Absolute Estimator Research Articles

Robust Regression with Covariate Filtering: Heavy Tails and Adversarial Contamination

We study the problem of linear regression where both covariates and responses are potentially (i) heavy-tailed and (ii) adversarially contaminated. Several computationally efficient estimators have been proposed for the simpler setting where the covariates are sub-Gaussian and uncontaminated; however, these estimators may fail when the covariates are either heavy-tailed or contain outliers. In this work, we show how to modify the Huber regression, least trimmed squares, and least absolute deviation estimators to obtain estimators which are simultaneously computationally and statistically efficient in the stronger contamination model. Our approach is quite simple, and consists of applying a filtering algorithm to the covariates, and then applying the classical robust regression estimators to the remaining data. We show that the Huber regression estimator achieves near-optimal error rates in this setting, whereas the least trimmed squares and least absolute deviation estimators can be made to achieve near-optimal error after applying a postprocessing step. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Quantile estimation of heterogenous panel quantile model with group structure

This paper introduces a heterogeneous panel quantile model whose coefficients have group structure. We propose a penalized minimum weighted absolute deviation estimator in order to estimate the coefficients and identify the group structure simultaneously at any quantiles.

Smooth least absolute deviation estimators for outlier-proof identification

The paper proposes to identify the parameters of linear dynamic models based on the original implementation of least absolute deviation estimators. It is known that the object estimation procedures synthesized in the sense of the least sum of absolute prediction errors are particularly resistant to occasional outliers and gaps in the analyzed system data series, while the classical least squares procedure unfortunately becomes of little use for reliably identifying systems in the presence of destructive measurement errors. Bearing in mind that the classic task of minimizing the quality functional of absolute deviations encounters fundamental analytical problems, it is proposed to use a dedicated iterative estimator for off-line evaluation of the parameters of the analyzed process. In addition, a simplified recursive version of the absolute deviation estimation procedure was developed, which allows for practical on-line tracking of the evolution of variable parameters of non-stationary systems. Importantly, a novel refinement of the discussed absolute deviation estimators was proposed to effectively overcome some inconvenient numerical effects. We also present an interesting comparison of the improved (by non-linear modification) iterative absolute-deviation estimator with the classical Gauss-Newton gradient algorithm, which leads to constructive conclusions. Finally, using computer simulations, the operation of the developed iterative and recursive estimators minimizing the absolute deviation is illustrated. The work ends with an indication of directions for further research.

Abstract PO4-19-02: Genetic risk estimation in breast cancer and assessing health disparities

Abstract Background: Breast Cancer (BC) incidence and the distribution of BC risk factors vary between racial and ethnic groups within the United States. It has been demonstrated that providing education on individualized breast cancer risk will improve patient uptake to recommended BC screening and prevention strategies. Most clinical risk assessment tools, including the validated IBIS model were derived from data using non-Hispanic white women. Consortia have also now identified over 300 common genetic susceptibility loci for BC, summarized by the polygenic risk score (PRS). When combined with clinical risk assessment tools, such as the IBIS or BCRAT models, the PRS can further refine risk estimation for patients. To date, most studies on the use of PRS have been done in non-Hispanic white women. While these PRS still perform well in racial minorities, there has been a recognized need to improve racial and ethnic diversity in genomic research cohorts. Herein, we aim to combine clinical risk assessment models that are already used in routine clinical practice with information derived from PRS testing in women of racial minorities to determine if this can improve risk estimation, patient understanding, and uptake to recommended breast cancer screening and prevention strategies. Design: This is a minimal risk prospective study with a single arm incorporating the PRS into a standard breast cancer risk reduction consultation, followed by annual surveys over 10 years to determine if and how the information provided by the PRS influenced patient decisions regarding recommended BC screening and prevention. Patients will have IBIS and BCRAT risks calculated at baseline visit. A survey of patient perceptions/understanding of risk, intention to undergo screening, and use of preventive medicine is completed after baseline visit prior to receiving the PRS results. Blood sample is obtained at baseline and DNA samples are analyzed for approximately 300 SNPs on a custom-designed, targeted SNP panel from ThermoFisher Scientific by the Mayo Clinic Genomics Laboratory. This PRS result is then combined with the clinical risk assessments (IBIS and BCRAT scores) using the R package Individualized Coherent Absolute Risk Estimators (iCare) tool to provide a new estimate of breast cancer risk for 5year, 10year, and lifetime risk. Patients are seen for follow-up to review results and then complete a second survey to assess their understanding of the results, BC risk, and how the PRS impacted their decision to undergo screening/prevention strategies. Eligibility Criteria: Women who self-identify as African American/Black or Hispanic/Latinx between 30-75 years old with any of the following: 1.)IBIS score of ≥5% for the 10 year risk OR BCRAT score of ≥ 3 % for the 5 year risk. 2.) History of biopsy proven atypical hyperplasia 3.) History of biopsy proven lobular carcinoma in situ. Specific Aims: Aim 1: The primary aim of this study is to explore if the addition of PRS to the BCRAT and IBIS score will improve intentions to undergo recommended breast cancer screening strategies such as mammography, MRI, or molecular breast imaging in women of underserved racial minorities Aim 2: To explore if the addition of the PRS to the BCRAT and IBIS risk score will aid women of racial minorities in deciding whether to take preventative endocrine therapy. Aim 3: To understand how individualized risk assessment and information on PRS may alter perceived risk of breast cancer. Statistical Methods Analysis will be mostly descriptive. Continuous variables will be summarized as mean (standard deviation) or median (range) and categorical variables will be reported as frequency (percentage). Kaplan-Meier method will be used to estimate the long-term cumulative risk of cancer. Current Accrual: 3. Target Accrual: 50 Citation Format: Lauren Cornell, Sandhya Pruthi, Linda Hasadsri, Sarah McLaughlin. Genetic risk estimation in breast cancer and assessing health disparities [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO4-19-02.

Iterative‐recursive estimation of parameters of regression models with resistance to outliers on practical examples

AbstractHere, identification of processes and systems in the sense of the least sum of absolute values is taken into consideration. The respective absolute value estimators are recognised as exceptionally insensitive to large measurement faults or other defects in the processed data, whereas the classical least squares procedure appears to be completely impractical for processing the data contaminated with such parasitic distortions. Since the absolute value quality index cannot be minimised analytically, an iterative solution is used to find optimal estimates of the parameters of the underlying regression model. In addition, an approximate recursive estimator is proposed and implemented for on‐line evaluation of system parameters. The convergence (basic property) of the iterative estimator is show to be proven and some aspects related to the absolute value criterion are explained. This allows for the formulation of practical conclusions and indication of directions for further research. In addition, the effectiveness of the described iterative‐recursive estimation procedures is practically verified by appropriate numerical experiments.

Abstract B052: Ovarian cancer risk prediction: a systematic assessment of the state of the art and development of a new model

Abstract B052: Ovarian cancer risk prediction: a systematic assessment of the state of the art and development of a new model

On robust method of estimation in polynomial-sinusoidal regression

In this paper we have considered the polynomial-sinusoidal regression model in presence of additive errors. The standard method of estimation of the unknown parameters is the least squares approach. It is well known that the least squares estimators are not robust in presence of outliers. We have proposed a weighted least squares method and it can be more robust than the least squares estimators in presence of outliers. The proposed weighted least squares method can be implemented very easily compared to some of the well known robust estimators like least absolute deviation estimators or Huber’s M (Huber-M) estimators. Theoretical properties of the proposed estimators have been obtained under the same set of assumptions as what are needed for the least squares estimators. Simulations have been performed to show the effectiveness of the proposed method, and it is observed that the weighted least squares estimators behave better than the least squares estimators in presence of outliers and they behave at par with the other robust estimators. Further, the implementation of the LAD and Huber-M estimators is computationally quite challenging when the number of components is large. It is observed that the performance of the weighted least squares estimators depends on the choice of the weight function. We have proposed a method to choose the proper weight function. We have analyzed one data set for illustrative purposes.

Polygenic Risk Score, Environmental Tobacco Smoke, and Risk of Lung Adenocarcinoma in Never-Smoking Women in Taiwan

Estimating absolute risk of lung cancer for never-smoking individuals is important to inform lung cancer screening programs. To integrate data on environmental tobacco smoke (ETS), a known lung cancer risk factor, with a polygenic risk score (PRS) that captures overall genetic susceptibility, to estimate the absolute risk of lung adenocarcinoma (LUAD) among never-smokers in Taiwan. The analyses were conducted in never-smoking women in the Taiwan Genetic Epidemiology Study of Lung Adenocarcinoma, a case-control study. Participants were recruited between September 17, 2002, and March 30, 2011. Data analysis was performed from January 17 to July 15, 2022. A PRS was derived using 25 genetic variants that achieved genome-wide significance (P < 5 × 10-8) in a recent genome-wide association study, and ETS was defined as never exposed, exposed at home or at work, and exposed at home and at work. The Individualized Coherent Absolute Risk Estimator software was used to estimate the lifetime absolute risk of LUAD in never-smoking women aged 40 years over a projected 40-year span among the controls by using the relative risk estimates for the PRS and ETS exposures, as well as age-specific lung cancer incidence rates for never-smokers in Taiwan. Likelihood ratio tests were conducted to assess an additive interaction between the PRS and ETS exposure. Data were obtained on 1024 women with LUAD (mean [SD] age, 59.6 [11.4] years, 47.9% ever exposed to ETS at home, and 19.5% ever exposed to ETS at work) and 1024 controls (mean [SD] age, 58.9 [11.0] years, 37.0% ever exposed to ETS at home, and 14.3% ever exposed to ETS at work). The overall average lifetime 40-year absolute risk of LUAD estimated using PRS alone was 2.5% (range, 0.6%-10.3%) among women never exposed to ETS. When integrating both ETS and PRS data, the estimated absolute risk was 3.7% (range, 0.6%-14.5%) for women exposed to ETS at home or work and 5.3% (range, 1.2%-12.1%) for women exposed to ETS at home and work. A super-additive interaction between ETS and the PRS (P = 6.5 × 10-4 for interaction) was identified. This study found differences in absolute risk of LUAD attributed to genetic susceptibility according to levels of ETS exposure in never-smoking women. Future studies are warranted to integrate these findings in expanded risk models for LUAD.

Distributed Least Absolute Deviations Estimation

Distributed algorithms are essential for reducing communication costs, computational complexity, and memory requirements while performing collaborative estimation using multi-agent systems. Additionally, robustness in estimators is important to prevent performance degradation when the measurement noise is non-Gaussian. Least absolute deviations estimators are known to be robust in the presence of gross errors or outliers in the measurements. To this end, we develop the distributed least absolute deviations (D-LAD) estimator for linear systems whereby the agents iteratively exchange information with their immediate neighbors via single-hop communications to gain a network-wide consensus on the estimates. Additionally, the D-LAD algorithm is implemented in a nonlinear framework to solve the problem of distributed orbit determination of a target body using a formation of spacecraft. Numerical simulations demonstrating the effectiveness of the D-LAD estimator in linear and nonlinear settings are provided.

Weighted least squares: A robust method of estimation for sinusoidal model

In this article, we consider the weighted least squares estimators (WLSEs) of the unknown parameters of a multiple sinusoidal model. Although, the least squares estimators (LSEs) are known to be the most efficient estimators in case of a multiple sinusoidal model, they are quite susceptible in presence of outliers. In presence of outliers, robust estimators like the least absolute deviation estimators (LADEs) or Huber’s M-estimators (HMEs) may be used. But implementation of the LADEs and HMEs are quite challenging in case of a sinusoidal model, the problem becomes more severe in case of multiple sinusoidal model. Moreover, to derive the theoretical properties of the robust estimators, one needs stronger assumptions on the error random variables than what are needed for the LSEs. The proposed WLSEs are used as robust estimators and they have the following two major advantages. First, they can be implemented very easily in case of multiple sinusoidal model, and their properties can be obtained under the same set of error assumptions as the LSEs. Extensive simulation results suggest that in presence of outliers, the WLSEs behave better than the LSEs, and at par with the LADEs and HEMs. It is observed that the performance of the WLSEs depend on the weight function, and we discuss how to choose a proper weight function for a given data set. We have analyzed one synthetic data set to show how the proposed methods can be implemented in practice.

Robust estimators of two dimensional sinusoidal model parameters

In this paper we consider the two dimensional (2-D) sinusoidal model. This particular model has several applications in the statistical signal processing and in texture analysis. Extensive work has been done in developing several efficient procedures and establishing their properties. The least squares estimators (LSEs) are known to be the most efficient estimators in presence of additive noise. But it is observed that the LSEs are quite sensitive in presence of outliers. In this paper we propose to use the weighted least squares estimators (WLSEs) of the unknown parameters in presence of additive white noise. It is observed that in presence of outliers, the WLSEs are more robust than the least squares estimators (LSEs) and they behave very similarly to some of the other well known robust estimators, for example the least absolute deviation estimators (LADEs). It is observed that developing the properties of the LADEs is not immediate. We derive the consistency and asymptotic normality properties of the WLSEs. Extensive simulations have been performed to show the effectiveness of the proposed method. One synthetic data set has been analyzed to illustrate how the proposed method can be used in practice.

APPLICATION OF ROBUST REGRESSION FOR PORTFOLIO OPTIMIZATION

The single-index model is a portfolio optimization method that uses each asset’s beta’. In general, the beta is estimated using the return data by the least square method. However, the return data frequently contains several outliers, so the estimation resulting from the least square method is inaccurate. This study examines several beta estimators from robust regression methods, namely the least absolute value estimator, M-estimator, LMS-estimator, LTS-estimator, MM-estimator, and Tau estimator to estimate the beta of each asset and make an optimal portfolio based on this estimated value. We also evaluate the effect of robust beta estimators on the stability and performance of each portfolio. We present the Sharpe ratio and some turnover measures, namely the l-period portfolio turnover, maximum turnover, lower bound single-asset turnover, and lower bound multiple-asset turnover. Among various estimators used here, the Tau estimator is the best estimator to replace the OLS for estimating the beta.

Exact Computation of Censored Least Absolute Deviations Estimator for Panel Data with Fixed Effects

Exact Computation of Censored Least Absolute Deviations Estimator for Panel Data with Fixed Effects

Parameter Estimation of Bernoulli-Gaussian Noise Using Fractional Order Absolute Moments

The prior information of impulsive noise at receiver will improve the performance of the communication systems impaired by impulsive noise. Therefore, parameters estimation techniques of Bernoulli-Gaussian noise are studied in this letter. A Fractional Order absolute Moment-based Estimator (FOME) is proposed and the Asymptotic Normalized Mean Square Error (ANMSE) of the FOME is derived. Both the ANMSE and simulation results show that the FOME performs better than the 1,2,3-order moment-based estimator, and the performance of the FOME will approach the expectation maximization estimator with the increase of the received signal length.

Distribution System Topology Identification via Efficient MILP-Based WLAV State Estimation

A common assumption in distribution system state estimation is that the system’s topology is perfectly known. However, it is difficult to ensure that the topology available is the actual system’s topology because many network equipment settings are unmonitored. Additionally, topology changes are performed by crews on-site, and the information sometimes is unreported to the distribution system operator. Topological errors can result in severe consequences if the obtained state estimates are used to control the system. In this context, this paper presents an efficient Weighted Least Absolute Value State Estimator (WLAV-SE) for the Topology Identification (TI) of distribution systems. Firstly, the traditional Linear Programming (LP)-based WLAV-SE is reformulated for computational efficiency. Secondly, supplementary variables and constraints are included in the reformulated WLAV-SE to suit TI problems. The resulting problem consists of a mixed-integer linear programming-based WLAV state estimation problem, which can be solved by well-established optimization software. The effectiveness of the proposed TI method is illustrated on diverse distribution systems considering various case studies and comparisons with a well-consolidated TI method. Results in scenarios considering different amounts of real-time measurements, high pseudo-measurement errors, measurements corrupted with bad data, and several unknown branch statuses show the excellent performance of the proposed method.

Optimal subsampling for least absolute relative error estimators with massive data

Due to the limitation of computational resources, traditional statistical methods are no longer applicable to large data sets. Subsampling is a popular method which can significantly reduce computational burden. This paper considers a subsampling strategy based on the least absolute relative error in the multiplicative model for massive data. In addition, we employ the random weighting and the least squares methods to handle the problem that the asymptotic covariance of the estimator is difficult to be estimated directly. Moreover, the comparison among the least absolute relative error, least absolute deviation and least squares under the optimal subsampling strategy are given in simulation studies and real examples.

Abstract 2256: Validation of an integrated breast cancer risk model with mammographic density in three prospective cohort studies

Abstract Introduction: Incorporation of mammographic density to breast cancer risk models could improve risk stratification to tailor screening and prevention strategies according to risk. However, robust validation of such models in prospective cohort studies is needed to determine their accuracy in identifying women at different risk levels. Methods: We incorporated Breast Imaging and Reporting Data System (BI-RADS) breast density to a literature-based model with questionnaire-based risk factors and a 313-variant polygenic risk score (PRS). The Individualized Coherent Absolute Risk Estimator (iCARE) tool was used to build and validate a 5-year absolute risk model for breast cancer. The model was evaluated for calibration and discrimination in three prospective cohorts of women of European ancestry (1,468 cases, 19,104 controls): US-based Nurses’ Health Study (NHS I and II) and Mayo Mammography Health Study (MMHS); and Sweden-based Karolinska Mammography Project for Risk Prediction of Breast Cancer (KARMA) study. Analyses were done separately for women younger (NHS II, KARMA) and older than 50 years (NHS I, MMHS, KARMA). Improvements in risk stratification were assessed among US non-Hispanic White women aged 50-70 years. Results: For women younger than 50 years, the model with questionnaire-based risk factors, PRS and BI-RADS was well calibrated across risk deciles in NHS II, but overestimated risk at the highest risk decile in KARMA. For women 50 years or older, the model showed good calibration in all studies, with evidence of slight overestimation at the highest risk decile and underestimation at the lowest risk decile. The model with PRS and BI-RADS was well calibrated for women at high-risk in both age groups. Incorporation of BI-RADS to questionnaire-based risk factors and PRS improved risk discrimination: area under the curves (AUC) 67.0% (95% CI: 63.5-70.6%) vs. 65.6% (95% CI: 61.9-69.3%) for models with and without BI-RADS for younger women and 66.1% (95% CI 64.4-67.8%) vs. 65.5% (95% CI: 63.8-67.2%) for older women. The model with BI-RADS identified 18.4% population of non-Hispanic US women 50-70 years old above 3% 5-year risk (used for recommending risk-reducing medication in the US), with 42.4% of future cases expected to occur in this group. Addition of BI-RADS led to the reclassification of ~8.0% of US non-Hispanic White women aged 50-70 years at the high-risk threshold (3% 5-year risk), resulting in identification of 4.2% of additional future cases. Conclusion: Integrating BI-RADS with questionnaire-based risk factors and PRS resulted in improved risk stratification among women of European ancestry, with evidence of slight overestimation of risk for women at elevated risk. Additional validation of the integrated model in diverse populations is needed prior to considering clinical applications. Citation Format: Charlotta V. Mulder, Yon Ho Jee, Xin Yang, Christopher G. Scott, Chi Gao, Amber N. Hurson, Mikael Eriksson, Per Hall, Peter Kraft, Celine M. Vachon, Antonis C. Antoniou, Gretchen Gierach, Montserrat Garcia-Closas, Parichoy Pal Choudhury. Validation of an integrated breast cancer risk model with mammographic density in three prospective cohort studies [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2022; 2022 Apr 8-13. Philadelphia (PA): AACR; Cancer Res 2022;82(12_Suppl):Abstract nr 2256.

Athree-dimensional method to calculate mechanical advantage in mandibular function : Intra- and interexaminer reliability study.

Masticatory muscles are physically affected by several skeletal features. The muscle performance depends on muscle size, intrinsic strength, fiber direction, moment arm, and neuromuscular control. To date, for the masticatory apparatus, only atwo-dimensional cephalometric method for assessing the mechanical advantage, which is ameasure for the ratio of the output force to the input force in asystem, is available. This study determined the reliability and errors of athree-dimensional (3D) mechanical advantage calculation for the masticatory system. Using cone-beam computed tomography images from teenage patients undergoing orthodontic treatments, 36craniofacial landmarks were identified, and the moment arms for seven muscles and their load moment arms (biomechanical variables) were determined. The 3D mechanical advantage for each muscle was calculated. This procedure was repeated by three examiners. Reliability was verified using the intraclass correlation coefficient (ICC) and the errors by calculating the absolute differences, variance estimator and coefficient of variation (CV). Landmark coordinates demonstrated excellent intra- and interexaminer reliability (ICC 0.998-1.000; p < 0.0001). Intraexaminer data showed errors < 1.5 mm. Unsatisfactory interexaminer errors ranged from 1.51-5.83 mm. All biomechanical variables presented excellent intraexaminer reliability (ICC 0.919-1.000, p < 0.0001; CV < 7%). Interexaminer results were almost excellent, but with lower values (ICC 0.750-1.000, p < 0.0001; CV < 10%). However, the muscle moment arm and 3D mechanical advantage of the lateral pterygoid muscles had ICCs < 0.500 (p < 0.05) and CV < 30%. Intra- and interexaminer errors were ≤ 0.01 and ≤ 0.05, respectively. Both landmarks and biomechanical variables showed high reliability and acceptable errors. The proposed method is viable for the 3D mechanical advantage measure.

Nonparametric multivariate covariance chart for monitoring individual observations

Parametric and nonparametric multivariate control charts that are proven very useful in monitoring the covariance matrix of multivariate normally or “nearly” normally distributed continuous datasets have been proposed in statistical process control (SPC) literature. However, in many recent practical applications of SPC, the underlying systems or processes are characterised by discrete or a mixture of discrete and continuous multivariate random variables. In such cases, the available multivariate control charts for monitoring the covariance matrix of continuous processes are inadequate. We propose a multivariate nonparametric Shewhart-type chart for monitoring shifts in the covariance matrix of multivariate discrete or mixture of discrete and continuous random variables. The proposed chart first projects the multivariate dataset into Euclidean space. It then uses the Alt’s likelihood ratio obtained from the least absolute shrinkage and selection operator estimator that guarantees a well-conditioned estimate of the covariance matrix as the monitoring statistic. The proposed scheme does not require any parametric model assumptions and can be based on any distance measure of choice. It has the advantage of handling multivariate datasets of any type, including discrete, continuous or a mixture of discrete and continuous random variables. It uses the relationships among the process variables to build new variables that capture the dominant structure among the original variables. A bootstrap procedure is employed to obtain the control limit of the proposed chart for a suitable distance-based model through time. Simulation results show the advantage of the proposed chart in monitoring shifts in the covariance matrix. An illustrative example involving monitoring covariance structures of the lapping process in wafer semiconductor manufacturing and diagnosis single-proton emission computed tomography are provided to show the applications of the proposed chart.

Misclassification-robust semiparametric estimation of single-index binary-choice models

Summary In this paper, a new class of semiparametric estimators for single-index binary-choice models is introduced. The proposed estimators are based on the semiparametric indirect inference that identifies and estimates the parameters of the model via possibly misspecified auxiliary criteria. A large class of considered auxiliary criteria includes the ordinary least squares, nonlinear least squares, and nonlinear least absolute deviations estimators. Besides deriving the consistency and asymptotic normality of the proposed methods, we demonstrate that the proposed indirect inference methodology—at least for selected auxiliary criteria—combines weak distributional assumptions, good estimation precision, and robustness to misclassification of responses. We conduct Monte Carlo experiments and an application study to compare the finite-sample performance of the proposed and existing estimators.