Skewed Error Research Articles

A Nonlinear Quantile Regression for Accelerated Destructive Degradation Testing Data

Traditional regression approaches to accelerated destructive degradation test (ADDT) data have modeled the mean curve as representative. However, maximum likelihood estimates (MLEs) of the mean model are likely to be biased when the data are non-Gaussian or highly skewed. The median model can be an alternative for skewed degradation data. In this work, we introduce a nonlinear quantile regression (QR) approach for estimating quantile curves of ADDT data. We propose an iterative QR algorithm that uses the generalized expectation-maximization (GEM) framework to estimate the parameters of the nonlinear QR ADDT model, based on the asymmetric Laplace distribution to accommodate non-Gaussian and skewed errors. Using the asymptotic properties of the QR parameter estimates, we estimate variance-covariance matrix for the τ th QR parameters using order statistics and bootstrap methods. We propose a new prediction method of the quantile of the failure-time distribution in the normal use condition. Confidence intervals for the quantiles of the failure-time distribution are constructed using the parametric bootstrap method. The proposed model is illustrated using an industrial application and compared with the existing model. Various quantile curve estimates derived using the QR ADDT model provide a more flexible modeling framework than the traditional mean ADDT modeling approach.

Rank-based regression with bootstrapping and the least square regression for analysis of small sample interrupted time-series data: simulation studies and application to illegal organ transplants

For interrupted time series with small samples and outliers, the standard least squares regression for estimating intervention effects can be biased when the assumption of normality is violated. This article proposes a rank-based regression method combined with bootstrapping to address outliers and non-normality. The proposed method minimizes the Jaeckel dispersion function based on ranked residuals, offering robustness against outliers, skewed error distributions, and correlated errors. Simulation studies and a real-world example compare the performance of the proposed method with the least squares method under various conditions. Both produce unbiased estimates for the intervention effects under normality. However, with non-normal errors, outliers, and correlated errors, the rank-based bootstrapping method outperforms least squares producing narrower confidence intervals, higher efficiency, lower type I error rates, and greater power. Application to the Istanbul declaration on illegal organ activities confirms the improved robustness and precision of the rank-based method with bootstrapping over least squares.

An Examination of Cryptocurrency Volatility: Insights from Skewed Error Innovation Distributions Within GARCH Model Frameworks

An Examination of Cryptocurrency Volatility: Insights from Skewed Error Innovation Distributions Within GARCH Model Frameworks

Sensitivity and robustness of randomization test and F‐test in some experimental designs

AbstractThis work examined the sensitivity of randomization test and F‐test in completely randomized design, randomized complete block design and Latin square design in order to ascertain which is more robust in terms of power (sensitivity) of the test and type I error rate in cases with normal error, skewed error (Chi‐square and Exponential distributions), by varying the sample sizes. Using Monte Carlo simulation, response observations were generated for normal error, skewed error at various sample sizes for each of the experimental designs. The results showed that for the normal error, the power and type I error rate of the F‐test were slightly smaller and greater respectively in each of the designs than that of the randomization test. Furthermore, for the skewed error, the randomization test was more sensitive than the F‐test, and also, the type I error was smaller in each of the designs.

Robust rank-based meta-analyses for two-sample designs with application to platelet counts of malaria infection data.

In this paper, we propose robust meta-analysis procedures for individual studies that report a broad range of robust summary statistics for a two-sample problem. Summary statistics of individual studies could be presented in different forms including full data, medians of the two samples, the Hodges-Lehman and Wilcoxon estimates of the location shift parameters. Data synthesis is made under both fixed-effect and random-effect meta-analysis models. We systematically compare these robust meta-analysis procedures via simulation studies to meta-analysis procedure based on sample means and variances from individual studies under a wide range of error distributions. We show that the coverage probabilities of the robust meta-analysis confidence intervals are quite close to the nominal confidence level. We also show that mean square error (MSE) of the robust meta-analysis estimator is considerably smaller than that of the non-robust meta-analysis estimator under the contaminated normal, heavy tailed and skewed error distributions. The robust meta-analysis procedures are then applied to platelet count reduction for malaria infected patients in Ghana.

Bayesian Spatio-Temporal Modeling for the Inpatient Hospital Costs of Alcohol-Related Disorders

AbstractUnderstanding how health care costs vary across different demographics and health conditions is essential to developing policies for health care cost reduction. It may not be optimal to apply the conventional mean regression due to its sensitivity to the high level of skewness and spatio-temporal heterogeneity presented in the cost data. To find an alternative method for spatio-temporal analysis with robustness and high estimation efficiency, we combine information across multiple quantiles and propose a Bayesian spatio-temporal weighted composite quantile regression (ST-WCQR) model. An easy-to-implement Gibbs sampling algorithm is provided based on the asymmetric Laplace mixture representation of the error term. Extensive simulation studies show that ST-WCQR outperforms existing methods for skewed error distributions. We apply ST-WCQR to investigate how patients’ characteristics affected the inpatient hospital costs for alcohol-related disorders and identify areas that could be targeted for cost reduction in New York State from 2015 to 2017.

Modeling Volatility of Daily Stock Return Using Generalized Autoregressive Conditional Heteroscadatic with Skewed Error Distribution and Stochastic Volatility Models

The problem of how best to model volatility of stock prices returns have continued to occupy the minds of researchers in this area. This study therefore compares the performance of two competitive volatility models: The Stochastic Volatility (SV) and Generalized Autoregressive Conditional Heteroscedasticity (GARCH) based on skewed error distributions models using daily stock returns of Fourteen Nigerian banks. The daily stock closing price of the selected banks were collected for the period 4/1/2007 to 31/12/2021 and the stationarity, normality and ARCH effect of the series were tested. The parameter of SV and different types of GARCH models based on skewed normal, skewed student t and skewed generalized error distribution were estimated and selection of the best model was done using Akaike Information Criterion (AIC). Post estimation and evaluation were carried out using the Mean Square Error (MSE) and Root Mean Square Error (RMSE). Results of analysis revealed a stationary but asymmetrically distributed return series with ARCH effect. The model forecasting performance proved APARCH (1,1) based on student t-distribution as the best among the competing GARCH models with the record of the least AIC value in ten of the fourteen daily bank stocks returns. Although, APARCH (1,1) and SV models were found to be comparable. SV model was however recommended as the best since it recorded the least RMSE value in 11 (79%) of the 14 bank stocks against 3 (21%) bank stock in favour of APARCH (1,1). This implies that SV model was found to be adequate in modeling volatility of the daily stock returns of ECO Bank, First City Monument bank, Fidelity bank, first bank, Guaranty Trust bank, Stanbic IBTC bank, Sterling bank, United Bank for Africa, Union bank, WEMA bank and Zenith bank; while APARCH (1,1) base on Skewed student ‘t’ distribution was preferred in modeling volatility of Access bank, Skye Bank and Unity bank. Results also showed the presence of volatility persistence, suggesting uncertainties and the risk of losses by investors. Therefore, it can be deduced that the application of a stochastic volatility model to the selected stock returns would drastically minimize the losses by investors.

Measurement error in linear regression models with fat tails and skewed errors

Linear regression models which account for skewed error distributions with fat tails have been previously studied. These two important features, skewness, and fat tails, are often observed in real data analyses. Covariates measured with an error also happen frequently in the observational data set-up. As a motivating example, wind speed as a covariate is usually used, among other covariates, to estimate the particulate matter (PM) which is one of the most critical air pollutants and has a major impact on human health and on the environment. However, the wind speed is measured with error and the distribution of PM is neither symmetric nor normally distributed (see Section “PM data application in Canada” for more details). Ignoring the issue of measurement error in covariates may produce bias in model parameters estimate and lead to wrong conclusions. In this paper, we propose an approach to study properly linear regression models where the covariates are measured with error and the error distribution is skewed with fat tails. We use a hierarchical Bayesian approach for inference, addressing also sensitivity of the results to priors. Performance of the proposed approach is evaluated through a simulation study and also by a real data application (PM in Canada).

Testing Error Distribution by Kernelized Stein Discrepancy in Multivariate Time Series Models

Knowing the error distribution is important in many multivariate time series applications. To alleviate the risk of error distribution mis-specification, testing methodologies are needed to detect whether the chosen error distribution is correct. However, the majority of existing tests only deal with the multivariate normal distribution for some special multivariate time series models, and thus cannot be used for testing the often observed heavy-tailed and skewed error distributions in applications. In this article, we construct a new consistent test for general multivariate time series models, based on the kernelized Stein discrepancy. To account for the estimation uncertainty and unobserved initial values, a bootstrap method is provided to calculate the critical values. Our new test is easy-to-implement for a large scope of multivariate error distributions, and its importance is illustrated by simulated and real data. As an extension, we also show how to test for the error distribution in copula time series models.

Development of a new modified hogg type adaptive scheme for multilevel models with diverse error distributions

The rank-based method is a well-known robust estimation technique in analyzing the linear models for non normal error distributions. The efficiency of rank-based analysis can be upgraded by selecting a suitable score function according to the probability distribution of the error term. In this study, a modified version of Hogg’s type adaptive scheme is developed by introducing a new set of cutoff values. The novel idea is to use the rank-based estimation by using the score functions selected through Hogg’s and modified Hogg’s schemes for multilevel models that generate cluster-correlated errors. The efficiency of both schemes is compared for symmetric, asymmetric, and light-tailed to heavy-tailed error distributions under various sample sizes through simulation study based on bias, variance, mean square error, the selected score function, and precision. The modified Hogg’s scheme produces a more efficient rank-based fit than Hogg’s scheme in case of skewed error distribution and produces equal efficiency for symmetric heavy, moderate, and light-tailed distributions. The empirical comparison of score selection through both schemes is also illustrated via a real example. The modified Hogg’s scheme considered the skewness and selected an appropriate score function, giving a more efficient fit than Hogg’s scheme that ignores the skewness.

Prioritizing of volatility models: a computational analysis using data envelopment analysis

AbstractEconomic crisis and uncertainty in global status quo affect stock markets around the world. This fact imposes improvement in the development of volatility models. However, the comparison among volatility models cannot be made based on a single‐error measure as a model can perform better in one‐error measure and worst in another. In this paper, we propose a two‐stage approach for prioritizing volatility models, where in the first stage we develop a novel slack‐based data envelopment analysis to rank volatility models. The robustness of the proposed approach has also been investigated using cluster analysis. In the second‐stage analysis, it is investigated whether the efficiency scores depend on model characteristics. These attributes concern the time needed in order to estimate the model, the value of Akaike Information Criterion, the number of models' significant parameters, groups and bias terms, and the error sum of squares (ESS). Further, dummy variables have been introduced to the regression model in order to find whether the employed model includes an in‐mean effect, whether the assumed distribution is skewed, and whether the employed model belongs to the generalized autoregressive conditional heteroskedasticity (GARCH) family. The main findings of this research show that the number of models' statistically significant coefficients, ESS, and in‐mean effects tend to increase the efficiency scores, while time elapsed, the number of statistically significant bias terms, and skewed error distributions tend to decrease the efficiency score.

Accounting for Skewed or One-Sided Measurement Error in the Dependent Variable

AbstractWhile classical measurement error in the dependent variable in a linear regression framework results only in a loss of precision, nonclassical measurement error can lead to estimates, which are biased and inference which lacks power. Here, we consider a particular type of nonclassical measurement error: skewed errors. Unfortunately, skewed measurement error is likely to be a relatively common feature of many outcomes of interest in political science research. This study highlights the bias that can result even from relatively “small” amounts of skewed measurement error, particularly, if the measurement error is heteroskedastic. We also assess potential solutions to this problem, focusing on the stochastic frontier model and Nonlinear Least Squares. Simulations and three replications highlight the importance of thinking carefully about skewed measurement error as well as appropriate solutions.

Unifying the U–Pb and Th–Pb methods: joint isochron regression and common Pb correction

Abstract. The actinide elements U and Th undergo radioactive decay to three isotopes of Pb, forming the basis of three coupled geochronometers. The 206Pb ∕238U and 207Pb ∕235U decay systems are routinely combined to improve accuracy. Joint consideration with the 208Pb ∕232Th decay system is less common. This paper aims to change this. Co-measured 208Pb ∕232Th is particularly useful for discordant samples containing variable amounts of non-radiogenic (“common”) Pb. The paper presents a maximum likelihood algorithm for joint isochron regression of the 206Pb ∕238Pb, 207Pb ∕235Pb and 208Pb ∕232Th chronometers. Given a set of cogenetic samples, this total-Pb/U-Th algorithm estimates the common Pb composition and concordia intercept age. U–Th–Pb data can be visualised on a conventional Wetherill or Tera–Wasserburg concordia diagram, or on a 208Pb ∕232Th vs. 206Pb ∕238U plot. Alternatively, the results of the new discordia regression algorithm can also be visualised as a 208Pbc ∕206Pb vs. 238U ∕206Pb or 208Pbc ∕207Pb vs. 235U ∕206Pb isochron, where 208Pbc represents the common 208Pb component. In its most general form, the total-Pb/U-Th algorithm accounts for the uncertainties of all isotopic ratios involved, including the 232Th ∕238U ratio, as well as the systematic uncertainties associated with the decay constants and the 238U ∕235U ratio. However, numerical stability is greatly improved when the dependency on the 232Th ∕238U-ratio uncertainty is dropped. For detrital minerals, it is generally not safe to assume a shared common Pb composition and concordia intercept age. In this case, the total-Pb/U-Th regression method must be modified by tying it to a terrestrial Pb evolution model. Thus, also detrital common Pb correction can be formulated in a maximum likelihood sense. The new method was applied to three published datasets, including low Th∕U carbonates, high Th∕U allanites and overdispersed monazites. The carbonate example illustrates how the total-Pb/U-Th method achieves a more precise common Pb correction than a conventional 207Pb-based approach does. The allanite sample shows the significant gain in both precision and accuracy that is made when the Th–Pb decay system is jointly considered with the U–Pb system. Finally, the monazite example is used to illustrate how the total-Pb/U-Th regression algorithm can be modified to include an overdispersion parameter. All the parameters in the discordia regression method (including the age and the overdispersion parameter) are strictly positive quantities that exhibit skewed error distributions near zero. This skewness can be accounted for using the profile log-likelihood method or by recasting the regression algorithm in terms of logarithmic quantities. Both approaches yield realistic asymmetric confidence intervals for the model parameters. The new algorithm is flexible enough that it can accommodate disequilibrium corrections and intersample error correlations when these are provided by the user. All the methods presented in this paper have been added to the IsoplotR software package. This will hopefully encourage geochronologists to take full advantage of the entire U–Th–Pb decay system.

Joint Response to the House Judiciary Committee on the State of Antitrust Law and Implications for Protecting Competition in Digital Markets

Economic research establishes that market power is now a serious problem. Growing market power harms consumers and workers, slows innovation, and limits productivity growth. Market power is on the rise in a number of major industries, including, for example, airlines, brewing, and hospitals, where multiple horizontal mergers that were allowed to proceed without antitrust challenge have markedly increased concentration in important markets and facilitated the exercise of market power. Exclusionary conduct by dominant companies that stifles competition from actual and potential rivals — including nascent rivals with capabilities for challenging a dominant firm’s market power and firms with competing R&D efforts — impairs what is often the most important economic force creating competitive pressure for dominant firms. This concern exists in digital marketplaces. Platforms are often insulated from platform competition to a substantial extent by substantial scale economies in supply and demand (network effects) combined with customer switching costs. Courts have contributed to increased monopoly power through decisions that have weakened the prohibitions against anticompetitive exclusionary conduct and anticompetitive mergers. The antitrust laws, as interpreted and enforced today, are inadequate to confront and deter growing market power in the U.S. economy and unnecessarily limit the ability of antitrust enforcers to address anticompetitive conduct. Many key antitrust precedents — particularly those precedents governing exclusionary conduct — rely on unsound economic theories or unsupported empirical claims about the competitive effects of certain practices. In part for this reason, the antitrust rules constructed by the courts reflect a systematically skewed error cost balance: they are too concerned to avoid both chilling procompetitive conduct and the high costs of litigation, and too dismissive of the costs of failing to deter harmful conduct. Excessively permissive precedents and unsound or unsupported economic claims have, in turn, encouraged overly cautious enforcement policies and overly demanding proof requirements and have discouraged government enforcers and private plaintiffs from bringing meritorious exclusionary conduct cases. The statement discusses a number of legal rules that are unsupported by or inconsistent with sound economic research that have contributed to overly permissive rules. The signatories to this letter strongly believe that antitrust enforcement has become too lax, in large part because of the courts and that Congress must act to correct this problem. The statement suggests a number of reforms could be considered. We do not collectively or unanimously endorse any of these, though some of us have done so in other contexts. We do believe that Congress has a historic opportunity to identify adverse trends in judicial interpretation of the antitrust and correct problems—not just by overriding damaging precedents, but also by reshaping the antitrust laws more broadly to enhance deterrence of anticompetitive conduct.

Gaussian process regression with skewed errors

In statistical literature, the Gaussian process is used as a prior process to treat a non-linear regression from a Bayesian viewpoint with normally distributed error terms. In this paper, we modify the Gaussian process regression (GPR) model by assuming the error terms of the GPR model follow a skew normal distribution instead of the normal distribution. We refer to this new modification as the Gaussian process regression with skewed errors (GPRSE). A key advantage of our model is that the well known GPR model is a subclass of the GPRSE model. Most importantly, we derive its predictive distribution for a new input location in a closed form. Moreover, the marginal likelihood of the GPRSE model is derived, and is used to train the model hyper-parameters. We also conduct simulation experiments to demonstrate the efficiency of our proposed approach compared to the GPR model.

An effective approach to linear calibration estimation with its applications

The availability of some extra information, along with the actual variable of interest, may be easily accessible in different practical situations. A sensible use of the additional source may help to improve the properties of statistical techniques. In this study, we focus on the estimators for calibration and intend to propose a setup where we reply only on first two moments instead of modeling the whole distributional shape. We have proposed an estimator for linear calibration problems and investigated it under normal and skewed environments. We have partitioned its mean squared error into intrinsic and estimation components. We have observed that the bias and mean squared error of the proposed estimator are function of four dimensionless quantities. It is to be noticed that both the classical and the inverse estimators become the special cases of the proposed estimator. Moreover, the mean squared error of the proposed estimator and the exact mean squared error of the inverse estimator coincide. We have also observed that the proposed estimator performs quite well for skewed errors as well. The real data applications are also included in the study for practical considerations.

Accounting for Skewed or One-Sided Measurement Error in the Dependent Variable

While classical measurement error in the dependent variable in a linear regression framework results only in a loss of precision, non-classical measurement error can lead to estimates which are biased and inference which lacks power. Here, we consider a particular type of non-classical measurement error: skewed errors. Unfortunately, skewed measurement error is likely to be a relatively common feature of many outcomes of interest in political science research. This study highlights the bias that can result even from relatively small amounts of skewed measurement error, particularly if the measurement error is heteroskedastic. We also assess potential solutions to this problem, focusing on the stochastic frontier model and nonlinear least squares. Simulations and three replications highlight the importance of thinking carefully about skewed measurement error, as well as appropriate solutions.

Does the Assumption on Innovation Process Play an Important Role for Filtered Historical Simulation Model?

Most of the financial institutions compute the Value-at-Risk (VaR) of their trading portfolios using historical simulation-based methods. In this paper, we examine the Filtered Historical Simulation (FHS) model introduced by Barone-Adesi et al. (1999) theoretically and empirically. The main goal of this study is to find an answer for the following question: “Does the assumption on innovation process play an important role for the Filtered Historical Simulation model?”. For this goal, we investigate the performance of FHS model with skewed and fat-tailed innovations distributions such as normal, skew normal, Student’s-t, skew-T, generalized error, and skewed generalized error distributions. The performances of FHS models are evaluated by means of unconditional and conditional likelihood ratio tests and loss functions. Based on the empirical results, we conclude that the FHS models with generalized error and skew-T distributions produce more accurate VaR forecasts.

Bayesian analysis of financial volatilities addressing long-memory, conditional heteroscedasticity and skewed error distribution

Bayesian analysis of financial volatilities addressing long-memory, conditional heteroscedasticity and skewed error distribution

Boundedness-preserving implicit correction of mesh-induced errors for VOF based heat and mass transfer

Spatial discretisation of geometrically complex computational domains often entails unstructured meshes of general topology for Computational Fluid Dynamics (CFD). Mesh skewness is then typically encountered causing severe deterioration of the formal order of accuracy of the discretisation, or boundedness of the solution, or both. Particularly methods inherently relying on the accurate and bounded transport of sharp fields suffer from all types of mesh-induced skewness errors, namely both non-orthogonality and non-conjunctionality errors.This work is devoted to a boundedness-preserving strategy to correct for skewness errors arising from discretisation of advection and diffusion terms within the context of interfacial heat and mass transfer based on the Volume-of-Fluid methodology. The implementation has been accomplished using a second-order finite volume method with support for unstructured meshes of general topology. We examine and advance suitable corrections for the finite volume discretisation of a consistent single-field model, where both accurate and bounded transport due to diffusion and advection is crucial. In order to ensure consistency of both the volume fraction and the species concentration transport, i.e. to avoid artificial heat or species transfer, corrections are studied for both cases using distorted 2D meshes. The cross interfacial jump and adjacent sharp gradients of species concentration render the correction for skewness-induced diffusion and advection errors additionally demanding and has not so far been addressed in the literature.