Bootstrap Intervals Research Articles

115 Confidence intervals for validation statistics with data truncation in genomic prediction

Abstract Validation by data truncation is a common practice in genetic evaluations because of the interest in predicting the genetic merit of a set of young selection candidates. Two of the most used validation methods in genetic evaluations are predictivity (correlation between pre-adjusted phenotypes and EBV) divided by the square root of the heritability) and the linear regression (LR) method. Both methods compare predictions with the whole and partial datasets obtained by removing the information related to a set of validation individuals. Confidence intervals (CI) for predictivity and the LR method can be obtained by k-fold validation or bootstrapping. Analytical or frequentist CI are unavailable for predictivity and the LR method and would be beneficial to avoid running several validations. The analytical CI can also help test the quality of bootstrap intervals. This study aimed to derive analytical CI for predictivity, and statistics included in the LR method (bias, dispersion, ratio of accuracies, and reliability). The CI for the bias, dispersion, and reliability depends on the (co)variances of the EBV across the individuals in the validation set. The CI for the ratio of accuracies and predictivity were obtained through the Fisher transformation. We showed the adequacy of the analytical CI using simulation. The analytical CI were closer to the simulated ones. Bootstrap CI tend to be narrower than the simulated ones. Estimating the sampling variation of predictivity and the statistics in the LR method without replication or bootstrap is possible for any dataset with the method proposed in this study. All the formulas derived in this study were implemented in a new program belonging to the BLUPF90 suite called Validationf90.

Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model

Accelerated life tests play a very critical role in reliability analysis because highly reliable products are being produced with recent advanced technologies to sustain the market demand and competition. A progressive-stress accelerated life test is one of the kinds of accelerated life tests that allows applied stress to change continuously. Considering the importance of progressive-stress accelerated life tests, this paper deals with progressive-stress accelerated life tests when the lifetimes of units follow the log-logistic distribution and are progressively type-II censored where the associated scale parameter conforms to the inverse power law. Different estimators of model parameters are derived using maximum likelihood and Bayesian methods. Interval estimation is also considered. In the sequel, approximate confidence, bootstrap, and Bayes credible intervals are constructed. Bayes estimators are obtained under squared error, LINEX, and entropy loss functions using proper and improper prior distributions. A simulation study is conducted based on various censoring schemes. The coverage percentages and interval widths are computed via Monte Carlo simulations. Bayes predictive estimates and intervals are also obtained. Finally, two different accelerated life test data are analyzed for illustration purposes.

Reliability estimation of s-out-of-k system with Kumaraswamy distribution based on partially constant stress accelerated life tests

In reliability theory, the reliability inference for s-out-of-k systems holds significant importance. In this paper, we explore the estimation of reliability for s-out-of-k systems based on partially constant stress accelerated life tests. Assume that the latent failure times of the components follow the Kumaraswamy distribution. Maximum likelihood estimates for the unknown parameters are established, and their uniqueness is demonstrated. In addition, confidence intervals for the unknown parameters are constructed using the covariance matrix. Confidence intervals for the reliability functions are determined by the Delta method, while Bootstrap intervals are provided for comparison purposes. Subsequently, Bayesian point and interval estimates based on MCMC techniques considering different loss functions are discussed. Lastly, we conduct an extensive simulation study and analyse one real data set, which reveals that the Bayesian approach yields the best results.

Electricity market price forecasting using ELM and Bootstrap analysis: A case study of the German and Finnish Day-Ahead markets

Electricity market liberalization and the absence of cost-efficient energy storage technologies have led to the transformation of state-owned electricity companies into complex electricity market entities, each having a different time horizon. Deregulation has intensified competition, giving rise to increased uncertainty caused by a multitude of interrelated exogenous factors, resulting in unexpected fluctuations in electricity prices. As a consequence, market participants encounter elevated risks and seek effective mitigation strategies. In this paper, the challenges described in the literature are addressed by studying price distribution histograms in the German and Finnish electricity markets. The objective is to identify normal price intervals that can serve as a foundation for an integrated Day-Ahead price forecasting methodology. A novel approach utilizing the Extreme Learning Machine in combination with Bootstrap intervals is proposed and applied to both markets. The findings demonstrate that Bootstrap intervals effectively capture normal prices, whereas extremely high prices typically align with the upper limits of Bootstrap intervals. Conversely, negative prices tend to fall outside the lower boundaries of the intervals. In order to assess the performance of the proposed methodology, a comparative analysis of its forecasting accuracy against the well-established Generalized AutoRegressive Conditional Heteroskedasticity and AutoRegressive Fractionally Integrated Moving Average models is conducted. In addition, both the computational efficiency and forecasting accuracy of the Extreme Learning Machine in comparison to the Artificial Neural Network are assessed. The results reveal the superior efficiency of the Extreme Learning Machine. The developed forecasting model could potentially assist market participants in making well-informed decisions and executing optimal bidding strategies in response to various scenarios before the Day-Ahead market closes. Notably, the proposed methodology transcends the limitations of fixed price thresholds and effectively addresses market nuances, including the occurrence of negative prices, thus offering a more comprehensive approach for electricity price forecasting.

Confidence intervals for validation statistics with data truncation in genomic prediction

BackgroundValidation by data truncation is a common practice in genetic evaluations because of the interest in predicting the genetic merit of a set of young selection candidates. Two of the most used validation methods in genetic evaluations use a single data partition: predictivity or predictive ability (correlation between pre-adjusted phenotypes and estimated breeding values (EBV) divided by the square root of the heritability) and the linear regression (LR) method (comparison of “early” and “late” EBV). Both methods compare predictions with the whole dataset and a partial dataset that is obtained by removing the information related to a set of validation individuals. EBV obtained with the partial dataset are compared against adjusted phenotypes for the predictivity or EBV obtained with the whole dataset in the LR method. Confidence intervals for predictivity and the LR method can be obtained by replicating the validation for different samples (or folds), or bootstrapping. Analytical confidence intervals would be beneficial to avoid running several validations and to test the quality of the bootstrap intervals. However, analytical confidence intervals are unavailable for predictivity and the LR method.ResultsWe derived standard errors and Wald confidence intervals for the predictivity and statistics included in the LR method (bias, dispersion, ratio of accuracies, and reliability). The confidence intervals for the bias, dispersion, and reliability depend on the relationships and prediction error variances and covariances across the individuals in the validation set. We developed approximations for large datasets that only need the reliabilities of the individuals in the validation set. The confidence intervals for the ratio of accuracies and predictivity were obtained through the Fisher transformation. We show the adequacy of both the analytical and approximated analytical confidence intervals and compare them versus bootstrap confidence intervals using two simulated examples. The analytical confidence intervals were closer to the simulated ones for both examples. Bootstrap confidence intervals tend to be narrower than the simulated ones. The approximated analytical confidence intervals were similar to those obtained by bootstrapping.ConclusionsEstimating the sampling variation of predictivity and the statistics in the LR method without replication or bootstrap is possible for any dataset with the formulas presented in this study.

Genetic features of the island populations of sockeye salmon Oncorhynchus nerka (Walbaum, 1792) in Russian part of the species range

Sockeye salmon genetic diversity in the Kuril Islands (Iturup, Urup, Paramushir, Shumshu) and the Commander Islands (Bering Island) was examined on 7 microsatellite loci. Average estimates of the heterozygocity observed on different loci vary from 0.300 to 0.858. General estimate of genetic differentiation θst is 15%, with a 95% confidence bootstrap interval [10.40–21.18%]. The results obtained in the program BOTTLENECK 1.2.02 made it possible to identify the passage of the “bottle neck” by all island populations. Unique properties and significant differences from each other are shown for allfivчe populations examined. The level of sockeye salmon intrapopulation diversity found in the Kuril and Commander Islands is significantly lower compared to that in the continental water bodies, what is generally typical for marginal populations of the species range.

The violent death toll from the Iraq War: 2003-2023.

From the beginning of the Iraq war, in March of 2003, to the present day, controversy has swirled around the death toll of the war. This paper narrows down the range of uncertainty for the numbers and trends in violent deaths in the war. I assemble and appraise all primary sources that cover the period from March of 2003 onwards-six sample surveys plus a casualty recording project (Iraq Body Count [IBC]). Data permitting, I present cumulative monthly figures with, for the surveys, 95% bootstrapped uncertainty intervals. The analysis uncovers a core of high-quality mainstream sources that are highly consistent with each another. In addition, there are three outlier surveys that are compromised by serious flaws and produce estimates far outside the mainstream. Discarding the outlying and flawed surveys reveals a clear picture of the violent death toll from the Iraq war. IBC figures, extended to include combatants, occupy a central position within the mainstream range of estimates. The strong consistency across the high-quality sources provides a rare validation of three war-death-measurement methodologies-household-based surveys, sibling-based surveys, and casualty recording. Methodological success notwithstanding, we must transcend the numbers to truly comprehend the human costs of the war.

Exponentiated half‐logistic Weibull distribution with reliability inference

AbstractThis paper presents an in‐depth analysis of the exponentiated half‐logistic Weibull (EHLW) distribution, investigating its fundamental statistical properties and exploring parameters estimation using both maximum likelihood and Bayesian approaches. Specifically, our focus centers on the analysis of stress–strength reliability, denoted as , where the strength variable follows EHLW distribution, and the stress variable follows either Weibull distribution or the EHLW distribution. The estimation of the parameter is discussed in both scenarios, employing both Maximum Likelihood and Bayesian approaches. Furthermore, we calculate that the asymptotic confidence interval, percentile bootstrap interval, and the highest probability density credible interval are obtained for the stress–strength parameter . In order to assess the efficiency of these estimation techniques, simulation studies are conducted, providing valuable insights into the performance of each estimation approach. Finally, the proposed reliability model is applied to real datasets, highlighting its practical significance.

Comparison of Mediation Effects on Interaction and Multigroup Approach in Structural Equation Modeling PLS in Case of Bank Mortgage

“Structural Equation Modeling is one of multivariate statistical method that used to explain multiple relationships between latent variables simultaneously to test a mediation model to conduct a formal test on mediation effects. Application PLS-SEM for exploratory research and theory development are increasing. Under certain conditions, the effect of exogenous variables on endogenous variable is also strengthened or weakened by moderating variable. In SEM, there are two approaches in analyzing moderation variables, namely the interaction method and the multigroup method. This article aims to compare the mediation effect on interaction approaches and multigroup approaches in Structural Equation Modeling. The data used is the case of timeliness of Bank X mortgage payments. In this article, statistical methods are evaluated to compare indirect effect between groups and examine indirect effect on each group. It was concluded that Collectability Status moderates the indirect relationship between Capital and the Timeliness of Payment through Willingness to Pay. Debtors with current collectability status more strongly effect the Timeliness of Payment than debtors with incurrect collectability status. Theresults of testing indirect effects on moderation with interaction and multigroup approaches are not much different. In the multigroup approach, the bootstrap interval bias is smaller than the bootstrap interval bias in the interaction approach. The Q-square Predictive Relevance value in both methods is quite high, indicating that the model is good. On the Current Collectibility Status group Q^2 is 89.3%, in the incurrect Collectibility Status Q^2 is 84.2%. While in the interaction approach, Q^2 is 70.4%. Researcher recommend a multigroup approach to data that has categorical moderation variables because differences between groups can be directly observed without adding interaction variables in the model.”

A residual bootstrap for conditional Value-at-Risk

A fixed-design residual bootstrap method is proposed for the two-step estimator of Francq and Zakoïan(2015) associated with the conditional Value-at-Risk. The bootstrap’s consistency is proven for a general class of volatility models and intervals are constructed for the conditional Value-at-Risk. A simulation study reveals that the equal-tailed percentile bootstrap interval tends to fall short of its nominal value. In contrast, the reversed-tails bootstrap interval yields accurate coverage. We also compare the theoretically analyzed fixed-design bootstrap with the recursive-design bootstrap. It turns out that the fixed-design bootstrap performs equally well in terms of average coverage, yet leads on average to shorter intervals in smaller samples. An empirical application illustrates the interval estimation.

Bootstrap Intervals for the Mean of the Weighted Mixture Generalized Gamma Distribution

Bootstrap Intervals for the Mean of the Weighted Mixture Generalized Gamma Distribution

Bayes Estimation for the Rayleigh–Weibull Distribution Based on Progressive Type-II Censored Samples for Cancer Data in Medicine

The aim of this study is to obtain the Bayes estimators and the maximum likelihood estimators (MLEs) for the unknown parameters of the Rayleigh–Weibull (RW) distribution based on progressive type-II censored samples. The approximate Bayes estimators are calculated using the idea of Lindley, Tierney–Kadane approximations, and also the Markov Chain Monte Carlo (MCMC) method under the squared-error loss function when the Bayes estimators are not handed in explicit forms. In this study, the approximate Bayes estimates are compared with the maximum likelihood estimates in the aspect of the estimated risks (ERs) using Monte Carlo simulation. The asymptotic confidence intervals for the unknown parameters are obtained using the MLEs of parameters. In addition, the coverage probabilities the parametric bootstrap estimates are computed. Real lifetime datasets related to bladder cancer, head and neck cancer, and leukemia are used to illustrate the empirical results belonging to the approximate Bayes estimates, the maximum likelihood estimates, and the parametric bootstrap intervals.

On the estimation of fuzzy stress–strength reliability parameter

In this paper, we introduce a novel, more interpretative, membership function that can be used for the definition of fuzzy stress–strength reliability parameter R of a two-component system. We find the MLEs and UMVUEs point estimators of R, as well as the asymptotic and bootstrap interval estimators. In the wide empirical study for the case when both components are exponentially distributed, we examine the properties of introduced estimators. We also explore the impact of the membership function on the quality of estimators.

Survival Analysis and Applications of Weighted NH Parameters Using Progressively Censored Data

A new weighted Nadarajah–Haghighi (WNH) distribution, as an alternative competitor model to gamma, standard half-logistic, generalized-exponential, Weibull, and other distributions, is considered. This paper explores both maximum likelihood and Bayesian estimation approaches for estimating the parameters, reliability, and hazard rate functions of the WNH distribution when the sample type is Type-II progressive censored order statistics. In the classical interval setup, both asymptotic and bootstrap intervals of each unknown parameter are constructed. Using independent gamma priors and symmetric squared-error loss, the Bayes estimators cannot be obtained theoretically. Thus, two approximation techniques, namely: Lindley and Markov-Chain Monte Carlo (MCMC) methods, are used. From MCMC variates, the Bayes credible and highest posterior density intervals of all unknown parameters are also created. Extensive Monte Carlo simulations are implemented to compare the performance of the proposed methodologies. Numerical evaluations showed that the estimates developed by the MCMC sampler performed better than the Lindley estimates, and both behaved significantly better than the frequentist estimates. To choose the optimal censoring scheme, several optimality criteria are considered. Three engineering applications, including vehicle fatalities, electronic devices, and electronic components data sets, are provided. These applications demonstrated how the proposed methodologies could be applied in real practice and showed that the proposed model provides a satisfactory fit compared to three new weighted models, namely: weighted exponential, weighted Gompertz, and new weighted Lindley distributions.

High-order Coverage of Smoothed Bayesian Bootstrap Intervals for Population Quantiles

We characterize the high-order coverage accuracy of smoothed and unsmoothed Bayesian bootstrap confidence intervals for population quantiles. Although the original (Rubin 1981) unsmoothed intervals have the same O(n−1/2) coverage error as the standard empirical bootstrap, the smoothed Bayesian bootstrap of Banks (1988) has much smaller O(n−3/2[log(n)]3) coverage error and is exact in special cases, without requiring any smoothing parameter. It automatically removes an error term of order 1/n that other approaches need to explicitly correct for. This motivates further study of the smoothed Bayesian bootstrap in more complex settings and models.

Accurate Confidence and Bayesian Interval Estimation for Non-centrality Parameters and Effect Size Indices

Reporting effect size index estimates with their confidence intervals (CIs) can be an excellent way to simultaneously communicate the strength and precision of the observed evidence. We recently proposed a robust effect size index (RESI) that is advantageous over common indices because it’s widely applicable to different types of data. Here, we use statistical theory and simulations to develop and evaluate RESI estimators and confidence/credible intervals that rely on different covariance estimators. Our results show (1) counter to intuition, the randomness of covariates reduces coverage for Chi-squared and F CIs; (2) when the variance of the estimators is estimated, the non-central Chi-squared and F CIs using the parametric and robust RESI estimators fail to cover the true effect size at the nominal level. Using the robust estimator along with the proposed nonparametric bootstrap or Bayesian (credible) intervals provides valid inference for the RESI, even when model assumptions may be violated. This work forms a unified effect size reporting procedure, such that effect sizes with confidence/credible intervals can be easily reported in an analysis of variance (ANOVA) table format.

Confidence Intervals of a Loss Based PCI S′pmk Using Exponentiated-Exponential Distributed Quality Characteristics

Process capability indices (PCIs) are most effective devices/techniques used in industries for determining the quality of products and performance of manufacturing processes. In this article, we consider the process capability index which is based on an asymmetric loss function (linear exponential) and is applicable to normally as well as non-normally distributed processes. In order to estimate the PCI when the process follows exponentiated exponential distribution, we have used ten classical methods of estimation and the performances of these classical estimates for the index are compared in terms of mean squared errors (MSEs) through simulation study. Also, the confidence intervals for the index are constructed based on four bootstrap confidence interval (BCIs) methods. A simulation study is performed in order to compare the performance of these four BCIs in terms of average width and coverage probabilities. We use two published data sets related to electronic and food industries to illustrate the performance of the proposed methods of estimation and BCIs. All the data sets show that width of bias-corrected accelerated bootstrap interval is the lowest among all other considered BCIs.

Modeling the Amount of Carbon Dioxide Emissions Application: New Modified Alpha Power Weibull-X Family of Distributions

The use of statistical distributions to model life phenomena has received considerable attention in the literature. Recent studies have shown the potential of statistical distributions in modeling data in applied sciences, especially in environmental sciences. Among them, the Weibull distribution is one of the most well-known models that can be used very effectively for modeling data in the fields of pollution and gas emissions, to name a few. In this paper, we introduce a family of distributions, which we call the modified Alpha-Power Weibull-X family of distributions. Based on the proposed family, we introduce a new model with five parameters, the modified Alpha-Power Weibull–Weibull distribution. Some mathematical properties were determined. Bayesian and maximum likelihood estimates for the model parameters were derived. The MLEs, bootstrap and Bayesian HPD credibility intervals for the unknown parameters were performed. A Monte Carlo simulation study was performed to evaluate the performance of the estimates. A simulation study was performed based on the parameters of the proposed model. An application to the carbon dioxide emissions dataset was performed to predict unique symmetric and asymmetric patterns and illustrate the applicability and potential of the model. For this data set, the proposed model is compared with the modified alpha power Weibull exponential distribution and the two-parameter Weibull distribution. To show which of the competing distributions is the best, we draw on certain analytical tools such as the Kolmogorov–Smirnov test. Based on these analytical measures, we found that the new model outperforms the competing models.

Percentile Bootstrap Interval on Univariate Local Polynomial Regression Prediction

This study offers a new technique for constructing percentile bootstrap intervals to predict the regression of univariate local polynomials. Bootstrap regression uses resampling derived from paired and residual bootstrap methods. The main objective of this study is to perform a comparative analysis between the two resampling methods by considering the nominal coverage probability. Resampling uses a nonparametric bootstrap technique with the return method, where each sample point has an equal chance of being selected. The principle of nonparametric bootstrapping uses the original sample data as a source of diversity in contrast to parametric bootstrapping, where the variety comes from generating a particular distribution. The simulation results show that the paired and residual bootstrap interval coverage probabilities are close to nominal coverage. The results showed no significant difference between paired bootstrap interval and percentile residual. Increasing the bootstrap sample size sufficiently large gives the scatterplot smoothness of the confidence interval. Applying the smoothing parameter by choice gives a second-order polynomial regression with a smoother distribution than the first-order polynomial regression. The scatterplot shows that the second-degree polynomial regression can capture the data curvature feature compared to the first-degree polynomial. The bands made from second-degree polynomials give a narrower width than first-degree polynomials. In contrast, applying optimal smoothing parameters to the model provides different conclusions by using smoothing parameters based on choice. In addition to the differences based on the scatterplot, the bootstrap estimates of the coverage probability are also other. Selecting smoothing parameters based on a particular value provides probability coverage with the paired bootstrap method for the first-degree local polynomial regression is 0.93, while the second-degree local polynomial is 0.96. The probability of coverage based on the residual bootstrap method for the first-degree local polynomial regression is 0.95, while the second-degree local polynomial is 0.96. The probability coverage based on the optimal parameters of the paired bootstrap method for the first-degree local polynomial regression is 0.945, while the second-degree local polynomial is 0.93. The residual bootstrap method gives the first-degree local polynomial regression of 0.95, while the second-degree local polynomial is 0.93. In general, both bootstrap methods work well for estimating prediction confidence intervals.

Measuring teachers' readiness to use ICT before the COVID-19 pandemic in Italy.

This study seeks to measure teachers' readiness to use ICT in Italy by exploiting the data collected by the National Institute for the evaluation of education and training system (INVALSI) in 2018-2019. We propose a fuzzy set approach to provide a multidimensional picture of how much teachers were ready to integrate ICTs into educational practice. In addition, we use empirical bootstrap intervals to test whether significant differences exist in teacher readiness over several personal and socioeconomic characteristics. The study reveals that teachers' readiness for ICT is composed of three dimensions and varies by teachers' characteristics and regions. These results are a useful tool for understanding the relationship between teachers and ICTs so as to develop more appropriate educational policies.