High Breakdown Point Research Articles

OGLE-IV Period–Luminosity Relation of the LMC: An Analysis Using Mean and Median Magnitudes

The Period–Luminosity (PL) relation for Cepheid variable stars in the LMC is crucial for distance measurements in astronomy. This study analyzes the impact of using the median rather than the mean of the PL relation’s slope and zero-point. It also examines the persistence of the break at approximately 10 days and addresses specification issues in the PL relation model. Using VI-band median and mean magnitudes from the OGLE-IV survey, corrected for extinction, we fit the PL relation employing robust MM-regression, which features a high breakdown point and robust standard errors. Statistical tests and residual analysis are conducted to identify and correct model deficiencies. Our findings indicate a significant change in the PL relation for Cepheids with periods of 10 days or longer, regardless of whether median or mean magnitudes are used. A bias in the zero-point and slope estimators is observed when using median magnitudes instead of mean magnitudes, especially in the V band. By identifying and correcting regression issues and considering the period break, our estimators for the slope and zero-point are more accurate for distance calculations. Comparative analysis of the models for each band quantifies the bias introduced by using median magnitudes, highlighting the importance of considering the Cepheids’ periods for accurate location measure results, similar to those obtained using mean magnitudes.

Robust and efficient estimation in ordinal response models using the density power divergence

In real life, we frequently encounter ordinal variables depending upon independent covariates. The latent linear regression model is useful for modelling such data. One can find the model's parameters' maximum likelihood estimate (MLE). Though noted for its optimum properties, a small proportion of outliers may destabilize the MLE. This paper uses the minimum density power divergence estimate (MDPDE) as a robust alternative. The roles of different link functions are analysed in this context. We discuss their asymptotic properties in this setup. Unlike the MLE, the MDPDEs are robust for– lower values of the gross error sensitivity, and very high breakdown point. Also, the slope’s MDPDEs never implode. In simulation studies for pure data, MDPDEs perform almost as good as the MLE. However, the MDPDEs outperform the MLE in data contamination. Moreover, MDPDEs are very competitive with the other robust alternatives. Finally, this article is wrapped up with a real-data example.

The Early Warning of Financial Failure for Iraqi Banks Based on Robust Adaptive Lasso Logistic Regression

It is well known that there are many mathematical financial failure models that have been proposedin the financial literature for specific stock markets. Some researchers are not aware these mathematical modelswere constructed to be fitted for that market data, not for other markets. Iraq stock market exchange is one of thesemarkets in which the researchers used imported models such as Kida, Sherrod, Altman, and others to predictfinancial failure. Therefore, the development of a financial failure warning model for banks has become verycrucial for the Iraqi bank sector in the stock market exchange. Unfortunately, there is no clear information aboutthe financial failure of Iraqi banks as a response variable, and the financial indicators contain outliers. Theobjective of this paper is to propose an algorithm to know the performance of efficient and inefficient banks basedon their indicators during specific time periods. The output of this algorithm will be considered as responsevariables. Then, a weighted adaptive lasso logistic regression algorithm that has a high breakdown point is used totackle outliers’ problem. Thirteen banks have been chosen as the most traded during the period (2010-2017), andfor each bank (27) financial indicators were collected. Our proposed model is compared with adaptive lassologistic regression by using Deviance, Misclassification, Area Under Curve, Mean Square Errors, and MeanAbsolute Errors. Consequently, the results showed that the weighted Adaptive Lasso Logistic Regression model ismore robust and relevant than others to be a financial model to warn of the failure of the banks in Iraq's stockmarket.

Patellar tendon shortening surgery restores the knee extensor mechanism in flexed knee gait in children with cerebral palsy

BackgroundThis study evaluated a patellar tendon shortening (PTS) surgical procedure that uses an overlapping repair combined with an additional Tycron non-absorbable suture to support the shortening in children with Cerebral Palsy (CP). This study aimed to outline this surgical technique and to evaluate its effectiveness in restoring the knee extensor mechanism. MethodsThe sagittal plane lower limb kinematics, peak knee extensor moment, gait deviation index (GDI), localised movement deviation profile (MDP), temporospatial parameters, passive knee extension ROM, quadriceps lag, and knee extensor strength were calculated pre- and postoperatively. To determine significant differences a robust linear regression model with high breakdown point and high efficiency was fitted to the data. ResultsIn this retrospective cohort study, a total of 41 patients with CP who were treated with unilateral or bilateral PTS in isolation or as part of single event multilevel surgery (SEMLS), with a mean age of 11.1 years were included. The knee extension angle improved at initial contact (p < 0.0001), and during stance phase (p < 0.0001). The peak internal knee extensor moment decreased during early (p = 0.0014) and late stance phase (p < 0.0001). The quadriceps lag decreased (p < 0.0001) and knee extensor strength increased (p < 0.0001). The GDI improved (p < 0.0001), as well as the localised MDP for sagittal angles (p < 0.0001) and moments (p = 0.0001). Walking speed (p = 1.0) remained unchanged, but the cadence decreased (p = 0.024) and step length increased (p = 0.0001). ConclusionsThe knee extension angle and moment during stance phase improved significantly. The children with CP in this study showed improvements in knee extensor strength and quadriceps lag. Thereby it can be concluded that the PTS procedure was able to restore the knee extensor mechanism effectively.

A comparative study between the two methods GM6.IDRGP(RMVN) and the GM6 method for analyzing the multiple linear regression model in the presence of outliers using the simulation method.

Multiple linear regression is one of the most widely used statistical analysis methods in many scientific fields. Its parameters are estimated based on the ordinary least squares method. Which gives the best unbiased linear estimate if its assumptions are met. The most important of these assumptions is that it has a normal distribution of error with a mean of zero and a constant variance. If the data does not meet certain assumptions, the sample estimates and results may be misleading. The linear regression model is sensitive to the appearance of outliers and leverage points. Therefore, statistical techniques have been developed capable of dealing with or detecting outliers. This led to the emergence of many alternative methods to the OLS method, such as M-Huber, s, LTS, and MM, which have high efficiency and breakdown points, but are affected by HLPs, which result in the problem of Masking and Swamping. The GM6 method is one of the methods that was developed in order to treat such problems through the use of a weight function, but the weight function depends on the individual diagnosis, which gives inaccurate results. In order to overcome this problem, comprehensive diagnostic methods have been proposed, such as the DRGP and IDRGP.RMVN methods. Therefore, in this study, we proposed to employ the IDRGP.RMVN method in the GM6 method algorithm. And comparing it with some hippocampal regression methods through a simulation study in determining the best methods.

Review of Some Robust Estimators in Multiple Linear Regressions in the Presence of Outlier(s)

Linear regression has been one of the most important statistical data analysis tools. Multiple regression is a type of regression where the dependent variable shows a linear relationship with two or more independent variables. OLS estimate is extremely sensitive to unusual observations (outliers), with low breakdown point and low efficiency. This paper reviews and compares some of the existing robust methods (Least Absolute Deviation, Huber M-Estimator, Bisquare M-Estimator, MM Estimator, Least Median Square, Least Trimmed Square, S-Estimator); a simulation method is used to compare the selected existing methods. It was concluded based on the results that for y direction outlier, the best estimator in terms of high efficiency and breakdown point of at most 0.3 is MM; for x direction outlier, the best estimator in term breakdown point of at most 0.4 is S; for x, y direction outlier, the best estimator in terms of high efficiency and breakdown point of at most 0.2 is MM.

Robust solution of coordinate transformation parameters with a high breakdown point

Because the M estimation method may lead to poor robustness or even failure owing to excessive outliers, a robust algorithm with a high breakdown point was proposed and applied to the estimation of coordinate transformation parameters. Firstly, the sampling method was used to calculate multiple sets of model parameters, and some sampling results were sifted according to posterior information. Then, the samples were sorted according to their number in the sampling results, and the F-test was adopted to screen and reserve valid information. Finally, the initial values of the reliable parameters were computed using the valid information, and the final parameters were obtained by the Institute of Geodesy and Geophysics III scheme. Monte Carlo method was adopted for the simulation test, and a case analysis was chosen for verification. The results show that the proposed method can identify and process outliers more accurately than those of Rousseeuw and Hubert (2011 Wires Data Min. Knowl. 1 73–79) and Tao et al (2016 Acta Geod. Cartogr. Sin. 45 297–301). When the proportion of outliers exceeded 50%, the proposed algorithm maintained a strong robustness and had a high breakdown point.

The effects of economic growth and fossil fuel consumption to climate change: Evidence from Mediterranean Europe by robust estimators

Climate change and global warming during the recent decades are posing formidable chal- lenges to ecosystems. Nevertheless, changing the climate system due to extreme weather events such as cold spells, high temperatures, droughts, and heat waves have been recorded all over the world. Particularly, it has become less accurate to predict the weather in some European regions using a short time series without considering the extreme values events in the estimated model. Thus, forecasting the behaviour of climate needs more accurate statisti- cal techniques to be used. The main objective of this experimental study is to detect the best robust scale or robust location estimator to model the relationship between CO2 emissions, fossil fuel consumption and gross domestic product by considering the influence of different types of extreme weather events in the panel data of Mediterranean Europe countries over the period 1960-2020. The findings show that the MM-estimator is the best robust estimator han- dling data with high efficiency and high breakdown point with the existence of different types of extreme weather events. In conclusion, the robust MM-estimator could be used to provide an innovative integrated climate-economic approach for the accurate prediction of carbon emissions.

Highly Efficient Estimators with High Breakdown Point for Linear Models with Structured Covariance Matrices

A unified approach is provided for a method of estimation of the regression parameter in balanced linear models with a structured covariance matrix that combines a high breakdown point with high asymptotic efficiency at models with multivariate normal errors. Of main interest are linear mixed effects models, but our approach also includes several other standard multivariate models, such as multiple regression, multivariate regression, and multivariate location and scatter. Sufficient conditions are provided for the existence of the estimators and corresponding functionals, strong consistency and asymptotic normality is established, and robustness properties are derived in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities. Some results are new and others are more general than existing ones in the literature. In this way, results on high breakdown estimation with high efficiency in a wide variety of multivariate models are completed and improved.

New approaches for outlier detection: The least trimmed squares adjustment

Classical outlier tests based on the least-squares (LS) have significant disadvantages in some situations. The adjustment computation and classical outlier tests deteriorate when observations include outliers. The robust techniques that are not sensitive to outliers have been developed to detect the outliers. Several methods use robust techniques such as M-estimators, L1- norm, the least trimmed squares etc. The least trimmed squares (LTS) among them have a high-breakdown point. After the theoretical explanation, the adjustment computation has been carried out in this study based on the least squares (LS) and the least trimmed squares (LTS). A certain polynomial with arbitrary values has been used for applications. In this way, the performances of these techniques have been investigated.

The trimmed mean in non-parametric regression function estimation

This article studies a trimmed version of the Nadaraya–Watson estimator for the unknown non-parametric regression function. The characterization of the estimator through the minimization problem is established, and its pointwise asymptotic distribution is derived. The robustness property of the proposed estimator is also studied through the breakdown point. Moreover, similar to the trimmed mean in the location model, and for a wide range of trimming proportion, the proposed estimator possesses good efficiency and high breakdown point, which is out of the ordinary properties for any estimator. Furthermore, the usefulness of the proposed estimator is shown for two benchmark real data and various simulated data.

Analisis Regresi Robust Estimasi Least Trimmed Square dan Estimasi Maximum Likelihood pada Pemodelan IPM di Pulau Sulawesi

Sulawesi Island is one of the islands in Indonesia with the second lowest HDI after Papua in 2011-2018, so it is necessary to study more deeply the factors that affect the HDI on Sulawesi Island. One of the methods in modeling is the robust regression method that can be used even though there are outliers in the data. This study aimed to determine the factors that affect HDI in Sulawesi Island using a robust regression model of Least Trimmed Square (LTS) estimation and Maximum Likelihood (M) estimation and to find out the best method for modeling HDI in Sulawesi Island. The robust regression of the LTS estimate has a reasonably high breakdown point value compared to other estimates, while the M-estimate has a high efficiency. The results showed that the factors that affect HDI on the island of Sulawesi are Life Expectancy (UHH), Expected Years of Schooling (HLS), Average Years of Schooling (RLS), and Real Expenditures per capita (PPP). The best model in HDI modeling in Sulawesi Island is a robust regression model with LTS estimation where the standard error is 0.1513, and the R-Square is 99.78%. A very high R-Square value indicates that the ability of the UHH, HLS, RLS, and PPP variables to explain HDI is very good.

All-in-one robust estimator of the Gaussian mean

The goal of this paper is to show that a single robust estimator of the mean of a multivariate Gaussian distribution can enjoy five desirable properties. First, it is computationally tractable in the sense that it can be computed in a time which is at most polynomial in dimension, sample size and the logarithm of the inverse of the contamination rate. Second, it is equivariant by translations, uniform scaling and orthogonal transformations. Third, it has a high breakdown point equal to $0.5$, and a nearly-minimax-rate-breakdown point approximately equal to $0.28$. Fourth, it is minimax rate optimal, up to a logarithmic factor, when data consists of independent observations corrupted by adversarially chosen outliers. Fifth, it is asymptotically efficient when the rate of contamination tends to zero. The estimator is obtained by an iterative reweighting approach. Each sample point is assigned a weight that is iteratively updated by solving a convex optimization problem. We also establish a dimension-free non-asymptotic risk bound for the expected error of the proposed estimator. It is the first result of this kind in the literature and involves only the effective rank of the covariance matrix. Finally, we show that the obtained results can be extended to sub-Gaussian distributions, as well as to the cases of unknown rate of contamination or unknown covariance matrix.

Robust interval estimators of weibull parameters with probability integral transformation

This article gives a construction of a weighted mean which has good robust properties (qualitative robustness, bounded influence function, high breakdown points). The construction is based on M-functionals with smooth defining functions which are used to control weighting. This method can be applied some other statistical problems but it is used for estimation of the parameters of Weibull-family.

Robust bootstrap regression testing in the presence of outliers

Bootstrap is one of the random sampling methods with replacement, that was proposed to address the problem of small samples whose distributions are difficult to derive. The distribution of bootstrap samples is empirical or free and due to its random sampling with replacement, the probability of choosing a specific observation may be equal to one. Unfortunately, when the original sample data contains an outlier, there is a serious problem that leads to a breakdown OLS (Ordinary Least Squares) estimator, and robust regression methods should be recommended. It is well known that the best robust regression method has a high breakdown point is not more than 0.50, so the robust regression method would break down when the percentage of outliers in the bootstrap sample exceeds 0.50. It is well known that fixed-x bootstrap is resampled the residuals which probably are having outliers. Moreover, the leverage point(s) is an outlier that occurs in X-direction, so the effects of it on fixed-x bootstrap samples would be existence. However, the decision-making about the null hypothesis of bootstrap regression coefficients could not be reliable. In this paper, we propose using weighted fixed-x bootstrap with a probability approach to guarantee the percentage of outliers in the bootstrap samples will be very low. And then weighted M-estimate should be to tackle the problem of outliers and leverage points and taking a more reliable decision about bootstrap regression coefficients hypothesis test. The performance of the suggested method has been tested with others by using real data and simulation. The results show our proposed method is more efficient and reliable than the others.

Stratified sampling in highly polluted data as an effective and reliable alternative to high breakdown point estimators

Observations on certain real-life cases include units that are incompatible with other data sets. Values that are extreme in nature do influence estimates obtained by conventional estimators. Robust estimators are therefore necessary for efficient estimation of parameters. This paper uses stratification with simple random sampling without replacement to optimize sample allocation in stratum for efficient parameter estimation as an alternative method of handling highly contaminated samples. Our proposed method stratifies the highly contaminated population into two non-overlapping sub-populations, and stratified samples of sizes 50, 200, and 500 was drawn. We estimate the model parameters form the contaminated sampled data using ordinary least squares under the proposed method, and using the two high breakdown point estimators; the Least Median of Squares and Least Trimmed Squares. Our findings shows that the proposed method did not perform well for low contamination levels (⩽ 30%) but outperformed Least Median of Squares and Least Trimmed Squares for higher contamination rates (⩾ 40%). This indicates that our proposed method compares well and compete favorably with the two high breakdown point estimators.

Response Surface Regression with LTS and MM-Estimator to Overcome Outliers on Red Roselle Flowers

The surface response method is similar to the regression analysis method which uses procedures or ways of estimating the response function regression model based on the Ordinary Least Square (OLS) method. Unfortunately, using the quadratic method has no drawbacks because it is easily sensitive to assumption deviations due to outlier cases. One of the solutions to the outlier problem is using robust regression. The method of parameters in the regression is very diverse, but the methods used in this study are the Least Trimmed Square (LTS) and MM-estimator methods because both methods have a high breakdown point of nearly 50%. The variables studied were the response variable consisting of red roselle plant height (Y1) and red roselle flower weight (Y2). While the independent variables were soil moisture factor (X1) and NPK fertilizer application factor (X2). The purpose of this study is to estimate the response surface regression parameters. using the LTS and MM-estimator methods on data that contains outliers. The resulting model in data analysis shows the same result that the best model is using the LTS estimation method. The modeling result of plant height obtained an R-Square value of 98,27% with an error is 1,243. Meanwhile, for the red rosella plant flower weight model, the R-Square value was 97,31% with an error is 0.6632.

Simple and Fast Generalized - M (GM) Estimator and Its Application to Real Data Set

It is now evident that some robust methods such as MM-estimator do not address the concept of bounded influence function, which means that their estimates still be affected by outliers in the X directions or high leverage points (HLPs), even though they have high efficiency and high breakdown point (BDP). The Generalized M(GM) estimator, such as the GM6 estimator is put forward with the main aim of making a bound for the influence of HLPs by some weight function. The limitation of GM6 is that it gives lower weight to both bad leverage points (BLPs) and good leverage points (GLPs) which make its efficiency decreases when more GLPs are present in a data set. Moreover, the GM6 takes longer computational time. In this paper, we develop a new version of GM-estimator which is based on simple and fast algorithm. The attractive feature of this method is that it only downs weights BLPs and vertical outliers (VOs) and increases its efficiency. The merit of our proposed GM estimator is studied by simulation study and well-known aircraft data set.

Iteratively reweighted total least squares for PEIV model

The weighted total least-squares (WTLS) estimate for the partial EIV (PEIV) model is very susceptible to outliers. In view of outliers in observations and coefficient matrix, a robust estimate of WTLS which makes using of iteratively reweighted technique (IRTLS) for the PEIV model is developed by combining a new two-step iterated algorithm of the WTLS estimate with the sensitivity-analysis based on the robust M-estimation. The uniformly most powerful test statistics are constructed to determine the down-weighting factor and the variance of unit weight is estimated by least median squares (LMS) method possessing high break-down point. Depended on the PEIV model, two different down-weighting schemes are presented. In the first scheme down-weighting is only implemented for the coefficient matrix and not for observations when the elements of the coefficient matrix are estimated, and the second scheme is contrary. The most attractive aspect is that the suggested computational formulae are the same with the traditional robust least-squares (LS) methods. The two-dimensional affine transformation and liner fitting example are analyzed, and some comparisons are performed with different available weight functions. The proposed approach (Scheme 1) proves to be a powerful tool in detecting outliers for the PEIV model.

Breakdown Analysis of Pearson Correlation Coefficient and Robust Correlation Methods

This paper discussed plug in robust procedure to robustify the Pearson product moment correlation coefficient (PPMCC). The mean of PPMCC is highly susceptible to influential observations hence the PPMCC is not robust against data set that contains substantial amount of influential observations. This study focused on robust plug in techniques with high breakdown points. The performance of these techniques are compared using real and simulated data set. The comparative analysis indicates different degrees of robustness and breakdown based on the percentage of contamination and data modification.