Trimmed Likelihood Estimator Research Articles

Image Segmentation Using a Trimmed Likelihood Estimator in the Asymmetric Mixture Model Based on Generalized Gamma and Gaussian Distributions

Finite mixture model (FMM) is being increasingly used for unsupervised image segmentation. In this paper, a new finite mixture model based on a combination of generalized Gamma and Gaussian distributions using a trimmed likelihood estimator (GGMM-TLE) is proposed. GGMM-TLE combines the effectiveness of Gaussian distribution with the asymmetric capability of generalized Gamma distribution to provide superior flexibility for describing different shapes of observation data. Another advantage is that we consider the spatial information among neighbouring pixels by introducing Markov random field (MRF); thus, the proposed mixture model remains sufficiently robust with respect to different types and levels of noise. Moreover, this paper presents a new component-based confidence level ordering trimmed likelihood estimator, with a simple form, allowing GGMM-TLE to estimate the parameters after discarding the outliers. Thus, the proposed algorithm can effectively eliminate the disturbance of outliers. Furthermore, the paper proves the identifiability of the proposed mixture model in theory to guarantee that the parameter estimation procedures are well defined. Finally, an expectation maximization (EM) algorithm is included to estimate the parameters of GGMM-TLE by maximizing the log-likelihood function. Experiments on multiple public datasets demonstrate that GGMM-TLE achieves a superior performance compared with several existing methods in image segmentation tasks.

Investigation of the performance of trimmed estimators of life time distributions with censoring

SummaryFor the lifetime (or negative) exponential distribution, the trimmed likelihood estimator has been shown to be explicit in the form of a β‐trimmed mean which is representable as an estimating functional that is both weakly continuous and Fréchet differentiable and hence qualitatively robust at the parametric model. It also has high efficiency at the model. The robustness is in contrast to the maximum likelihood estimator (MLE) involving the usual mean which is not robust to contamination in the upper tail of the distribution. When there is known right censoring, it may be perceived that the MLE which is the most asymptotically efficient estimator may be protected from the effects of ‘outliers’ due to censoring. We demonstrate that this is not the case generally, and in fact, based on the functional form of the estimators, suggest a hybrid defined estimator that incorporates the best features of both the MLE and the β‐trimmed mean. Additionally, we study the pure trimmed likelihood estimator for censored data and show that it can be easily calculated and that the censored observations are not always trimmed. The different trimmed estimators are compared by a modest simulation study.

Robust MR image segmentation using the trimmed likelihood estimator in asymmetric Student's-t mixture model.

Finite mixture model (FMM) has been widely used for unsupervised segmentation of magnetic resonance (MR) images in recent years. However, in real applications, the distribution of the observed data usually contains an unknown fraction of outliers, which would interfere with the estimation of the parameters of the mixture model. The statistical model-based technique which provides a theoretically well segmentation criterion in presence of outliers is the mixture modeling and the trimming approach. Therefore, in this paper, a robust estimation of asymmetric Student's-t mixture model (ASMM) using the trimmed likelihood estimator for MR image segmentation has been proposed. The proposed method is supposed to discard the outliers, and then to estimate the parameters of the ASMM with the remaining samples. The advantages of the proposed algorithm are that its robustness to dispose the disturbance of outliers and its flexibility to describe various shapes of data. Finally, expectation-maximization (EM) algorithm is adopted to maximize the log-likelihood and to obtain the estimation of the parameters. The experimental results show that the proposed method has a better performance on the segmentation of synthetic data and real MR images.

Robust fitting of mixtures of factor analyzers using the trimmed likelihood estimator

ABSTRACTMixtures of factor analyzers (MFAs) have been popularly used to cluster the high-dimensional data. However, the traditional estimation method is based on the normality assumptions of random terms and thus is sensitive to outliers. In this article, we introduce a robust estimation procedure of MFAs using the trimmed likelihood estimator. We use a simulation study and a real data application to demonstrate the robustness of the trimmed estimation procedure and compare it with the traditional normality-based maximum likelihood estimate.

Trimmed likelihood estimators for lifetime experiments and their influence functions

We study the behaviour of trimmed likelihood estimators (TLEs) for lifetime models with exponential or lognormal distributions possessing a linear or nonlinear link function. In particular, we investigate the difference between two possible definitions for the TLE, one called original trimmed likelihood estimator (OTLE) and one called modified trimmed likelihood estimator (MTLE) which is the finite sample version of a form for location and linear regression used by Bednarski and Clarke [Trimmed likelihood estimation of location and scale of the normal distribution. Aust J Statist. 1993;35:141–153, Asymptotics for an adaptive trimmed likelihood location estimator. Statistics. 2002;36:1–8] and Bednarski et al. [Adaptive trimmed likelihood estimation in regression. Discuss Math Probab Stat. 2010;30:203–219]. The OTLE is always an MTLE but the MTLE may not be unique even in cases where the OLTE is unique. We compare especially the functional forms of both types of estimators, characterize the difference with the implicit function theorem and indicate situations where they coincide and where they do not coincide. Since the functional form of the MTLE has a simpler form, we use it then for deriving the influence function, again with the help of the implicit function theorem. The derivation of the influence function for the functional form of the OTLE is similar but more complicated.

Automatic segmentation and volumetric quantification of white matter hyperintensities on fluid-attenuated inversion recovery images using the extreme value distribution.

This study aims to develop an automatic segmentation framework on the basis of extreme value distribution (EVD) for the detection and volumetric quantification of white matter hyperintensities (WMHs) on fluid-attenuated inversion recovery (FLAIR) images. Two EVD-based segmentation methods, namely the Gumbel and Fréchet segmentation, were developed to detect WMHs on FLAIR (slice thickness = 5 mm; TR/TE/TI = 11,000/120/2,800 ms; flip angle = 90°) images. Another automatic segmentation method using a trimmed likelihood estimator (TLE) was implemented for comparison with our proposed segmentation framework. The performances of the three automatic segmentation methods were evaluated by comparing with the manual segmentation method. The Dice similarity coefficients (DSCs) of the two EVD-based segmentation methods were larger than those of the TLE-based segmentation method (Gumbel, 0.823 ± 0.063; Fréchet, 0.843 ± 0.057; TLE, 0.817 ± 0.068), demonstrating that the EVD-based segmentation outperformed the TLE-based segmentation. The Fréchet segmentation obtained larger DSCs on patients with moderate to severe lesion loads and a comparable performance on patients with mild lesion loads, indicating that the Fréchet segmentation was superior to the Gumbel segmentation. The Gumbel segmentation underestimated the lesion volumes of all patients, whereas the Fréchet and TLE-based segmentation methods obtained overestimated lesion volumes (Manual, 13.71 ± 14.02 cc; Gumbel, 12.73 ± 13.21 cc; Fréchet, 13.88 ± 13.96 cc; TLE, 13.54 ± 12.27 cc). Moreover, the EVD-based segmentation was demonstrated to be comparable to other state-of-the-art methods on a publicly available dataset. The proposed EVD-based segmentation framework is a promising, effective, and convenient tool for volumetric quantification and further study of WMHs in aging and dementia.

3D+t brain MRI segmentation using robust 4D Hidden Markov Chain.

In recent years many automatic methods have been developed to help physicians diagnose brain disorders, but the problem remains complex. In this paper we propose a method to segment brain structures on two 3D multi-modal MR images taken at different times (longitudinal acquisition). A bias field correction is performed with an adaptation of the Hidden Markov Chain (HMC) allowing us to take into account the temporal correlation in addition to spatial neighbourhood information. To improve the robustness of the segmentation of the principal brain structures and to detect Multiple Sclerosis Lesions as outliers the Trimmed Likelihood Estimator (TLE) is used during the process. The method is validated on 3D+t brain MR images.

Erratum to: Ultrahigh dimensional variable selection through the penalized maximum trimmed likelihood estimator

The penalized maximum likelihood estimator (PMLE) has been widely used for variable selection in high-dimensional data. Various penalty functions have been employed for this purpose, e.g., Lasso, weighted Lasso, or smoothly clipped absolute deviations. However, the PMLE can be very sensitive to outliers in the data, especially to outliers in the covariates (leverage points). In order to overcome this disadvantage, the usage of the penalized maximum trimmed likelihood estimator (PMTLE) is proposed to estimate the unknown parameters in a robust way. The computation of the PMTLE takes advantage of the same technology as used for PMLE but here the estimation is based on subsamples only. The breakdown point properties of the PMTLE are discussed using the notion of $$d$$ -fullness. The performance of the proposed estimator is evaluated in a simulation study for the classical multiple linear and Poisson linear regression models.

Ultrahigh dimensional variable selection through the penalized maximum trimmed likelihood estimator

Ultrahigh dimensional variable selection through the penalized maximum trimmed likelihood estimator

회귀모형에서 이상치 검색을 이용한 로버스트 변수변환방법

Transforming response variable is a general tool to adapt data to a linear regression model. However, it is well known that response transformations in linear regression are very sensitive to one or a few outliers. Many methods have been suggested to develop transformations that will not be influenced by potential outliers. Recently Cheng (2005) suggested to using a trimmed likelihood estimator based on the idea of the least trimmed squares estimator(LTS). However, the method requires presetting the number of outliers and needs many computations. A new method is proposed, that can solve the problems addressed and improve the robustness of the estimates. The method uses a stepwise procedure, suggested by Hadi and Simonoff (1993), to detect outliers that determine response transformations.

Robust joint modeling of mean and dispersion through trimming

The Maximum Likelihood Estimator (MLE) and Extended Quasi-Likelihood (EQL) estimator have commonly been used to estimate the unknown parameters within the joint modeling of mean and dispersion framework. However, these estimators can be very sensitive to outliers in the data. In order to overcome this disadvantage, the usage of the maximum Trimmed Likelihood Estimator (TLE) and the maximum Extended Trimmed Quasi-Likelihood (ETQL) estimator is recommended to estimate the unknown parameters in a robust way. The superiority of these approaches in comparison with the MLE and EQL estimator is illustrated by an example and a simulation study. As a prominent measure of robustness, the finite sample Breakdown Point (BDP) of these estimators is characterized in this setting.

Trimmed-Likelihood Estimation for Focal Lesions and Tissue Segmentation in Multisequence MRI for Multiple Sclerosis

We present a new automatic method for segmentation of multiple sclerosis (MS) lesions in magnetic resonance images. The method performs tissue classification using a model of intensities of the normal appearing brain tissues. In order to estimate the model, a trimmed likelihood estimator is initialized with a hierarchical random approach in order to be robust to MS lesions and other outliers present in real images. The algorithm is first evaluated with simulated images to assess the importance of the robust estimator in presence of outliers. The method is then validated using clinical data in which MS lesions were delineated manually by several experts. Our method obtains an average Dice similarity coefficient (DSC) of 0.65, which is close to the average DSC obtained by raters (0.66).

Adaptive trimmed likelihood estimation in regression

Summary In this paper we derive an asymptotic normality result for an adaptive trimmed likelihood estimator of regression starting from initial high breakdownpoint robust regression estimates. The approach leads to quickly and easily computed robust and efficient estimates for regression. A highlight of the method is that it tends automatically in one algorithm to expose the outliers and give least squares estimates with the outliers removed. The idea is to begin with a rapidly computed consistent robust estimator such as the least median of squares (LMS) or least trimmed squares (LTS) or for example the more recent MM estimators of Yohai. Such estimators are now standard in statistics computing packages, for example as in SPLUS or R. In addition to the asymptotics we provide data analyses supporting the new adaptive approach. This approach appears to work well on a number of data sets and is quicker than the related brute force adaptive regression approach described in Clarke (2000). This current approach builds on the work of Bednarski and Clarke (2002) which considered the asymptotics for the location estimator only.

MS Lesion Segmentation based on Hidden Markov Chains

In this paper, we present a new automatic robust algorithm to segment multimodal brain MR images with Multiple Sclerosis (MS) lesions. The method performs tissue classification using a Hidden Markov Chain (HMC) model and detects MS lesions as outliers to the model. For this aim, we use the Trimmed Likelihood Estimator (TLE) to extract outliers. Furthermore, neighborhood information is included using the HMC model and we propose to incorporate a priori information brought by a probabilistic atlas.

Robust and Efficient Adaptive Estimation of Binary-Choice Regression Models

The binary-choice regression models, such as probit and logit, are used to describe the effect of explanatory variables on a binary response variable. Typically estimated by the maximum likelihood method, estimates are very sensitive to deviations from a model, such as heteroscedasticity and data contamination. At the same time, the traditional robust (high-breakdown point) methods, such as the maximum trimmed likelihood, are not applicable because, by trimming observations, they induce nonidentification of parameter estimates. To provide a robust estimation method for binary-choice regression, we consider a maximum symmetrically trimmed likelihood estimator (MSTLE) and design a parameter-free adaptive procedure for choosing the amount of trimming. The proposed adaptive MSTLE preserves the robust properties of the original MSTLE, significantly improves the finite-sample behavior of MSTLE, and also ensures the asymptotic equivalence of the MSTLE and maximum likelihood estimator under no contamination. The results concerning the trimming identification, robust properties, and asymptotic distribution of the proposed method are accompanied by simulation experiments and an application documenting the finite-sample behavior of some existing and the proposed methods.

Maximum trimmed likelihood estimator for multivariate mixed continuous and categorical data

In this article, we apply the maximum trimmed likelihood (MTL) approach [Hadi, A.S., Luceño, A., 1997. Maximum trimmed likelihood estimators: a unified approach, examples, and algorithms. Comput. Statist. Data Anal. 25, 251–272] to obtain the robust estimators of multivariate location and shape, especially for data mixed with continuous and categorical variables. The forward search algorithm [Atkinson, A.C., 1994. Fast very robust methods for the detection of multiple outliers. J. Amer. Statist. Assoc. 89, 1329–1339] is adapted to compute the proposed MTL estimates. A simulation study shows that the proposed estimator outperforms the classical maximum likelihood estimator when outliers exist in data. Real data sets are also used to illustrate the method and results of the detection of the outliers.

Robust fitting of mixtures using the trimmed likelihood estimator

The maximum likelihood estimator (MLE) has commonly been used to estimate the unknown parameters in a finite mixture of distributions. However, the MLE can be very sensitive to outliers in the data. In order to overcome this the trimmed likelihood estimator (TLE) is proposed to estimate mixtures in a robust way. The superiority of this approach in comparison with the MLE is illustrated by examples and simulation studies. Moreover, as a prominent measure of robustness, the breakdown point (BDP) of the TLE for the mixture component parameters is characterized. The relationship of the TLE with various other approaches that have incorporated robustness in fitting mixtures and clustering are also discussed in this context.

AN ADAPTIVE TRIMMED LIKELIHOOD ALGORITHM FOR IDENTIFICATION OF MULTIVARIATE OUTLIERS

This article describes an algorithm for the identification of outliers in multivariate data based on the asymptotic theory for location estimation as described typically for the trimmed likelihood estimator and in particular for the minimum covariance determinant estimator. The strategy is to choose a subset of the data which minimizes an appropriate measure of the asymptotic variance of the multivariate location estimator. Observations not belonging to this subset are considered potential outliers which should be trimmed. For α less than about 0.5, the correct trimming proportion is taken to be that α > 0 for which the minimum of any minima of this measure of the asymptotic variance occurs. If no minima occur for an α > 0 then the data set will be considered outlier free.

Trimmed Likelihood-Based Estimation in Binary Regression Models

The binary-choice regression models such as probit and logit are typically estimated by the maximum likelihood method. To improve its robustness, various M-estimation based procedures were proposed, which however require bias corrections to achieve consistency and their resistance to outliers is relatively low. On the contrary, traditional high-breakdown point methods such as maximum trimmed likelihood are not applicable since they induce the separation of data and thus non-identification of estimates by trimming observations. We propose a new robust estimator of binary-choice models based on a maximum symmetrically trimmed likelihood estimator. It is proved to be identified and consistent, and additionally, it does not create separation in the space of explanatory variables as the existing maximum trimmed likelihood. We also discuss asymptotic and robust properties of the proposed method and compare all methods by means of Monte Carlo simulations.

Robust regression diagnostics with data transformations

The problems of non-normality or functional relationships between variables may often be simplified by an appropriate transformation. However, the evidence for transformations may sometimes depend crucially on one or a few observations. Therefore, the purpose of the paper is to develop a method that will not be influenced by potential outliers during the process of data transformations. The concepts of the least trimmed squares estimator and the trimmed likelihood estimator are used to obtain the robust transformation parameters. Furthermore, the proposed procedure unifies robust statistics and a diagnostic approach to deal with the outlier problem in the regression transformation.