Bias correction for estimated distortion risk measure using the bootstrap

The bias of the empirical estimate of a given risk measure has recently been of interest in the risk management literature. In particular, Kim and Hardy (2007) showed that the bias can be corrected for the Conditional Tail Expectation (CTE, a.k.a. Tail-VaR or Expected Shortfall) using the bootstrap. This article extends their result to the distortion risk measure (DRM) class where the CTE is a special case. In particular, through the exact bootstrap, it is analytically proved that the bias of the empirical estimate of DRM with concave distortion function is negative and can be corrected on the bootstrap, using the fact that the bootstrapped loss is majorized by the original loss vector. Since the class of DRM is a subset of the L-estimator class, the result provides a sufficient condition for the bootstrap bias correction for L-estimators. Numerical examples are presented to show the effectiveness of the bootstrap bias correction. Later a practical guideline to choose the estimate with a lower mean squared error is also proposed based on the analytic form of the double bootstrapped estimate, which can be useful in estimating risk measures where the bias is non-cumulative across loss portfolio.     

Cross-validated bagged learning

Many applications aim to learn a high dimensional parameter of a data generating distribution based on a sample of independent and identically distributed observations. For example, the goal might be to estimate the conditional mean of an outcome given a list of input variables. In this prediction context, bootstrap aggregating (bagging) has been introduced as a method to reduce the variance of a given estimator at little cost to bias. Bagging involves applying an estimator to multiple bootstrap samples and averaging the result across bootstrap samples. In order to address the curse of dimensionality, a common practice has been to apply bagging to estimators which themselves use cross-validation, thereby using cross-validation within a bootstrap sample to select fine-tuning parameters trading off bias and variance of the bootstrap sample-specific candidate estimators. In this article we point out that in order to achieve the correct bias variance trade-off for the parameter of interest, one should apply the cross-validation selector externally to candidate bagged estimators indexed by these fine-tuning parameters. We use three simulations to compare the new cross-validated bagging method with bagging of cross-validated estimators and bagging of non-cross-validated estimators.     

BOOTSTRAP WAVELET IN THE NONPARAMETRIC REGRESSION MODEL WITH WEAKLY DEPENDENT PROCESSES

This paper introduces a method of bootstrap wavelet estimation in a nonparametric regression model with weakly dependent processes for both fixed and random designs. The asymptotic bounds for the bias and variance of the bootstrap wavelet estimators are given in the fixed design model. The conditional normality for a modified version of the bootstrap wavelet estimators is obtained in the fixed model. The consistency for the bootstrap wavelet estimator is also proved in the random design model. These results show that the bootstrap wavelet method is valid for the model with weakly dependent processes.     

Bootstrapped efficiency estimates for a model for groups and hierarchies in DEA

We apply some recently introduced bootstrap techniques to derive bias corrected efficiency scores for a model for groups and hierarchies in DEA. The use of the bootstrap makes it possible to overcome some deficiencies of the original formulation of this model, which rests on rescaling individual efficiency scores using average efficiencies calculated from different subsets of the data. These average or structural efficiencies are differently biased and bias varies with sample size when standard DEA techniques are used. Bias correction makes it possible to identify the true differences in efficiency and thus to compare DMUs belonging to different groups via their rescaled individual efficiency scores on one common basis. Moreover, this type of bias problem is present in other DEA applications. Therefore, the method proposed to deal with it has many potential applications beyond the groups and hierarchies model.     

Estimation and Bootstrap with Censored Data in Fixed Design Nonparametric Regression

We study Beran's extension of the Kaplan-Meier estimator for thesituation of right censored observations at fixed covariate values. Thisestimator for the conditional distribution function at a given value of thecovariate involves smoothing with Gasser-Müller weights. We establishan almost sure asymptotic representation which provides a key tool forobtaining central limit results. To avoid complicated estimation ofasymptotic bias and variance parameters, we propose a resampling methodwhich takes the covariate information into account. An asymptoticrepresentation for the bootstrapped estimator is proved and the strongconsistency of the bootstrap approximation to the conditional distributionfunction is obtained.     

交替迭代最大似然估计量的偏倚校正

In this paper, we give a definition of the alternating iterative maximum likelihood estimator (AIMLE) which is a biased estimator. Furthermore we adjust the AIMLE to result in asymptotically unbiased and consistent estimators by using a bootstrap iterative bias correction method as in Kuk (1995). Two examples and simulation results reported illustrate the performance of the bias correction for AIMLE.     

Confidence Intervals and Accuracy Estimation for Heavy-Tailed Generalized Pareto Distributions

The generalized Pareto distribution (GPD) is a two-parameter family of distributions which can be used to model exceedances over a threshold. We compare the empirical coverage of some standard bootstrap and likelihood-based confidence intervals for the parameters and upper p-quantiles of the GPD. Simulation results indicate that none of the bootstrap methods give satisfactory intervals for small sample sizes. By applying a general method of D. N. Lawley, correction factors for likelihood ratio statistics of parameters and quantiles of the GPD have been calculated. Simulations show that for small sample sizes accuracy of confidence intervals can be improved by incorporating the computed correction factors to the likelihood-based confidence intervals. While the modified likelihood method has better empirical coverage probability, the mean length of produced intervals are not longer than corresponding bootstrap confidence intervals. This article also investigates the performance of some bootstrap methods for estimation of accuracy measures of maximum likelihood estimators of parameters and quantiles of the GPD.     

定数截尾时可靠性指标的 Bootstrap 区间估计

在工程可靠性试验数据的统计分析中,关于可靠性指标的置信区间的获得是一个有实际意义的课题.这个问题历史上通常是对特定的分布作大量模拟计算,这种做法不但计算量大,而且随着寿命分布的不同必须另行重复模拟,使用极为不便,特别对某些我们所关心的指标,至今缺少较为理想的求置信区间的办法.例如,某些电子元件的寿命服从     

样本函数条件极值中减低偏差的方法

对样本函数条件极值中偏差项的阶进行了分析,探讨了减低偏差项的方法,分析表明古典折刀法、减-d折刀法均不能减低偏差项;在此基础上,提出了减低偏差项的自助法,并论证了在均方误差意义下,θnab是一种较优的估计.     

Efficient construction of a smooth nonparametric family of empirical distributions and calculation of bootstrap likelihood

This paper considers the efficient construction of a nonparametric family of distributions indexed by a specified parameter of interest and its application to calculating a bootstrap likelihood for the parameter. An approximate expression is obtained for the variance of log bootstrap likelihood for statistics which are defined by an estimating equation resulting from the method of selecting the first-level bootstrap populations and parameters. The expression is shown to agree well with simulations for artificial data sets based on quantiles of the standard normal distribution, and these results give guidelines for the amount of aggregation of bootstrap samples with similar parameter values required to achieve a given reduction in variance. An application to earthquake data illustrates how the variance expression can be used to construct an efficient Monte Carlo algorithm for defining a smooth nonparametric family of empirical distributions to calculate a bootstrap likelihood by greatly reducing the inherent variability due to first-level resampling.     

Estimating the intensity of a cyclic Poisson process in the presence of linear trend

We construct and investigate a consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process in the presence of linear trend. It is assumed that only a single realization of the Poisson process is observed in a bounded window. We prove that the proposed estimator is consistent when the size of the window indefinitely expands. The asymptotic bias, variance, and the mean-squared error of the proposed estimator are also computed. A simulation study shows that the first order asymptotic approximations to the bias and variance of the estimator are not accurate enough. Second order terms for bias and variance were derived in order to be able to predict the numerical results in the simulation. Bias reduction of our estimator is also proposed.     

Application of special reduction procedures to metrological data

To improve the reduction of metrological data, that are typically grouped in series and cannot be considered as replicated data, a modelling procedure has been obtained by adding to the model representing the physical behaviour, common to all data, a specific term for each series. Such a procedure combines both the advantages of preserving the individuality of each series and of improving the variance estimate which arises from fitting the overall data. A non-parametric bootstrap method for the error analysis has been developed, which does not imply the assumption of the Normal distribution in the least squares estimation. Two examples of application of the method to thermodynamic data series are reported.     

Risk Estimation for Classification Trees

This article is a study of techniques for bias reduction of estimates of risk both globally and within terminal nodes of CART^R classification trees. In Section 5.4 of Classification and Regression Trees, Leo Breiman presented an estimator that has two free parameters. An empirical Bayes method was put forth for estimating them. Here we explain why the estimator should be successful in the many examples for which it is. We give numerical evidence from simulations in the two-class case with attention to ordinary resubstitution and seven other methods of estimation. There are 14 sampling distributions, all but one simulated and the remaining concerning E. coli promoter regions. We report on varying minimum node sizes of the trees; prior probabilities and misclassification costs; and, when relevant, the numbers of bootstraps or cross-validations. A variation of Breiman's method in which repeated cross-validation is employed to estimate global rates of misclassification was the most accurate from among the eight methods. Exceptions are cases for which the Bayes risk of the Bayes rule is small. For them, either a local bootstrap .632 estimate or Breiman's method modified to use a bootstrap estimate of the global misclassification rate is most accurate.     

The exact bootstrap method shown on the example of the mean and variance estimation

The bootstrap method is based on resampling of the original randomsample drawn from a population with an unknown distribution. In the article it was shown that because of the progress in computer technology resampling is actually unnecessary if the sample size is not too large. It is possible to automatically generate all possible resamples and calculate all realizations of the required statistic. The obtained distribution can be used in point or interval estimation of population parameters or in testing hypotheses. We should stress that in the exact bootstrap method the entire space of resamples is used and therefore there is no additional bias which results from resampling. The method was used to estimate mean and variance. The comparison of the obtained distributions with the limit distributions confirmed the accuracy of the exact bootstrap method. In order to compare the exact bootstrap method with the basic method (with random sampling) probability that 1,000 resamples would allow for estimating a parameter with a given accuracy was calculated. There is little chance of obtaining the desired accuracy, which is an argument supporting the use of the exact method. Random sampling may be interpreted as discretization of a continuous variable.     

Correcting for biases in affected sib-pair linkage analysis caused by uncertainty in sibling relationship

When there is uncertainty in sibling relationship,the classical affected sib-pair(ASP) linkage tests may be severely biased.This can happen,for example,if some of the half sib-pairs are mixed with full sib-pairs.The genomic control method has been used in association analysis to adjust for population structures.We show that the same idea can be applied to ASP linkage analysis with uncertainty in sibling relationship.Assuming that,in addition to the candidate marker,null markers that are unlinked to the disease locus are also genotyped,we may use the information on these loci to estimate the proportion of half sib-pairs and to correct for the bias and variance distortion caused by the heterogeneity of sibling relationship.Unlike in association studies,the null loci are not required to be matched with the candidate marker in allele frequency for ASP linkage analysis.This makes our approach flexible in selecting null markers.In our simulations,using a number of 30 or more null loci can effectively remove the bias and variance distortion.It is also shown that,even the null loci are weakly linked to the disease locus,the proposed method can also provide satisfactory correction.     

Bias-corrected AIC for selecting variables in multinomial logistic regression models

In this paper, we consider the bias correction of Akaike’s information criterion (AIC) for selecting variables in multinomial logistic regression models. For simplifying a formula of the bias-corrected AIC, we calculate the bias of the AIC to a risk function through the expectations of partial derivatives of the negative log-likelihood function. As a result, we can express the bias correction term of the bias-corrected AIC with only three matrices consisting of the second, third, and fourth derivatives of the negative log-likelihood function. By conducting numerical studies, we verify that the proposed bias-corrected AIC performs better than the crude AIC.     

RESAMPLING METHOD UNDER DEPENDENT MODELS

As well known,the jackknife and the bootstrap methods fail for the mean of thedependent observations.Recently,the moving blocks jackknife and bootstrap havebeen proposed in the case of the dependent observations.For the mean of the strictlystationary and m-dependent observations,it has been proved that the proposeddistribution and variance estimators are weakly consistent.This paper proves that thedistribution and variance estimators are strongly consistent for the mean(and theregular functions of mean)of the strictly stationary and m-dependent or(?)-mixingobservations.     

The impact of sampling methods on bias and variance in stochastic linear programs

Stochastic linear programs can be solved approximately by drawing a subset of all possible random scenarios and solving the problem based on this subset, an approach known as sample average approximation (SAA). The value of the objective function at the optimal solution obtained via SAA provides an estimate of the true optimal objective function value. This estimator is known to be optimistically biased; the expected optimal objective function value for the sampled problem is lower (for minimization problems) than the optimal objective function value for the true problem. We investigate how two alternative sampling methods, antithetic variates (AV) and Latin Hypercube (LH) sampling, affect both the bias and variance, and thus the mean squared error (MSE), of this estimator. For a simple example, we analytically express the reductions in bias and variance obtained by these two alternative sampling methods. For eight test problems from the literature, we computationally investigate the impact of these sampling methods on bias and variance. We find that both sampling methods are effective at reducing mean squared error, with Latin Hypercube sampling outperforming antithetic variates. For our analytic example and the eight test problems we derive or estimate the condition number as defined in Shapiro et al. (Math. Program. 94:1–19, 2002). We find that for ill-conditioned problems, bias plays a larger role in MSE, and AV and LH sampling methods are more likely to reduce bias.     

A Retinex-based total variation approach for image segmentation and bias correction

Image segmentation methods usually suffer from intensity inhomogeneity problem caused by many factors such as spatial variations in illumination (or bias fields of imaging devices). In order to address this problem, this paper proposes a Retinex-based variational model for image segmentation and bias correction. According to Retinex theory, the input inhomogeneous image can be decoupled into illumination bias and reflectance parts. The main contribution of this paper is to consider piecewise constant of the reflectance, and thereby introduce the total variation term in the proposed model for correcting and segmenting the input image. This is different from the existing model in which the spatial smoothness of the illumination bias is employed only. The existence of the minimizers to the variational model is established. Furthermore, we develop an efficient algorithm to solve the model numerically by using the alternating minimization method. Our experimental results are reported to demonstrate the effectiveness of the proposed method, and its performance is competitive with that of the other testing methods.     

Bootstrap Algorithms for Risk Models with Auxiliary Variable and Complex Samples

Resampling methods are often invoked in risk modelling when the stability of estimators of model parameters has to be assessed. The accuracy of variance estimates is crucial since the operational risk management affects strategies, decisions and policies. However, auxiliary variables and the complexity of the sampling design are seldom taken into proper account in variance estimation. In this paper bootstrap algorithms for finite population sampling are proposed in presence of an auxiliary variable and of complex samples. Results from a simulation study exploring the empirical performance of some bootstrap algorithms are presented.