Unbiasedness Research Articles

On the bias in the AUC variance estimate

The area under the Receiver Operating Characteristic (ROC) curve (AUC) is a standard metric for quantifying and comparing binary classifiers. A popular approach to estimating the AUCs and the associated variabilities – the variance of the AUC or the full covariance matrix of multiple correlated AUCs – is the one proposed by DeLong et al. (1988), which is based on the Mann Whitney two-sample U-statistics. The bias of a variance estimator is an important factor in applications such as hypothesis testing and construction of confidence intervals – a negatively biased variance estimator may lead to incorrect conclusions, and a positive bias is conservative hence preferable. In this work, we show that the (co-)variance estimate in DeLong’s approach is always positively biased. More specifically, the difference matrix between the expectation of the estimated covariance and the true covariance is a positive semi-definite matrix. This bias is non-negligible when the sample size is small, and quickly diminishes as the sample size increases. Our method relies on constructing, from the AUC kernel, a random variable whose (co-)variance matrix coincides with the bias, thereby establishing the claim. We also discuss alternative approaches to AUC variance estimation that may potentially reduce the bias.

A New biased estimator and variations based on the Kibria Lukman Estimator

A New biased estimator and variations based on the Kibria Lukman Estimator

Kibria–Lukman estimator for the zero inflated negative binomial regression model: theory, simulation and applications

The zero inflated negative binomial model is an appropriate choice to model count response variables with excessive zeros and over-dispersion simultaneously. This article addresses the parameter estimation for the zero-inflated negative binomial model when there are many predictors and the problems of multicollinearity are present. Since, under multicollinearity, the widely used maximum likelihood estimator becomes unstable, this article proposes an alternative estimator, called Kibria–Lukman estimator for the zero inflated negative binomial model and provided some new biasing parameters. Then compared the performance of Kibria–Lukman estimator with some of the traditional biased estimators, that is, ridge and Liu estimators. The superiority of the proposed estimator is systematically scrutinized both theoretically and numerically. For numerical assessment, we conduct a Monte Carlo simulation study under different controlled factors. A numerical example is also applied to appraise the performance of proposed estimator. Based on the simulation and example results, we observed that the performance of the proposed estimator under different biasing parameters was better than that of the maximum likelihood estimator and other involved biased estimation methods when there exists a high but imperfect multicollinearity.

BETA-BINOMIAL MODEL IN SMALL AREA ESTIMATION USING HIERARCHICAL LIKELIHOOD APPROACH

Small Area Estimation is a statistical method used to estimate parameters in sub-populations with small or even no sample sizes. This research aims to evaluate the Beta-Binomial model's performance for estimating small areas at the area level. The estimation method used is Hierarchical Likelihood (HL). The data used are simulation data and empirical data. Simulation studies were used to investigate the proposed model. The estimator's Mean Squared Error of Prediction (MSEP) and Absolute Bias (AB) estimator values determine the best estimation criteria. An empirical study using data on the illiteracy rate at the sub-district level in Bengkulu Province. The results of the simulation study show that, in general, the parameter estimators are nearly unbiased. Proportion prediction has the same tendency as parameters. Finally, the HL estimator has a small MSEP estimator. The results of an empirical study show that the average illiteracy rate in Bengkulu province is quite diverse. Kepahiang District has the highest average illiteracy rate in Bengkulu Province in 2021.

Bias reduction for semi-competing risks frailty model with rare events: application to a chronic kidney disease cohort study in South Korea.

In a semi-competing risks model in which a terminal event censors a non-terminal event but not vice versa, the conventional method can predict clinical outcomes by maximizing likelihood estimation. However, this method can produce unreliable or biased estimators when the number of events in the datasets is small. Specifically, parameter estimates may converge to infinity, or their standard errors can be very large. Moreover, terminal and non-terminal event times may be correlated, which can account for the frailty term. Here, we adapt the penalized likelihood with Firth's correction method for gamma frailty models with semi-competing risks data to reduce the bias caused by rare events. The proposed method is evaluated in terms of relative bias, mean squared error, standard error, and standard deviation compared to the conventional methods through simulation studies. The results of the proposed method are stable and robust even when data contain only a few events with the misspecification of the baseline hazard function. We also illustrate a real example with a multi-centre, patient-based cohort study to identify risk factors for chronic kidney disease progression or adverse clinical outcomes. This study will provide a better understanding of semi-competing risk data in which the number of specific diseases or events of interest is rare.

The odd log- logistic Power Inverse Lindley distribution: Model, Properties and Applications

In this article, we introduce a new three-parameter odd log-logistic power inverse Lindley distribution and discuss some of its properties. These include the shapes of the density and hazard rate functions, mixture representation, the moments, the quantile function, and order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and root mean square error of the maximum likelihood estimators of the parameters. An application of the model to three real data sets is presented finally and compared with the fit attained by some other well-known two and three-parameter distributions for illustrative purposes. It is observed that the proposed model has some advantages in analyzing lifetime data as compared to other popular models in the sense that it exhibits varying shapes and shows more flexibility than many currently available distributions.

Magnification bias estimators for realistic surveys: an application to the BOSS survey

ABSTRACT In addition to the intrinsic clustering of galaxies themselves, the spatial distribution of galaxies observed in surveys is modulated by the presence of weak lensing due to matter in the foreground. This effect, known as magnification bias, is a significant contaminant to analyses of galaxy-lensing cross-correlations and must be carefully modelled. We present a method to estimate the magnification bias in spectroscopically confirmed galaxy samples based on finite differences of galaxy catalogues while marginalizing over errors due to finite step size. We use our estimator to measure the magnification biases of the CMASS and LOWZ samples in the SDSS BOSS galaxy survey, analytically taking into account the dependence on galaxy shape for fibre and PSF magnitudes, finding αCMASS = 2.71 ± 0.02 and αLOWZ = 2.45 ± 0.02 and quantify modelling uncertainties in these measurements. Finally, we quantify the redshift evolution of the magnification bias within the CMASS and LOWZ samples, finding a difference of up to a factor of three between the lower and upper redshift bounds for the former. We discuss how to account for this evolution in modelling and its interaction with commonly applied redshift-dependent weights. Our method should be readily applicable to upcoming surveys and we make our code publicly available as part of this work.

Inferring unknown unknowns: Regularized bias-aware ensemble Kalman filter

Because of physical assumptions and numerical approximations, low-order models are affected by uncertainties in the state and parameters, and by model biases. Model biases, also known as model errors or systematic errors, are difficult to infer because they are ‘unknown unknowns’, i.e., we do not necessarily know their functional form a priori. With biased models, data assimilation methods may be ill-posed because either (i) they are ‘bias-unaware’ because the estimators are assumed unbiased, (ii) they rely on an a priori parametric model for the bias, or (iii) they can infer model biases that are not unique for the same model and data. First, we design a data assimilation framework to perform combined state, parameter, and bias estimation. Second, we propose a mathematical solution with a sequential method, i.e., the regularized bias-aware ensemble Kalman Filter (r-EnKF), which requires a model of the bias and its gradient (i.e., the Jacobian). Third, we propose an echo state network as the model bias estimator. We derive the Jacobian of the network, and design a robust training strategy with data augmentation to accurately infer the bias in different scenarios. Fourth, we apply the r-EnKF to nonlinearly coupled oscillators (with and without time-delay) affected by different forms of bias. The r-EnKF infers in real-time parameters and states, and a unique bias. The applications that we showcase are relevant to acoustics, thermoacoustics, and vibrations; however, the r-EnKF opens new opportunities for combined state, parameter and bias estimation for real-time and on-the-fly prediction in nonlinear systems.

Understanding implications of detection heterogeneity in wildlife abundance estimation

AbstractNegative bias in mark‐recapture abundance estimators due to heterogeneity in detection (capture) probability is a well‐known problem, but we believe most biologists do not understand why heterogeneity causes bias and how bias can be reduced. We demonstrate how heterogeneity creates dependence and bias in mark‐recapture approaches to abundance estimation. In comparison, heterogeneity, and hence estimator bias, is not as problematic for distance sampling and mark‐resight methods because both techniques estimate detection probabilities based on a known quantity. We show how the introduction of a known number of individuals planted into a study population prior to a mark‐recapture survey can reduce bias from heterogeneity in detection probability. We provide examples with simulation and an analysis of motion‐sensitive camera data from a study population of introduced eastern wild turkeys (Meleagris gallopavo silvestris) of known size with a subset of telemetered birds. In choosing a method for abundance estimation, careful consideration should be given to assumptions and how heterogeneity in detection probability can be accommodated for each application.

A more accurate estimation with kernel machine for nonparametric spatial lag models

Ignoring potential spatial autocorrelation in georeferenced data may cause biased estimators. Furthermore, existing studies assume insufficiently flexible structure of spatial lag model for some practical applications, which makes it difficult to portray the complex relationship between responses and covariates. Thus, we propose a novel garrotized kernel machine estimation method for the nonparametric spatial lag model and develop an eigenvector spatial filtering algorithm with sparse regression to filter spatial autocorrelation out of the residuals. The “one-group-at-a-time” cyclical coordinate descent algorithm is introduced for a solution path of tuning parameters. Our method can better describe the potential nonlinear relationship between responses and covariates, making it possible to model high-order interaction effects among covariates. Numerical results and the analysis of commodity residential house prices in large and medium-sized Chinese cities indicate that the proposed method achieves better prediction performance compared with competing ones. The result of real data analysis can provide guidance for the government to take targeted suppression measures of house prices for different areas.

Distance sampling and spatial capture-recapture for estimating density of Northern Bobwhite

Obtaining accurate density estimates is critical for managers making decisions regarding wildlife populations. A variety of methods have been used to estimate population density, with two of the most common being distance sampling (DS) and spatial capture recapture (SCR). We evaluated precision and bias of density estimators through a simulation study of DS using data collected from either human observer point counts or automated recording units (ARUs), and SCR using trapping data or trapping data augmented with telemetry data. We then fit each of the four models to empirical data collected during autumn 2016–2018 for a population of northern bobwhite Colinus virginianus on Di-Lane Wildlife Management Area in Georgia. Density estimates in our simulation study were relatively unbiased using all four methods but were most accurate for the two SCR methods. Empirical density estimates were similar across methods and years, with annual averages of 0.41 birds per ha in 2016, 0.44 birds per ha in 2017 and 0.31 birds per ha in 2018. In general, both SCR methods were also the most precise using the empirical data, although ARU distance sampling did also produce comparable density estimates and precision values. Although SCR methods performed the best overall, cost and increased labor of these monitoring programs should be evaluated in relation to the relative ease of ARU deployment or point count surveys, which also provided adequate, and often similar, density estimates. Integrating these constraints into a structured decision-making framework will aid managers weighing decisions regarding how to monitor, and ultimately make management decisions.

A New Biased Two-Parameter Estimator in Linear Regression Model

The most frequently used estimation technique in the linear regression model is the ordinary least squares (OLS) estimator. The presence of multicollinearity makes the technique inefficient and gives misleading results. This study proposed a new biased two-parameter estimator to deal with the multicollinearity problem. Theory and simulation results show that this estimator outperforms existing estimators considered under some conditions, according to the mean squares error (MSE) criterion. Finally, the real-life dataset illustrates the paper's findings, which agree with the theoretical and simulation results.

Covariate selection in causal learning under non-Gaussianity.

Understanding causal mechanisms is a central goal in the behavioral, developmental, and social sciences. When estimating and probing causal effects using observational data, covariate adjustment is a crucial element to remove dependencies between focal predictors and the error term. Covariate selection, however, constitutes a challenging task because availability alone is not an adequate criterion to decide whether a covariate should be included in the statistical model. The present study introduces a non-Gaussian method for covariate selection and provides a forward selection algorithm for linear models (i.e., non-Gaussian forward selection; nGFS) to select appropriate covariates from a set of potential control variables to avoid inconsistent and biased estimators of the causal effect of interest. Further, we demonstrate that the forward selection algorithm has properties compatible with principles of direction of dependence, i.e., probing whether the causal target model is correctly specified with respect to the causal direction of effects. Results of a Monte Carlo simulation study suggest that the selection algorithm performs well, in particular when sample sizes are large (i.e., n ≥ 250) and data strongly deviate from Gaussianity (e.g., distributions with skewness beyond 1.5). An empirical example is given for illustrative purposes.

Extremal index: estimation and resampling

The duration of extremes in time leads to a phenomenon known as clustering of high values, with a strong impact on risk assessment. The extremal index is a measure developed within Extreme Value Theory that quantifies the degree of clustering of high values. In this work we will consider the cycles estimator introduced in Ferreira and Ferreira (Ann Inst Henri Poincare Probab Stat 54(2):587–605, 2018). A reduced bias estimator based on the Jackknife methodology will be presented. The bootstrap technique will also be considered in the inference and will allow to obtain confidence intervals. The performance will be analyzed based on simulation. We found our proposal effective in reducing bias and it compares favorably with some well-known methods. An application of the methods to real data will also be presented.

A new improvement Liu-type estimator for the Bell regression model

The Poisson Regression Model (PRM) is a well-known model in applications when the response variable consists of count data. However, Bell Regression Model (BRM) is proposed recently as an alternative to the PRM in some cases where the data is over-dispersed. But, multicollinearity between explanatory variables negatively affects traditional estimation methods, such as MLE. Therefore, to avoid this problem, several shrinkage estimators are proposed in the BRM. In this study, a new improved Liu-type estimator is proposed as an alternative to the other proposed biased estimators for the BRM to model count data with over-dispersion. Furthermore, the Monte Carlo simulation studies are executed to compare the performances of the proposed biased estimators. Finally, the obtained results are illustrated in real data.

Minmers are a generalization of minimizers that enable unbiased local Jaccard estimation.

The Jaccard similarity on k-mer sets has shown to be a convenient proxy for sequence identity. By avoiding expensive base-level alignments and comparing reduced sequence representations, tools such as MashMap can scale to massive numbers of pairwise comparisons while still providing useful similarity estimates. However, due to their reliance on minimizer winnowing, previous versions of MashMap were shown to be biased and inconsistent estimators of Jaccard similarity. This directly impacts downstream tools that rely on the accuracy of these estimates. To address this, we propose the minmer winnowing scheme, which generalizes the minimizer scheme by use of a rolling minhash with multiple sampled k-mers per window. We show both theoretically and empirically that minmers yield an unbiased estimator of local Jaccard similarity, and we implement this scheme in an updated version of MashMap. The minmer-based implementation is over 10 times faster than the minimizer-based version under the default ANI threshold, making it well-suited for large-scale comparative genomics applications. MashMap3 is available at https://github.com/marbl/MashMap.

Improving SUVR quantification by correcting for radiotracer clearance in tissue

Standardized Uptake Value Ratio (SUVR) is a widely reported semi-quantitative positron emission tomography (PET) outcome measure, partly because of its ease of measurement from short scan durations. However, in brain, SUVR is often a biased estimator of the gold-standard distribution volume ratio (DVR) due to non-equilibrium conditions, i.e., clearance of the radiotracer in relevant tissues. Factors that affect radiotracer metabolism and clearance such as medication or subject groups could lead to artificial differences in SUVR. This work developed a correction that reduces the bias in SUVR (estimated from a short 15–30 min PET imaging session) by accounting for the effects of tracer clearance observed during the late SUVR time window. The proposed correction takes the form of a one-step non-linear algebraic transform of SUVR that is a function of radiotracer dependent parameters such as clearance rates from the reference and target tissues, and population averaged reference region clearance rate ( k 2 , ref ). An important observation was the need for accurate estimation of radiotracer clearance rate in target tissue, which was addressed with a regression based model. Simulations and human data from two different radiotracers (healthy controls for [11C]LSN3172176, healthy controls and Parkinson’s disease subjects for [18F]FE-PE2I) were used to validate the correction and evaluate its benefits and limitations. SUVR correction in human data significantly reduced mean SUVR bias across brain regions and subjects (from ∼25% for SUVR to <10% for corrected SUVR). This correction also significantly reduced the variability of this bias across brain regions for both tracers (approximately 50% for [11C]LSN3172176, 20% for [18F]FE-PE2I). Future work should investigate the benefits of using corrected SUVR in other populations and with different tracers.

Inflation risk and stock returns: Evidence from US aggregate and sectoral markets

This study examines the relation between inflation risk and stock returns and finds a negative relation in US aggregate data. A comparable result is found in other G7 economies. Introducing the inflation-induced equity market volatility as an incremental variable consistently produces a negative effect on stock returns. The same finding holds true for monetary policy uncertainty, which also produces a negative relation. Ignoring these two elements will produce a biased estimator of inflation’s effect on stock returns. Testing of sectoral stocks in the US market indicates that most sectors find a significant number of sectors exhibit negative inflation coefficients, confirming the Fama proxy hypothesis. The exception is the energy sector that presents a positive sign, indicating a hedging capacity against inflation. Including the interacting term between a change in monetary policy uncertainty and lagged equity market volatility, the evidence suggests an enhanced negative effect on stock returns.

A phase II multivariate EWMA chart for monitoring multi-dimensional ratios of process means with individual observations

In recent years, there has been a resurgence in the development of control charts for monitoring the mean of the ratio of two correlated variables. However, most of the existing research has focused on the univariate mean of the ratio of two correlated variables under the assumption that the process follows a bivariate normal distribution. Furthermore, most of the existing research utilize biased estimators of the mean of the ratio of two correlated variables to develop control charts. More importantly, in certain applications, critical quality characteristics to be closely monitored are actually the ratios of the means of correlated variables, in that the process is considered to be stable as long as the ratios of the means of correlated variables remain constant at given levels, regardless of how each variable changes. We are thus motivated in this study to develop an exponentially weighted moving average (EWMA) based Phase II control chart for monitoring multi-dimensional ratios of the means of correlated variables.In this study, we provide a general framework for estimating parameters and control limits which is applicable without having to assume that the process follows a multivariate normal distribution. The performance of the proposed chart is evaluated under different multivariate distributions. Finally, a real data application of the proposed chart is presented to illustrate the practicality of the proposed chart.

Bootstrap bandwidth selection for the pair correlation function of inhomogeneous spatial point processes

This work focuses on kernel estimation of the pair correlation function (PCF) for inhomogeneous spatial point processes. We propose a bootstrap bandwidth selector based on minimizing the mean integrated squared error (MISE). The variance term is estimated by nonparametric bootstrap, and the bias by a plug-in approach using a pilot estimator of the PCF. Kernel estimators of the PCF also require a pilot estimator of the first-order intensity. We test the performance of the bandwidth selector and the role of the pilot intensity estimator in a simulation study. The bootstrap bandwidth selector is competitive with cross-validation procedures, but the contribution of the bandwidth parameter to the goodness-of-fit of the kernel PCF estimator is minor in comparison with that of the pilot intensity function. The data-based kernel intensity estimator leads to biased kernel PCF estimators, while both kernel and parametric covariate-based intensities provide accurate estimators of the PCF.