On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions*

Abstract

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.     

Multivariate τ-estimators for location and scatter

We discuss the robustness and asymptotic behaviour of τ-estimators for multivariate location and scatter. We show that τ-estimators correspond to multivariate M-estimators defined by a weighted average of redescending ψ-functions, where the weights are adaptive. We prove consistency and asymptotic normality under weak assumptions on the underlying distribution, show that τ-estimators have a high breakdown point, and obtain the influence function at general distributions. In the special case of a location-scatter family, τ-estimators are asymptotically equivalent to multivariate S-estimators defined by means of a weighted ψ-function. This enables us to combine a high breakdown point and bounded influence with good asymptotic efficiency for the location and covariance estimator.     

A note on finite-sample efficiencies of estimators for the minimum volume ellipsoid

Among the most well known estimators of multivariate location and scatter is the Minimum Volume Ellipsoid (MVE). Many algorithms have been proposed to compute it. Most of these attempt merely to approximate as close as possible the exact MVE, but some of them led to the definition of new estimators which maintain the properties of robustness and affine equivariance that make the MVE so attractive. Rousseeuw and van Zomeren (1990) used the <$>(p+1)<$>- subset estimator which was modified by Croux and Haesbroeck (1997) to give rise to the averaged <$>(p+1)<$>- subset estimator . This note shows by means of simulations that the averaged <$>(p+1)<$>-subset estimator outperforms the exact estimator as far as finite-sample efficiency is concerned. We also present a new robust estimator for the MVE, closely related to the averaged <$>(p+1)<$>-subset estimator, but yielding a natural ranking of the data.     

Convergence behavior of an iterative reweighting algorithm to compute multivariate M-estimates for location and scatter

The iteratively reweighting algorithm is one of the widely used algorithm to compute the M-estimates for the location and scatter parameters of a multivariate dataset. If the M estimating equations are the maximum likelihood estimating equations from some scale mixture of normal distributions (e.g. from a multivariate t-distribution), the iteratively reweighting algorithm is identified as an EM algorithm and the convergence behavior of which is well established. However, as Tyler (J. Roy. Statist. Soc. Ser. B 59 (1997) 550) pointed out, little is known about the theoretical convergence properties of the iteratively reweighting algorithms if it cannot be identified as an EM algorithm. In this paper, we consider the convergence behavior of the iteratively reweighting algorithm induced from the M estimating equations which cannot be identified as an EM algorithm. We give some general results on the convergence properties and, we show that convergence behavior of a general iteratively reweighting algorithm induced from the M estimating equations is similar to the convergence behavior of an EM algorithm even if it cannot be identified as an EM algorithm.     

Location adjustment for the minimum volume ellipsoid estimator

Estimating multivariate location and scatter with both affine equivariance and positive breakdown has always been difficult. A well-known estimator which satisfies both properties is the Minimum Volume Ellipsoid Estimator (MVE). Computing the exact MVE is often not feasible, so one usually resorts to an approximate algorithm. In the regression setup, algorithms for positive-breakdown estimators like Least Median of Squares typically recompute the intercept at each step, to improve the result. This approach is called intercept adjustment. In this paper we show that a similar technique, called location adjustment, can be applied to the MVE. For this purpose we use the Minimum Volume Ball (MVB), in order to lower the MVE objective function. An exact algorithm for calculating the MVB is presented. As an alternative to MVB location adjustment we propose L ₁ location adjustment, which does not necessarily lower the MVE objective function but yields more efficient estimates for the location part. Simulations compare the two types of location adjustment. We also obtain the maxbias curves of L ₁ and the MVB in the multivariate setting, revealing the superiority of L ₁.     

On the Breakdown Properties of Some Multivariate M‐Functionals*

Abstract. For probability distributions on ? ^q, a detailed study of the breakdown properties of some multivariate M‐functionals related to Tyler's [Ann. Statist. 15 (1987) 234] ‘distribution‐free’ M‐functional of scatter is given. These include a symmetrized version of Tyler's M‐functional of scatter, and the multivariate t M‐functionals of location and scatter. It is shown that for ‘smooth’ distributions, the (contamination) breakdown point of Tyler's M‐functional of scatter and of its symmetrized version are 1/q and , respectively. For the multivariate t M‐functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/( q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low‐dimensional subspaces.     

Robust Estimation of Multivariate Linear Model Based on Depth Weighted Mean and Scatter

Based on the projection depth weighted mean and scatter estimation of the joint distribution of (x, y), we introduce a robust estimator of the regression coefficients for the multivariate linear model. The new estimator possesses desirable properties including affine invariance, Fisher consistency, and asymptotic normality. Also, we study the robustness of the estimator in terms of breakdown point and influence function. Extensive simulation studies are performed to investigate the finite sample behavior of robustness and efficiency. The methodology is illustrated with a real data example.     

The competing risks analysis for parallel and series systems using Type-II progressive censoring

Abstract

Recently, the study of the lifetime of systems in reliability and survival analysis in the presence of several causes of failure (competing risks) has attracted attention in the literature. In this paper, series and parallel systems with exponential lifetime for each item of the system are considered. Several causes of failure independently affect lifetime distributions and observations of failure times of the systems are considered under progressive Type-II censored scheme. For series systems, the maximum likelihood estimates of parameters are computed and confidence intervals for parameters of the model are obtained using Fisher information matrix. For parallel systems, the generalized EM algorithm which uses the Newton-Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. Also, the standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. The simulation study confirms the good performance of the introduced approach.     

Estimating finite mixture of continuous trees using penalized mutual information

Abstract

In this paper we introduce continuous tree mixture model that is the mixture of undirected graphical models with tree structured graphs and is considered as multivariate analysis with a non parametric approach. We estimate its parameters, the component edge sets and mixture proportions through regularized maximum likalihood procedure. Our new algorithm, which uses expectation maximization algorithm and the modified version of Kruskal algorithm, simultaneosly estimates and prunes the mixture component trees. Simulation studies indicate this method performs better than the alternative Gaussian graphical mixture model. The proposed method is also applied to water-level data set and is compared with the results of Gaussian mixture model.     

On multivariate separating Hill estimator under estimated location and scatter

We consider estimating the tail-index of a distribution under the assumption of multivariate ellipticity. Recently, a separating Hill estimator for multivariate elliptical distributions was proposed. This estimator is an affine invariant alternative to using the marginal observations in tail-index estimation and is hence unaffected by, e.g. change of units of measurement. However, the separating Hill estimator depends on the location and scatter of the elliptical distribution, which, in practice, have to be estimated. The effect of replacing the true location and scatter of the distribution by estimates has previously been only examined through simulations. In this article we show that the error caused by replacing the location and scatter of the distribution by estimates indeed is asymptotically negligible. This fact is essential for the practicality of the separating Hill estimator. In addition to providing the theoretical results, we present simulation results on the asymptotic behaviour of the estimators.     

Computing projection depth and its associated estimators

To facilitate the application of projection depth, an exact algorithm is proposed from the view of cutting a convex polytope with hyperplanes. Based on this algorithm, one can obtain a finite number of optimal direction vectors, which are x-free and therefore enable us (Liu et al., Preprint, 2011) to compute the projection depth and most of its associated estimators of dimension p≥2, including Stahel-Donoho location and scatter estimators, projection trimmed mean, projection depth contours and median, etc. Both real and simulated examples are also provided to illustrate the performance of the proposed algorithm.     

Sensitivity Coefficient in Principal Component Analysis: Robust Case

In the classical principal component analysis (PCA), the empirical influence function for the sensitivity coefficient ρ is used to detect influential observations on the subspace spanned by the dominants principal components. In this article, we derive the influence function of ρ in the case where the reweighted minimum covariance determinant (MCD¹) is used as estimator of multivariate location and scatter. Our aim is to confirm the reliability in terms of robustness of the MCD¹ via the approach based on the influence function of the sensitivity coefficient.     

Multivariate outlier detection in incomplete survey data: the epidemic algorithm and transformed rank correlations

Summary.  As a part of the EUREDIT project new methods to detect multivariate outliers in incomplete survey data have been developed. These methods are the first to work with sampling weights and to be able to cope with missing values. Two of these methods are presented here. The epidemic algorithm simulates the propagation of a disease through a population and uses extreme infection times to find outlying observations. Transformed rank correlations are robust estimates of the centre and the scatter of the data. They use a geometric transformation that is based on the rank correlation matrix. The estimates are used to define a Mahalanobis distance that reveals outliers. The two methods are applied to a small data set and to one of the evaluation data sets of the EUREDIT project.     

Robust M-estimators of multivariate location and scatter in the presence of asymmetry

Robust estimation of location vectors and scatter matrices is studied under the assumption that the unknown error distribution is spherically symmetric in a central region and completely unknown in the tail region. A precise formulation of the model is given, an analysis of the identifiable parameters in the model is presented, and consistent initial estimators of the identifiable parameters are constructed. Consistent and asymptotically normal M-estimators are constructed (solved iteratively beginning with the initial estimates) based on “influence functions” which vanish outside specified compact sets. Finally M-estimators which are asymptotically minimax (in the sense of Huber) are derived.     

A small-sample correction factor for S-estimators

S-estimators are frequently used as robust estimators of regression and of location and dispersion. Under certain differentiability conditions, S-estimators of multivariate location and dispersion parameters are consistent [Davies PL. Asymtotic behaviour of S-estimators of multivariate location parameters and dispersion matrices. Ann Stat. 1987;15(3):1269–1292]. However, it has been observed that the S-estimators of dispersion parameters give biased results in the case of small-sample data sets. In this work, we constructed formulas based on simulation studies, which allow us to compute small-sample correction factors for all sample sizes and dimensions for S-estimators of dispersion parameters without having to carry out any new simulations. We considered real data to illustrate the effects of the small-sample correction factor.     

Computing location depth and regression depth in higher dimensions

The location depth (Tukey 1975) of a point relative to a p-dimensional data set Z of size n is defined as the smallest number of data points in a closed halfspace with boundary through . For bivariate data, it can be computed in O(nlogn) time (Rousseeuw and Ruts 1996). In this paper we construct an exact algorithm to compute the location depth in three dimensions in O(n2logn) time. We also give an approximate algorithm to compute the location depth in p dimensions in O(mp3+mpn) time, where m is the number of p-subsets used.Recently, Rousseeuw and Hubert (1996) defined the depth of a regression fit. The depth of a hyperplane with coefficients (1,...,p) is the smallest number of residuals that need to change sign to make (1,...,p) a nonfit. For bivariate data (p=2) this depth can be computed in O(nlogn) time as well. We construct an algorithm to compute the regression depth of a plane relative to a three-dimensional data set in O(n2logn) time, and another that deals with p=4 in O(n3logn) time. For data sets with large n and/or p we propose an approximate algorithm that computes the depth of a regression fit in O(mp3+mpn+mnlogn) time. For all of these algorithms, actual implementations are made available.     

An application of ranked set sampling when observations from several distributions are to be included in the sample

ABSTRACT

In this article, we propose a method to estimate the common location and common scale parameters of several distributions using suitably defined ranked set sampling. Efficiency comparison of the obtained estimators with some of the standard estimators is made. Illustration of the results to real life data sets is also described.     

Convergence Behavior of the em algorithm for the multivariate t -distribution

Iterative reweighting (IR) is a popular method for computing M-estimates of location and scatter in multivariate robust estimation. When the objective function comes from a scale mixture of normal distributions the iterative reweighting algorithm can be identified as an EM algorithm. The purpose of this paper is to show that in the special case of the multivariate t-distribution, substantial improvements to the convergence rate can be obtained by modifying the EM algorithm.     

A relaxed approach to combinatorial problems in robustness and diagnostics

A range of procedures in both robustness and diagnostics require optimisation of a target functional over all subsamples of given size. Whereas such combinatorial problems are extremely difficult to solve exactly, something less than the global optimum can be ‘good enough’ for many practical purposes, as shown by example. Again, a relaxation strategy embeds these discrete, high-dimensional problems in continuous, low-dimensional ones. Overall, nonlinear optimisation methods can be exploited to provide a single, reasonably fast algorithm to handle a wide variety of problems of this kind, thereby providing a certain unity. Four running examples illustrate the approach. On the robustness side, algorithmic approximations to minimum covariance determinant (MCD) and least trimmed squares (LTS) estimation. And, on the diagnostic side, detection of multiple multivariate outliers and global diagnostic use of the likelihood displacement function. This last is developed here as a global complement to Cook’s (in J. R. Stat. Soc. 48:133–169, 1986) local analysis. Appropriate convergence of each branch of the algorithm is guaranteed for any target functional whose relaxed form is—in a natural generalisation of concavity, introduced here—‘gravitational’. Again, its descent strategy can downweight to zero contaminating cases in the starting position. A simulation study shows that, although not optimised for the LTS problem, our general algorithm holds its own with algorithms that are so optimised. An adapted algorithm relaxes the gravitational condition itself.