Parametric Bootstrap Method Research Articles

Bayesian parameter estimation using truncated normal distributions as priors for parameters in fundamental models of chemical processes

AbstractModellers of chemical processes with knowledge about plausible parameter values use Bayesian parameter estimation methods to account for their prior beliefs. Some modellers specify prior distributions with finite parameter ranges, such as uniform distributions and truncated normal distributions, because they better account for knowledge about realistic parameter ranges than normal prior distributions with parameter values ranging between and . We derive closed‐form objective functions for Bayesian parameter estimation with truncated normal priors and uniform priors, for the first time, so that parameter estimation can be performed by solving simple optimization problems rather than using complex sampling‐based techniques. A parametric bootstrapping method that considers truncated normal priors and model nonlinearity is proposed to determine 95% confidence intervals and joint confidence regions. A pharmaceutical case study is used to show the effectiveness of the proposed objective functions and bootstrapping methodology. Confidence regions from bootstrapping are similar to linearization‐based confidence regions that do not account for truncation when truncated areas in normal prior distributions are relatively small. More truncation, which corresponds to more‐precise prior knowledge about the parameters, results in smaller joint confidence regions. The proposed methods will be attractive for parameter estimation in complex process models because they can be less computationally intensive than Markov chain Monte Carlo methods that provide similar results.

Another market segment: sport event tourism by disabled athletes

Sports and tourism are connected in various parts of economies, cultures, and nations. However, there has been a limited understanding of sports event tourism involving disabled athletes. This study explored the behavioral and socio-demographic implications of the disabled sports event tourism and investigated the motivational factors involved. An in-person survey was administered to 108 disabled athletes who attended the summer and winter Paralympics. A Structural Equation Model was used to determine the association between the factors of life satisfaction, emotional happiness, escapism, perceived value, and future intention, using the Monte Carlo parametric bootstrapping method to test significance of direct and indirect effects. Cronbach was acceptable because it exceeds 0.70 which satisfies the cut-off of confirmatory factor analysis. In addition, the individual values of average variance extracted (AVE), were greater than 0.50 (0.72) which meets the requirement and the convergent validity of all the constructs. The results showed that life satisfaction had a significant direct effect on future intentions. Emotional happiness had a significant direct effect on the perceived value of an event. Escapism had significant direct effects on perceived value and future intentions. Perceived value significantly influenced future intentions. The relationship between emotional happiness and future intention was fully mediated by perceived value. However, the relationship between escapism and future intention was only partially mediated by perceived value. The results of this study are valuable for developing future management and marketing policies for disabled athletes and tourists to advance the existing sports event tourism and disabled athlete’s behavior studies.

Application of the Parametric Bootstrap Method for Confidence Interval Estimation and Statistical Analysis of PM2.5 in Bangkok

Research in epidemiology and health science indicates that exposure to particles with an aerodynamic diameter of less than 2.5 µm (PM2.5) causes harmful health consequences. Probability density functions (pdf) are utilized to analyze the distribution of pollutant data and study the occurrence of high-concentration occurrences. In this study, PM2.5 concentrations (in μg/m^3 ) were recorded daily from January 2011 to December 2022 at 12 air quality monitoring locations in Bangkok. The study utilized two-parameter distributions such as gamma, inverse Gaussian, lognormal, log-logistic, Weibull, and Pearson type V to identify the most suitable statistical distribution model for PM2.5 in Bangkok. The Anderson-Darling test result indicates that the inverse Gaussian and Pearson type V distributions are the most appropriate probability density functions for the daily average PM2.5 concentration at stations in Bangkok. The projected 98th percentile of daily PM2.5 levels at two locations is higher than the 24-hour threshold for daily PM2.5 concentrations in Thailand, posing significant health risks. Additionally, the two parametric bootstrap methods used to estimate confidence intervals for the median, namely percentile bootstrap and simple bootstrap, indicate that two stations have poor air quality for those with sensitive health conditions.

Reliability evaluation in the zero‐failure Weibull case based on double‐modified hierarchical Bayes

AbstractHierarchical Bayes, or E‐Bayes, is frequently used to estimate the failure probability when solving a zero‐failure reliability evaluation model; however, the accuracy of the reliability estimation using these methods is not very good in practice. Due to this, a novel double‐modified hierarchical Bayes (DMH‐Bayes) is proposed for Weibull characteristic data in this study to enhance failure probability estimation and improve reliability point estimation accuracy. Meanwhile, in order to guarantee the preservation of the assessment findings' consistency and confidence level, the parametric Bootstrap method (P‐Bootstrap) and the L‐moment estimation method based on point estimation are introduced to obtain reliability confidence interval estimates. Based on Monte–Carlo simulation testing and analysis of a gyroscope bearing, the new model is confirmed to have better applicability and robustness while improving the accuracy of reliability assessment.

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.

Multichannel detection of evoked responses using critical values corrected by a parametric bootstrap: Frequency-domain cholesky correction

Multivariate Objective Response Detection (MORD) techniques aim to detect evoked responses in multichannel electroencephalographic (EEG) recordings. They provide enhanced statistical power, allowing the detection of small signals in shorter or noisy recordings. However, the correlation between the signals in multichannel recordings can lead to false positive rates greater than the nominal significance level of the tests. To address this, we propose a parametric bootstrap approach that adjusts the critical values based on the correlation between EEG channels in the time domain, a method called time-domain Cholesky correction (TDCC). In that first approach, we assumed that correlation (or, more precisely, coherence) is constant across all frequency bands. However, this is unlikely to hold true, as signal-to-noise ratios (where the signal is the evoked response and noise all other signal components) may vary across frequencies. Thus, in the current work, we propose an alternative parametric bootstrap method for estimating the critical values of MORD techniques based on the correlation in the frequency domain (FDCC, frequency-domain Cholesky-corrected critical values). The proposed methods are evaluated using simulated data and an auditory steady-state response (ASSR) database in the 40 Hz range. The proposed method controlled the false positive rate well, with increased sensitivity compared to single-channel methods. When compared to TDCC, FDCC achieved similar performance but with the advantage of being, on average, 27 times faster in terms of computational time required to estimate the critical values.

Mean Square Error Estimation of Small Area Predictors by Use of Parametric and Nonparametric Bootstrap

In this article, we propose and compare some old and new parametric and nonparametric bootstrap methods for MSE estimation in small area estimation, restricting to the case of the widely used Fay-Herriot model. The parametric method consists of generating parametrically a large number of area bootstrap samples from the model fitted to the original data, re-estimating the model parameters for each bootstrap sample and then estimating the separate components of the MSE. The use of double-bootstrap is also considered. The nonparametric method generates the samples by bootstrapping standardized residuals, estimated from the original sample data. The bootstrap procedures are compared to other methods proposed in the literature in a simulation study, which also examines the robustness of the various methods to non-normality of the model error terms. A design-based MSE estimator for the Fay-Herriot model-dependent predictor is also described and its performance is investigated in a separate simulation study. AMS subject classification: 62F10, 62F40

SumVg: Total Heritability Explained by All Variants in Genome-Wide Association Studies Based on Summary Statistics with Standard Error Estimates.

Genome-wide association studies (GWAS) are commonly employed to study the genetic basis of complex traits/diseases, and a key question is how much heritability could be explained by all single nucleotide polymorphisms (SNPs) in GWAS. One widely used approach that relies on summary statistics only is linkage disequilibrium score regression (LDSC); however, this approach requires certain assumptions about the effects of SNPs (e.g., all SNPs contribute to heritability and each SNP contributes equal variance). More flexible modeling methods may be useful. We previously developed an approach recovering the "true" effect sizes from a set of observed z-statistics with an empirical Bayes approach, using only summary statistics. However, methods for standard error (SE) estimation are not available yet, limiting the interpretation of our results and the applicability of the approach. In this study, we developed several resampling-based approaches to estimate the SE of SNP-based heritability, including two jackknife and three parametric bootstrap methods. The resampling procedures are performed at the SNP level as it is most common to estimate heritability from GWAS summary statistics alone. Simulations showed that the delete-d-jackknife and parametric bootstrap approaches provide good estimates of the SE. In particular, the parametric bootstrap approaches yield the lowest root-mean-squared-error (RMSE) of the true SE. We also explored various methods for constructing confidence intervals (CIs). In addition, we applied our method to estimate the SNP-based heritability of 12 immune-related traits (levels of cytokines and growth factors) to shed light on their genetic architecture. We also implemented the methods to compute the sum of heritability explained and the corresponding SE in an R package SumVg. In conclusion, SumVg may provide a useful alternative tool for calculating SNP heritability and estimating SE/CI, which does not rely on distributional assumptions of SNP effects.

Generalized Fiducial Inference for the Generalized Rayleigh Distribution

This article focuses on the interval estimation of the generalized Rayleigh distribution with scale and shape parameters. The generalized fiducial method is used to construct the fiducial point estimators as well as the fiducial confidence intervals, and then their performance is compared with other methods such as the maximum likelihood estimation, Bayesian estimation and parametric bootstrap method. Monte Carlo simulation studies are carried out to examine the efficiency of the methods in terms of the mean square error, coverage probability and average length. Finally, two real data sets are presented to demonstrate the applicability of the proposed method.

A Wald test on the problem of homogeneity of variances

In this article, we propose a Wald test for homogeneity of variance in several normal distributions. The Wald test is an asymptotic test, and it is known that asymptotic tests perform well in large sample sizes, but not in small sample sizes. For this reason, we modified this test using a parametric bootstrap method called the computational approach test. To evaluate the efficiency of the proposed test against the existing tests in the literature, we compared this test in terms of the estimated size and power of tests. The results of our simulation study show that the proposed test performs well in especially small and different sample sizes under normal distributions. Besides, we illustrate the proposed test using a numerical example.

Analysis of the Stress–Strength Model Using Uniform Truncated Negative Binomial Distribution under Progressive Type-II Censoring

In this study, we introduce a novel estimation technique for assessing the reliability parameter R=P(Y<X) of the uniform truncated negative binomial distribution (UTNBD) in the context of stress–strength analysis. We base our inferences on the assumption that both the strength (X) and stress (Y) random variables follow a UTNBD with identical first shape and scale parameters. In the presence of a progressive type-II censoring scheme, we employ maximum likelihood, two parametric bootstrap methods, and Bayesian estimation approaches to derive the estimators. Due to the complexity introduced by censoring, the estimators are not available in explicit forms and are instead obtained through numerical approximation techniques. Furthermore, we compute the highest posterior density credible intervals and determine the asymptotic variance-covariance matrix. To assess the performance of our proposed estimators, we conduct a Monte Carlo simulation study and provide a comparative analysis. Finally, we illustrate the practical applicability of our study with an engineering application.

A Parametric Bootstrap Approach for a One-Way Error Component Regression Model with Measurement Errors

In this paper, a one-way error component regression model with measurement errors is considered. The unknown parameter vector is estimated by using the bias-corrected method, and its corresponding asymptotic properties are also developed. For the hypothesis testing problem of the vector of the coefficient parameter in the model, a parametric bootstrap (PB) method is proposed. Under various sample sizes and parameter configurations, the effectiveness of our proposed PB test method is discussed by using some numerical simulations and a real data analysis.

Multiple comparisons of treatment against control under unequal variances using parametric bootstrap

ABSTRACT In one-way analysis of variance models, performing simultaneous multiple comparisons of treatment groups with a control group may be of interest. Dunnett's test is used to test such differences and assumes equal variances of the response variable for each group. This assumption is not always met even after transformation. A parametric bootstrap (PB) method is developed here for comparing multiple treatment group means against the control group with unequal variances and unbalanced data. In simulation studies, the proposed method outperformed Dunnett's test in controlling the type I error under various settings, particularly when data have heteroscedastic variance and unbalanced design. Simulations show that power is often lower for the PB method than for Dunnett's test under equal variance, balanced data, or smaller sample size, but similar to or higher than for Dunnett's test with unequal variance, unbalanced data and larger sample size. The method is applied to a dataset concerning isotope levels found in elephant tusks from various geographical areas. These data have very unbalanced group sizes and unequal variances. This example illustrates that the PB method is easy to implement and avoids the need for transforming data to meet the equal variance assumption, simplifying interpretation of results.

A radiomics nomogram for preoperative prediction of nephron-sparing surgery in patients with bilateral Wilms tumor.

Bilateral Wilms tumor (BWT) is a relatively rare malignant renal tumor in children. Nephron-sparing surgery (NSS) is the preferred surgical approach for treating BWT, but lacks uniform surgical indications worldwide. This study aimed to summarize the clinical and imaging features of BWT children, establish a radiomics nomogram, and predict the feasibility of NSS for improving outcomes. A 12-year retrospective single-center review was conducted on clinical data and preoperative imaging features of BWT patients. The tumor kidneys were divided into NSS and non-NSS groups. Logistic regression analysis was performed to identify independent predictors and develop a prediction model of the feasibility of NSS in BWT patients. A radiomics nomogram was constructed and internally validated by the parametric bootstrapping method. A total of 58 BWT patients (115 renal units) were included in this study. After evaluations based on preoperative imaging and clinical data, 94 renal units underwent NSS with negative resection margins and were included in the NSS group, whereas 16 renal units with positive resection margins, macroscopic residual, or total nephrectomies were included in the non-NSS group. Tumor size [odds ratio (OR): 0.540, 95% confidence interval (CI): 0.308-0.945], relationship with the collecting system (OR: 0.013, 95% CI: 0.0004-0.370), and remaining renal parenchyma (RRP) proportion (OR: 71.23, 95% CI: 1.632-3108.8) were identified as independent predictors for NSS. A nomogram was constructed based on these factors, which demonstrated great consistency between the predicted and observed feasibility of NSS. The model presented with good discriminative ability [area under the curve (AUC), 0.982]. The decision curve analysis (DCA) revealed the clinical usefulness of the model. This study analyzed the clinical and preoperative imaging data of BWT patients and identified three independent predictors for the feasibility of NSS, including tumor size, relationship with the collecting system, and residual renal parenchyma proportion. The radiomics nomogram established in this study can provide individualized predictions to assist clinicians in making better decisions and improving patient outcomes.

Simulating Parametric and Nonparametric Models

The purpose of this paper was to investigate the performance of the parametric bootstrap data generating processes (DGPs) methods and to compare the parametric and nonparametric bootstrap (DGPs) methods for estimating the standard error of simple linear regression (SLR) under various assessment conditions. When the performance of the parametric bootstrap method was investigated, simple linear models were employed to fit the data. With the consideration of the different bootstrap levels and sample sizes, a total of twelve parametric bootstrap models were examined. Three hypothetical and one real datasets were used as the basis to define the population distributions and the “true” SEEs. A bootstrap paper was conducted on different parametric and nonparametric bootstrap (DGPs) methods reflecting three levels for group proficiency differences, three levels of sample sizes, three test lengths and three bootstrap levels. Bias of the SLR, standard errors of the SLR, root mean square errors of the SLR, were calculated and used to evaluate and compare the bootstrap results. The main findings from this bootstrap paper were as follows: (i) The parametric bootstrap DGP models with larger bootstrap levels generally produced smaller bias likewise a larger sample size. (ii) The parametric bootstrap models with a higher bootstrap level generally yielded more accurate estimates of the standard error than the corresponding models with lower bootstrap level. (iii) The nonparametric bootstrap method generally produced less accurate estimates of the standard error than the parametric bootstrap method. However, as the sample size increased, the differences between the two bootstrap methods became smaller. When the sample size was equal to or larger than 3,000, say 10000, the differences between the nonparametric bootstrap DGP method and the parametric bootstrap DGP model that produced the smallest RMSE were very small. (4) Of all the models considered in this paper, parametric bootstrap DGP models with higher bootstrap performed better under most bootstrap conditions. (5) Aside from method effects, sample size and test length had the most impact on estimating the Simple Linear Regression.

Confidence Interval Estimation for the Ratio of the Percentiles of Two Delta-Lognormal Distributions with Application to Rainfall Data

The log-normal distribution (skewed distribution or asymmetry distribution) is used to describe random variables comprising positive real values. It is well known that the logarithm values of these are normally distributed (symmetry distribution). Positively right-skewed data applicable to the log-normal distribution are frequently observed in the fields of environmental studies, biology, and medicine. The number of zero observations follows a binomial distribution. However, problems can arise in the analysis of data containing zero observations along with log-normally distributed data, for which the delta-lognormal distribution is often referred to for using the analysis of the data. In statistics, the percentile provides the relative standing of a numerical data point when compared to all of the others in a distribution with reference to the observations at or below it. In this study, estimates for the confidence interval for the ratio of the percentiles of two delta-lognormal distributions are constructed using fiducial generalized confidence interval approaches based on the fiducial quantity and the optimal generalized fiducial quantity, the Bayesian approach, and the parametric bootstrap method. As assessed by Monte Carlo simulations using the RStudio programming in terms of the coverage probability and the average length, the Bayesian approach performed quite well by providing adequate coverage probabilities along with the shortest average lengths in all of the scenarios tested. Daily rainfall data contain both zero and positive values. The daily rainfall data can usually be fitted to the delta-lognormal distribution. Their application to rainfall data is also provided to illustrate their efficacies with real data. The efficacy of the approach is used to compare two rainfall dispersion populations.

A nested error regression model with high-dimensional parameter for small area estimation

Abstract In this paper, we propose a flexible nested error regression small area model with high-dimensional parameter that incorporates heterogeneity in regression coefficients and variance components. We develop a new robust small area-specific estimating equations method that allows appropriate pooling of a large number of areas in estimating small area-specific model parameters. We propose a parametric bootstrap and jackknife method to estimate not only the mean squared errors but also other commonly used uncertainty measures such as standard errors and coefficients of variation. We conduct both model-based and design-based simulation experiments and real-life data analysis to evaluate the proposed methodology.

Wildfire prediction using zero-inflated negative binomial mixed models: Application to Spain

Wildfires have changed in recent decades. The catastrophic wildfires make it necessary to have accurate predictive models on a country scale to organize firefighting resources. In Mediterranean countries, the number of wildfires is quite high but they are mainly concentrated around summer months. Because of seasonality, there are territories where the number of fires is zero in some months and is overdispersed in others. Zero-inflated negative binomial mixed models are adapted to this type of data because they can describe patterns that explain both number of fires and their non-occurrence and also provide useful prediction tools. In addition to model-based predictions, a parametric bootstrap method is applied for estimating mean squared errors and constructing prediction intervals. The statistical methodology and developed software are applied to model and to predict number of wildfires in Spain between 2002 and 2015 by provinces and months.

Multivariate Mixture Model for Small Area Estimation of Poverty Indicators

Abstract When disaggregation of national estimates in several domains or areas is required, direct survey estimators, which use only the domain-specific survey data, are usually design-unbiased even under complex survey designs (at least approximately) and require no model assumptions. Nevertheless, they are appropriate only for domains or areas with sufficiently large sample size. For example, when estimating poverty in a domain with a small sample size (small area), the volatility of a direct estimator might make that area seems like very poor in one period and very rich in the next one. Small area (or indirect) estimators have been developed in order to avoid such undesired instability. Small area estimators borrow strength from the other areas so as to improve the precision and therefore obtain much more stable estimators. However, the usual model-based assumptions, which include some kind of area homogeneity, may not hold in real applications. A more flexible model based on multivariate mixtures of normal distributions that generalises the usual nested error linear regression model is proposed for estimation of general parameters in small areas. This flexibility makes the model adaptable to more general situations, where there may be areas with a different behaviour from the other ones, making the model less restrictive (hence, more close to nonparametric) and more robust to outlying areas. An expectation-maximisation (E-M) method is designed for fitting the proposed mixture model. Under the proposed mixture model, two different new predictors of general small area indicators are proposed. A parametric bootstrap method is used to estimate the mean squared errors of the proposed predictors. Small sample properties of the new predictors and of the bootstrap procedure are analysed by simulation studies and the new methodology is illustrated with an application to poverty mapping in Palestine.

Testing components of two-way interaction in multi-environment trials

Experiments with two factors are commonly analyzed using two-way analysis of variance, where testing significance of interaction is straightforward. However, using bilinear models, interaction can be analyzed further. The additive main effects and multiplicative interaction (AMMI) model uses singular value decomposition for partitioning interaction into multiplicative terms, such that the first terms typically account for a large portion of the sum of squares, whereas the last terms are of minor importance. This model is used extensively for analysis of genotype-by-environment interaction in multi-environment trials. A recurring question is how to determine the number of terms to retain in the model. If data is replicated, which is usually the case, the F R test can be used for this purpose. The simple parametric bootstrap method is another option, although this test was developed for unreplicated data. Since both of these tests of significance may be applied in cases with replication, researchers need advice on which of the methods to use. We discuss several statistical models and show that the two methods address different questions.