eliminated over the past decade in the United States because vaccines have been reformulated in single‐dose vials that do not require preservative. An exception is the inactivated influenza vaccines, which are formulated in both multidose vials requiring preservative and preservative‐free single‐dose vials. As part of an ongoing evaluation by USFDA of the safety of biologics throughout their lifecycle, the infant body burden of mercury following scheduled exposures to thimerosal preservative in inactivated influenza vaccines in the United States was estimated and compared to the infant body burden of mercury following daily exposures to dietary methylmercury at the reference dose established by the USEPA. Body burdens were estimated using kinetic parameters derived from experiments conducted in infant monkeys that were exposed episodically to thimerosal or MeHg at identical doses. We found that the body burden of mercury (AUC) in infants (including low birth weight) over the first 4.5 years of life following yearly exposures to thimerosal was two orders of magnitude lower than that estimated for exposures to the lowest regulatory threshold for MeHg over the same time period. In addition, peak body burdens of mercury following episodic exposures to thimerosal in this worst‐case analysis did not exceed the corresponding safe body burden of mercury from methylmercury at any time, even for low‐birth‐weight infants. Our pharmacokinetic analysis supports the acknowledged safety of thimerosal when used as a preservative at current levels in certain multidose infant vaccines in the United States.