Skewed Error Distribution Research Articles

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.

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.

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.

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

Performance of nonparametric multiple comparison tests under heteroscedasticity, dependency, and skewed error distribution

ABSTRACTIn this article, an extensive Monte Carlo simulation study is conducted to evaluate and compare nonparametric multiple comparison tests under violations of classical analysis of variance assumptions. Simulation space of the Monte Carlo study is composed of 288 different combinations of balanced and unbalanced sample sizes, number of groups, treatment effects, various levels of heterogeneity of variances, dependence between subgroup levels, and skewed error distributions under the single factor experimental design. By this large simulation space, we present a detailed analysis of effects of the violations of assumptions on the performance of nonparametric multiple comparison tests in terms of three error and four power measures. Observations of this study are beneficial to decide the optimal nonparametric test according to requirements and conditions of undertaken experiments. When some of the assumptions of analysis of variance are violated and number of groups is small, use of stepwise Steel-Dwass procedure with Holm's approach is appropriate to control type I error at a desired level. Dunn's method should be employed for greater number of groups. When subgroups are unbalanced and number of groups is small, Nemenyi's procedure with Duncan's approach produces high power values. Conover's procedure successfully provides high power values with a small number of unbalanced groups or with a greater number of balanced or unbalanced groups. At the same time, Conover's procedure is unable to control type I error rates.

A Monte Carlo study of traditional and rank-based picked-points analyses

ABSTRACTIn analysis of covariance with heteroscedastic slopes a picked-points analysis is often performed. Least-squares based picked-points analyses often lose efficiency (at times substantial) for nonnormal error distributions. Robust rank-based picked-points analyses are developed which are optimizable for heavy-tailed and/or skewed error distributions. The results of a Monte Carlo investigation of these analyses are presented. The situations include the normal model and models which violate it in one or several ways. Empirically the rank-based analyses appear to be valid over all these situations and more powerful than the least squares analysis for all the nonnormal models, while losing little efficiency at the normal model.

An evaluation and regional error modeling methodology for near-real-time satellite rainfall data over Australia

In providing uniform spatial coverage, satellite-based rainfall estimates can potentially benefit hydrological modeling, particularly for flood prediction. Maximizing the value of information from such data requires knowledge of its error. The most recent Tropical Rainfall Measuring Mission (TRMM) 3B42RT (TRMM-RT) satellite product version 7 (v7) was used for examining evaluation procedures against in situ gauge data across mainland Australia at a daily time step, over a 9 year period. This provides insights into estimating uncertainty and informing quantitative error model development, with methodologies relevant to the recently operational Global Precipitation Measurement mission that builds upon the TRMM legacy. Important error characteristics highlighted for daily aggregated TRMM-RT v7 include increasing (negative) bias and error variance with increasing daily gauge totals and more reliability at detecting larger gauge totals with a probability of detection of <0.5 for rainfall < ~3 mm/d. Additionally, pixel location within clusters of spatially contiguous TRMM-RT v7 rainfall pixels (representing individual rain cloud masses) has predictive ability for false alarms. Differences between TRMM-RT v7 and gauge data have increasing (positive) bias and error variance with increasing TRMM-RT estimates. Difference errors binned within 10 mm/d increments of TRMM-RT v7 estimates highlighted negatively skewed error distributions for all bins, suitably approximated by the generalized extreme value distribution. An error model based on this distribution enables bias correction and definition of quantitative uncertainty bounds, which are expected to be valuable for hydrological modeling and/or merging with other rainfall products. These error characteristics are also an important benchmark for assessing if/how future satellite rainfall products have improved.

A new multiple imputation method for bounded missing values

A proportioned residual draw method is introduced to impute bounded missing values. It is shown that this new regression-based imputation method is least biased and robust for heavy-tailed and skewed error distributions, regardless of number of boundaries and missing mechanisms.

Bayesian partial linear model for skewed longitudinal data.

Unlike majority of current statistical models and methods focusing on mean response for highly skewed longitudinal data, we present a novel model for such data accommodating a partially linear median regression function, a skewed error distribution and within subject association structures. We provide theoretical justifications for our methods including asymptotic properties of the posterior and associated semiparametric Bayesian estimators. We also provide simulation studies to investigate the finite sample properties of our methods. Several advantages of our method compared with existing methods are demonstrated via analysis of a cardiotoxicity study of children of HIV-infected mothers.

Confidence intervals for ARMA–GARCH Value-at-Risk: The case of heavy tails and skewness

It is a well-known result that, when the ARMA–GARCH model errors lack a finite fourth moment, the asymptotic distribution of the quasi-maximum likelihood estimator may not be Normal. In such a scenario the conventional bootstrap turns out inconsistent. Surprisingly, simulations show that the conventional bootstrap, despite its inconsistency, provides accurate confidence intervals for ARMA–GARCH Value-at-Risk (VaR) in case of various symmetric error distributions without finite fourth moment. The usual bootstrap does fail, however, in the presence of skewed error distributions without finite fourth moment. In this case several other methods for estimating confidence intervals fail as well. A residual subsample bootstrap is proposed to obtain confidence intervals for ARMA–GARCH VaR. According to theory, this ‘omnibus’ method produces confidence intervals with asymptotically correct coverage rates under very mild conditions. By means of a simulation study the favorable finite-sample properties of the residual subsample bootstrap are illustrated. Confidence intervals for ARMA–GARCH VaR with good coverage rates are established, even when other methods fail in the presence of skewed model errors without finite fourth moment. The estimation of confidence intervals by means of the residual subsample bootstrap is illustrated in an empirical application to daily stock returns.

Confidence Intervals for ARMA-GARCH Value-at-Risk

When the ARMA-GARCH model errors lack a finite fourth moment, the asymptotic distribution of the quasi-maximum likelihood estimator may not be Normal. In such a scenario the conventional bootstrap turns out inconsistent. Surprisingly, simulations show that the conventional bootstrap, despite its inconsistency, provides accurate confidence intervals for ARMA-GARCH Value-at-Risk (VaR) in case of various symmetric error distributions without finite fourth moment. The usual bootstrap does fail, however, in the presence of skewed error distributions without finite fourth moment. In this case several other methods for estimating confidence intervals fail as well. We propose a residual subsample bootstrap to obtain confidence intervals for ARMA-GARCH VaR. According to theory, this `omnibus' method produces confidence intervals with asymptotically correct coverage rates under very mild conditions. By means of a simulation study we illustrate the favorable finite-sample properties of the residual subsample bootstrap. We establish confidence intervals for ARMA-GARCH VaR with good coverage rates, even when other methods fail in the presence of skewed model errors without finite fourth moment. We illustrate the estimation of confidence intervals by means of the residual subsample bootstrap in an empirical application to daily stock returns.

Evaluation of a metofluthrin fan vaporizer device against phlebotomine sand flies (Diptera: Psychodidae) in a cutaneous leishmaniasis focus in the Judean Desert, Israel

The OFF! Clip-On fan vaporizer device releasing metofluthrin was evaluated against phletobomine sand flies in the Judean Desert, Israel, in October, 2009. A total of 76,400 sand flies was collected, with male flies representing 98.3%Phlebotomus sergenti and 1.7%P. papatasi. Females comprised 43.0% of the total catch and included 6.7% blood-fed females. Similar proportions of flies were collected in both suction and sticky traps. In trials with unbaited suction traps, similar numbers of sand flies were collected in traps with a metofluthrin device, blank device, or no device (i.e., suction only). In suction traps baited with CO(2) , higher numbers of P. sergenti males and blood-fed females were collected in traps with a blank device compared to traps with a metofluthrin device. In sticky traps baited with CO(2) , there were no significant differences between catches in traps with a metofluthrin device, blank device, or no device. The results suggest metofluthrin from the device is not repellent against sand flies in a field environment despite showing insecticidal activity against flies collected in suction traps.

A formal likelihood function for parameter and predictive inference of hydrologic models with correlated, heteroscedastic, and non‐Gaussian errors

Estimation of parameter and predictive uncertainty of hydrologic models has traditionally relied on several simplifying assumptions. Residual errors are often assumed to be independent and to be adequately described by a Gaussian probability distribution with a mean of zero and a constant variance. Here we investigate to what extent estimates of parameter and predictive uncertainty are affected when these assumptions are relaxed. A formal generalized likelihood function is presented, which extends the applicability of previously used likelihood functions to situations where residual errors are correlated, heteroscedastic, and non‐Gaussian with varying degrees of kurtosis and skewness. The approach focuses on a correct statistical description of the data and the total model residuals, without separating out various error sources. Application to Bayesian uncertainty analysis of a conceptual rainfall‐runoff model simultaneously identifies the hydrologic model parameters and the appropriate statistical distribution of the residual errors. When applied to daily rainfall‐runoff data from a humid basin we find that (1) residual errors are much better described by a heteroscedastic, first‐order, auto‐correlated error model with a Laplacian distribution function characterized by heavier tails than a Gaussian distribution; and (2) compared to a standard least‐squares approach, proper representation of the statistical distribution of residual errors yields tighter predictive uncertainty bands and different parameter uncertainty estimates that are less sensitive to the particular time period used for inference. Application to daily rainfall‐runoff data from a semiarid basin with more significant residual errors and systematic underprediction of peak flows shows that (1) multiplicative bias factors can be used to compensate for some of the largest errors and (2) a skewed error distribution yields improved estimates of predictive uncertainty in this semiarid basin with near‐zero flows. We conclude that the presented methodology provides improved estimates of parameter and total prediction uncertainty and should be useful for handling complex residual errors in other hydrologic regression models as well.