On some Sufficient Conditions for High Breakdown Point of ML Estimators

High breakdown point estimators LME(k) and LT E(k) for location and scale are obtained for symmetrical exponentially decreasing density family.

Application of the D-Fullness Technique for Breakdown Point Study of the Trimmed Likelihood Estimator to a Generalized Logistic Model

2000 Mathematics Subject Classification: 62J12, 62F35

On The Breakdown Properties of some Multivariate M-Functionals

For probability distributions on ℝq, 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 inline image, 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.

Adaptively weighted maximum likelihood estimation of discrete distributions

SummaryDiscrete data arise in various research fields, typically when the observations are count data.I propose a robust and efficient parametric procedure for estimation of discrete distributions. The estimation is done in two phases. First, a very robust, but possibly inefficient, estimate of the model parameters is computed and used to indentify outliers. Then the outliers are either removed from the sample or given low weights, and a weighted maximum likelihood estimate (WML) is computed.The weights are determined via an adaptive process such that if the data follow the model, then asymptotically no observation is downweighted.I prove that the final estimator inherits the breakdown point of the initial one, and that its influence function at the model is the same as the influence function of the maximum likelihood estimator, which strongly suggests that it is asymptotically fully efficient.The initial estimator is a minimum disparity estimator (MDE). MDEs can be shown to have full asymptotic efficiency, and some MDEs have very high breakdown points and very low bias under contamination. Several initial estimators are considered, and the performances of the WMLs based on each of them are studied.It results that in a great variety of situations the WML substantially improves the initial estimator, both in terms of finite sample mean square error and in terms of bias under contamination. Besides, the performances of the WML are rather stable under a change of the MDE even if the MDEs have very different behaviors.Two examples of application of the WML to real data are considered. In both of them, the necessity for a robust estimator is clear: the maximum likelihood estimator is badly corrupted by the presence of a few outliers.This procedure is particularly natural in the discrete distribution setting, but could be extended to the continuous case, for which a possible procedure is sketched.RésuméLes données discrètes sont présentes dans différents domaines de recherche, en particulier lorsque les observations sont des comptages.Je propose une méthode paramétrique robuste et efficace pour l'estimation de distributions discrètes. L'estimation est faite en deux phases. Tout d'abord, un estimateur très robuste des paramètres du modèle est calculé, et utilisé pour la détection des données aberrantes (outliers). Cet estimateur n'est pas nécessairement efficace. Ensuite, soit les outliers sont retirés de l'échantillon, soit des faibles poids leur sont attribués, et un estimateur du maximum de vraisemblance pondéré (WML) est calculé.Les poids sont déterminés via un processus adaptif, tel qu'asymptotiquement, si les données suivent le modèle, aucune observation n'est dépondérée.Je prouve que le point de rupture de l'estimateur final est au moins aussi élevé que celui de l'estimateur initial, et que sa fonction d'influence au modèle est la même que celle du maximum de vraisemblance, ce qui suggère que cet estimateur est pleinement efficace asymptotiquement.L'estimateur initial est un estimateur de disparité minimale (MDE). Les MDE sont asymptotiquement pleinement efficaces, et certains d'entre eux ont un point de rupture très élevé et un très faible biais sous contamination. J'étudie les performances du WML basé sur différents MDEs.Le résultat est que dans une grande variété de situations le WML améliore largement les performances de l'estimateur initial, autant en terme du carré moyen de l'erreur que du biais sous contamination. De plus, les performances du WML restent assez stables lorsqu'on change l'estimateur initial, même si les différents MDEs ont des comportements très différents.Je considère deux exemples d'application du WML à des données réelles, où la nécessité d'un estimateur robuste est manifeste : l'estimateur du maximum de vraisemblance est fortement corrompu par la présence de quelques outliers.La méthode proposée est particulièrement naturelle dans le cadre des distributions discrètes, mais pourrait être étendue au cas continu.

The truncated mean of an asymmetric distribution

This paper investigates a simple procedure to estimate robustly the mean of an asymmetric distribution. The procedure removes the observations which are larger or smaller than certain limits and takes the arithmetic mean of the remaining observations, the limits being determined with the help of a parametric model, e.g., the Gamma, the Weibull or the Lognormal distribution. The breakdown point, the influence function, the (asymptotic) variance, and the contamination bias of this estimator are explored and compared numerically with those of competing estimates.

Robust accelerated failure time regression

Robust estimators for accelerated failure time models with asymmetric (or symmetric) error distribution and censored observations are proposed. It is assumed that the error model belongs to a log-location-scale family of distributions and that the mean response is the parameter of interest. Since scale is a main component of mean, scale is not treated as a nuisance parameter. A three steps procedure is proposed. In the first step, an initial high breakdown point S estimate is computed. In the second step, observations that are unlikely under the estimated model are rejected or down weighted. Finally, a weighted maximum likelihood estimate is computed. To define the estimates, functions of censored residuals are replaced by their estimated conditional expectation given that the response is larger than the observed censored value. The rejection rule in the second step is based on an adaptive cut-off that, asymptotically, does not reject any observation when the data are generat ed according to the model. Therefore, the final estimate attains full efficiency at the model, with respect to the maximum likelihood estimate, while maintaining the breakdown point of the initial estimator. Asymptotic results are provided. The new procedure is evaluated with the help of Monte Carlo simulations. Two examples with real data are discussed.

Robust Estimates of the Negative Binomial Model

We consider robust parametric procedures for univariate discrete distributions, focusing on the negative binomial model. The procedures are based on three steps: ?First, a very robust, but possibly inefficient, estimate of the model parameters is computed. ?Second, this initial model is used to identify outliers, which are then removed from the sample. ?Third, a corrected maximum likelihood estimator is computed with the remaining observations. The final estimate inherits the breakdown point (bdp) of the initial one and its efficiency can be significantly higher. Analogous procedures were proposed in [1], [2], [5] for the continuous case. A comparison of the asymptotic bias of various estimates under point contamination points out the minimum Neyman's chi-squared disparity estimate as a good choice for the initial step. Various minimum disparity estimators were explored by Lindsay [4], who showed that the minimum Neyman's chi-squared estimate has a 50% bdp under point contamination; in addition, it is asymptotically fully efficient at the model. However, the finite sample efficiency of this estimate under the uncontaminated negative binomial model is usually much lower than 100% and the bias can be strong. We show that its performance can then be greatly improved using the three step procedure outlined above. In addition, we compare the final estimate with the procedure described in

ROBREG: Stata module providing robust regression estimators

robreg provides a number of robust estimators for linear regression models. Among them are the high breakdown-point and high efficiency MM-estimator, the Huber and bisquare M-estimator, and the S-estimator, each supporting classic or robust standard errors. Furthermore, basic versions of the LMS/LQS (least median of squares) and LTS (least trimmed squares) estimators are provided. Note that the moremata package, also available from SSC, is required.

Local-Median Method of Forecasting for Regression Time Series under Outliers

2002 Mathematics Subject Classification: 62M20, 62-07, 62J05, 62P20.

Least Median Of Squares Estimation In Power Systems

Static state estimators currently in use in power systems are prone to masking by multiple bad data. This is mainly because the power system regression model contains many leverage points; typically they have a cluster pattern. As reported recently in the statistical literature, only high breakdown point estimators are robust enough to cope with gross errors corrupting such a model. This paper deals with one such estimator, the least median of squares estimator, developed by Rousseeuw in 1984. The robustness of this method is assessed while applying it to power systems. Resampling methods are developed, and simulation results for IEEE test systems discussed. © 1991 IEEE.

Femtosecond imaging-mode laser-induced breakdown spectroscopy

Many nonlinear optical microscopy techniques based on the high-intensity nonlinear phenomena were developed recent years. A new technique based on the minimal-invasive in-situ analysis of the specific bound elements in biological samples is described in the present work. The imaging-mode Laser-Induced Breakdown Spectroscopy (LIBS) is proposed as a combination of LIBS, femtosecond laser material processing and microscopy. The Calcium distribution in the peripheral cell wall of the sunflower seedling (Helianthus Annuus L.) stem is studied as a first application of the imaging-mode LIBS. At first, several nonlinear optical microscopy techniques are overviewed. The spatial resolution of the imaging-mode LIBS microscope is discussed basing on the Point-Spread Function (PSF) concept. The primary processes of the Laser-Induced Breakdown (LIB) are overviewed. We consider ionization, breakdown, plasma formation and ablation processes. Water with defined Calcium salt concentration is used as a model of the biological object in the preliminary experiments. The transient LIB spectra are measured and analysed for both nanosecond and femtosecond laser excitation. The experiment on the local Calcium concentration measurements in the peripheral cell wall of the sunflower seedling stem employing nanosecond LIBS shows, that nanosecond laser is not a suitable excitation source for the biological applications. In case of the nanosecond laser the ablation craters have random shape and depth over 20 µm. The analysis of the femtosecond laser ablation craters shows the reproducible circle form. At 3.5 µJ laser pulse energy the diameter of the crater is 4 µm and depth 140 nm for single laser pulse, which results in 1 femtoliter analytical volume. The experimental result of the 2 dimensional and surface sectioning of the bound Calcium concentrations is presented in the work.

On the Electric Breakdown in Liquid Argon at Centimeter Scale

We present a study on the dependence of electric breakdown discharge properties on electrode geometry and the breakdown field in liquid argon near its boiling point. The measurements were performed with a spherical cathode and a planar anode at distances ranging from 0.1 mm to 10.0 mm. A detailed study of the time evolution of the breakdown volt-ampere characteristics was performed for the first time. It revealed a slow streamer development phase in the discharge. The results of a spectroscopic study of the visible light emission of the breakdowns complement the measurements. The light emission from the initial phase of the discharge is attributed to electro-luminescence of liquid argon following a current of drifting electrons. These results contribute to set benchmarks for breakdown-safe design of ionization detectors, such as Liquid Argon Time Projection Chambers (LAr TPC).

Marriage breakdown in Australia - The social correlates of separation and divorce

Marriage breakdown through separation and divorce is a pervasive feature of Australian society. But little research investigates the social factors associated with marital breakdown in Australia. This study builds on and extends Australian research by using survival analysis models to examine patterns of association among temporal, life-course, attitudinal and economic factors associated with marital breakdown. Using data from the Household Income and Labour Dynamics in Australia (HILDA) survey, we find marital breakdown in Australia is socially patterned in similar ways to other Western countries. But our findings point to several directions for future research into marriage breakdown in Australia, and we identify certain unique features of Australian marriage breakdown that warrant a more detailed investigation, such as the relationship between ethnic origin and the risk of marital breakdown.

Air-sea fluxes of CO2 and CH4 from the Penlee Point Atmospheric Observatory on the south-west coast of the UK

We present air–sea fluxes of carbon dioxide (CO2), methane (CH4), momentum, and sensible heat measured by the eddy covariance method from the recently established Penlee Point Atmospheric Observatory (PPAO) on the south-west coast of the United Kingdom. Measurements from the south-westerly direction (open water sector) were made at three different sampling heights (approximately 15, 18, and 27m above mean sea level, a.m.s.l.), each from a different period during 2014–2015. At sampling heights ≥18ma.m.s.l., measured fluxes of momentum and sensible heat demonstrate reasonable (≤ ±20% in the mean) agreement with transfer rates over the open ocean. This confirms the suitability of PPAO for air–sea exchange measurements in shelf regions. Covariance air–sea CO2 fluxes demonstrate high temporal variability. Air-to-sea transport of CO2 declined from spring to summer in both years, coinciding with the breakdown of the spring phytoplankton bloom. We report, to the best of our knowledge, the first successful eddy covariance measurements of CH4 emissions from a marine environment. Higher sea-to-air CH4 fluxes were observed during rising tides (20±3; 38±3; 29±6 μmolem-2 d-1 at 15, 18, 27ma.m.s.l.) than during falling tides (14±2; 22±2; 21±5 μmolem-2 d-1), consistent with an elevated CH4 source from an estuarine outflow driven by local tidal circulation. These fluxes are a few times higher than the predicted CH4 emissions over the open ocean and are significantly lower than estimates from other aquatic CH4 hotspots (e.g. polar regions, freshwater). Finally, we found the detection limit of the air–sea CH4 flux by eddy covariance to be 20 μmolem-2 d-1 over hourly timescales (4 μmolem-2 d-1 over 24 h).

Air-sea fluxes of CO2 and CH4 from the Penlee Point Atmospheric Observatory on the south-west coast of the UK

We present air–sea fluxes of carbon dioxide (CO2), methane (CH4), momentum, and sensible heat measured by the eddy covariance method from the recently established Penlee Point Atmospheric Observatory (PPAO) on the south-west coast of the United Kingdom. Measurements from the south-westerly direction (open water sector) were made at three different sampling heights (approximately 15, 18, and 27m above mean sea level, a.m.s.l.), each from a different period during 2014–2015. At sampling heights ≥18ma.m.s.l., measured fluxes of momentum and sensible heat demonstrate reasonable (≤ ±20% in the mean) agreement with transfer rates over the open ocean. This confirms the suitability of PPAO for air–sea exchange measurements in shelf regions. Covariance air–sea CO2 fluxes demonstrate high temporal variability. Air-to-sea transport of CO2 declined from spring to summer in both years, coinciding with the breakdown of the spring phytoplankton bloom. We report, to the best of our knowledge, the first successful eddy covariance measurements of CH4 emissions from a marine environment. Higher sea-to-air CH4 fluxes were observed during rising tides (20±3; 38±3; 29±6 μmolem-2 d-1 at 15, 18, 27ma.m.s.l.) than during falling tides (14±2; 22±2; 21±5 μmolem-2 d-1), consistent with an elevated CH4 source from an estuarine outflow driven by local tidal circulation. These fluxes are a few times higher than the predicted CH4 emissions over the open ocean and are significantly lower than estimates from other aquatic CH4 hotspots (e.g. polar regions, freshwater). Finally, we found the detection limit of the air–sea CH4 flux by eddy covariance to be 20 μmolem-2 d-1 over hourly timescales (4 μmolem-2 d-1 over 24 h).