Bootstrap-t Confidence Intervals Research Articles

Simulation Techniques for Strength Component Partially Accelerated to Analyze Stress–Strength Model

Based on independent progressive type-II censored samples from two-parameter Burr-type XII distributions, various point and interval estimators of δ=P(Y<X) were proposed when the strength variable was subjected to the step–stress partially accelerated life test. The point estimators computed were maximum likelihood and Bayesian under various symmetric and asymmetric loss functions. The interval estimations constructed were approximate, bootstrap-P, and bootstrap-T confidence intervals, and a Bayesian credible interval. A Markov Chain Monte Carlo approach using Gibbs sampling was designed to derive the Bayesian estimate of δ. Based on the mean square error, bias, confidence interval length, and coverage probability, the results of the numerical analysis of the performance of the maximum likelihood and Bayesian estimates using Monte Carlo simulations were quite satisfactory. To support the theoretical component, an empirical investigation based on two actual data sets was carried out.

Bootstrap-t Confidence Interval on Local Polynomial Regression Prediction

In local polynomial regression, prediction confidence interval estimation using standard theory will give coverage probability close to exact coverage probability. However, if the normality assumption is not met, the bootstrap method makes it possible to apply it. The working principle of the bootstrap method uses the resampling method where the sample data becomes a population and there is no need to know the distribution of the sample data is normal or not. Indiscriminate selection of smoothing parameters allows scatterplot results from local polynomial regressions to be rough and can even lead to misleading statistical conclusions. It is necessary to consider the optimal smoothing parameters to get local polynomial regression predictions that are not overfitting or underfitting. We offer two new algorithms based on the nested bootstrap resampling method to determine the bootstrap-t confidence interval in predicting local polynomial regression. Both algorithms consider the search for optimal smoothing parameters. The first algorithm performs paired and residual bootstrap samples, and the second algorithm performs based on residuals with residuals. The first algorithm provides a scatterplot and reasonable coverage probability on relatively large sample data. In contrast, the second algorithm is more powerful for each data size, including for relatively small sample data sizes. The mean of the bootstrap-t confidence interval coverage probability shows that the second algorithm for second-degree local polynomial regression is better than the other three. However, the larger the sample data size gives, the closer the closer the average coverage probability of the two algorithms is to the nominal coverage probability.

Better confidence intervals for the slope parameters of the quadratic regression model when the error term is not normally distributed

When the error term is not normally distributed, the confidence intervals for the slope parameters of the quadratic regression model, based on the ordinary t-statistic, are not appropriate. In this paper, we consider the interval based on the ordinary t-statistic and the intervals based on the resampling methods. Further, we derive the Edgeworth expansion of the distribution of the t-statistic. Based on that expansion we propose a new transformation of the t-statistic which is used for the construction of the bootstrap-t confidence interval for the slope parameter β 2 . The obtained results for the Gamma, Weibull, and Exponential distributions and for the real data set show that the intervals based on the resampling methods and the interval based on the new transformation of the t-statistic provide better coverage accuracy than the interval based on the ordinary t-statistic.

Bootstrap-based inferential improvements to the simplex nonlinear regression model.

In this paper we evaluate the performance of point and interval estimators based on the maximum likelihood(ML) method for the nonlinear simplex regression model. Inferences based on traditional maximum likelihood estimation have good asymptotic properties, but their performance in small samples may not be satisfactory. At out set we consider the maximum likelihood estimation for the parameters of the nonlinear simplex regression model, and so we introduced a bootstrap-based correction for such estimators of this model. We also develop the percentile and bootstrapt confidence intervals for those parameters as competitors to the traditional approximate confidence interval based on the asymptotic normality of the maximum likelihood estimators (MLEs). We then numerically evaluate the performance of these different methods for estimating the simplex regression model. The numerical evidence favors inference based on the bootstrap method, in special the bootstrapt interval, which was decisive in an application to real data.

Bivariate Weibull-G Family Based on Copula Function: Properties, Bayesian and non-Bayesian Estimation and Applications

This paper aims to obtain a new flexible bivariate generalized family of distributions based on FGM copula, which is called bivariate FGM Weibull-G family. Some of its statistical properties are studied as marginal distributions, product moments, and moment generating functions. Some dependence measures as Kendall’s tau and median regression model are discussed. After introducing the general class, four special sub models of the new family are introduced by taking the baseline distributions as Pareto, inverted Topp-Leone, exponential, and Rayleigh distributions. Maximum likelihood and Bayesian approaches are used to estimate the model unknown parameters. Further, percentile bootstrap confidence interval and bootstrap-t confidence interval are estimated for the model’s parameters. A Monte-Carlo simulation study is carried out of the maximum likelihood and Bayesian estimators. Finally, we illustrate the importance of the proposed bivariate family using two real data sets in medical field.

A fuzzy penalized regression model with variable selection

In the classical multiple regression modeling, there might be some insignificant input variables. These variables can be eliminated by automatic selectors, known as penalized methods. We propose a penalized estimation method for the coefficients of a linear regression model for studying the dependence of a LR fuzzy response (output) variable on a set of crisp explanatory (input) variables. To show the performances of the proposed model a simulation study was utilized under three scenarios of multicollinear and sparse data. The model demonstrates better performances in comparison to another three models on the basis of specified goodness of fit measures, under three variants of the penalty function. Evaluation of the method has been conducted on real data. The results demonstrate superior performances in terms of the goodness of fit measures in comparison to the other models. To take into account the imprecision due to the lack of knowledge about the data generation process, in the applications to real data bootstrap-t confidence intervals were also utilized.

A New Transmuted Generalized Lomax Distribution: Properties and Applications to COVID-19 Data.

A new five-parameter transmuted generalization of the Lomax distribution (TGL) is introduced in this study which is more flexible than current distributions and has become the latest distribution theory trend. Transmuted generalization of Lomax distribution is the name given to the new model. This model includes some previously unknown distributions. The proposed distribution's structural features, closed forms for an rth moment and incomplete moments, quantile, and Rényi entropy, among other things, are deduced. Maximum likelihood estimate based on complete and Type-II censored data is used to derive the new distribution's parameter estimators. The percentile bootstrap and bootstrap-t confidence intervals for unknown parameters are introduced. Monte Carlo simulation research is discussed in order to estimate the characteristics of the proposed distribution using point and interval estimation. Other competitive models are compared to a novel TGL. The utility of the new model is demonstrated using two COVID-19 real-world data sets from France and the United Kingdom.

Inferência bootstrap para intervalo de confiança de modelo de regressão quadrática

This study was carried out with the purpose of proposing a construction of confidence intervals for the critical point of a second degree regression model using a parametric bootstrap methodology. To obtain the distribution of the critical point, height growth data of the plants were used. From the analysis, the theoretical variables for the error and the confidence intervals were constructed. In addition, we examined different variance expressions with the purpose of the bootstrap-t confidence interval. The point estimate of the critical point was 10.7423 g L-1 of fertilizer doses without growth of C. canjerana plants. It was verified that the confidence intervals that considered the expression of the variance with the covariance between the regression models, present more satisfactory results, that is, results with more precision.

Inference for two Lomax populations under joint type-II censoring

Lomax distribution has been widely used in economics, business and actuarial sciences. Due to its importance, we consider the statistical inference of this model under joint type-II censoring scenario. In order to estimate the parameters, we derive the Newton-Raphson(NR) procedure and we observe that most of the times in the simulation NR algorithm does not converge. Consequently, we make use of the expectation-maximization (EM) algorithm. Moreover, Bayesian estimations are also provided based on squared error, linear-exponential and generalized entropy loss functions together with the importance sampling method due to the structure of posterior density function. In the sequel, we perform a Monte Carlo simulation experiment to compare the performances of the listed methods. Mean squared error values, averages of estimated values as well as coverage probabilities and average interval lengths are considered to compare the performances of different methods. The approximate confidence intervals, bootstrap-p and bootstrap-t confidence intervals are computed for EM estimations. Also, Bayesian coverage probabilities and credible intervals are obtained. Finally, we consider the Bladder Cancer data to illustrate the applicability of the methods covered in the paper.

Bivariate odd Weibull-G family of distributions: properties, Bayesian and non-Bayesian estimation with bootstrap confidence intervals and application

ABSTRACT The aim of our paper is to introduce a new flexible bivariate generalized family of distributions based on Marshall and Olkin shock model, in the so-called the bivariate odd Weibull-G family. The proposed family is constructed from three independent odd Weibull-G families using a minimization process. Creating a bivariate distribution using shock model technique is nothing new, but here, we used only this technique to propose a flexible bivariate family, which has not been considered in the statistical literature yet. Some of its statistical properties are studied. After introducing the general class, one special model of the new family is discussed in detail. Maximum likelihood and Bayesian approaches are used to estimate the model parameters. Further, percentile bootstrap confidence interval and bootstrap-t confidence interval are estimated for the model parameters. A detailed simulation study is carried out to examine the bias and mean square error of the maximum likelihood and Bayesian estimators. Finally, we illustrate the importance of the proposed bivariate family by means of real data set.

On Bayesian parameter estimation of beta log Weibull distribution under type-II censoring

This article presents analysis of Type-II censored data when the lifetime distribution follows beta log Weibull distribution. Parameter estimation under classical and Bayesian set up is undertaken. Asymptotic confidence intervals are obtained using observed information matrix. For comparison purpose bootstrap-p and bootstrap-t confidence intervals are also constructed. Bayes estimates of the unknown parameters and the reliability function are obtained using Tierney–Kadane’s approximation and Markov Chain Monte Carlo methods. A real data set is analyzed to illustrate the proposed strategies.

Statistical Inference of the Reliability for Generalized Exponential Distribution Under Progressive Type-II Censoring Schemes

This paper discusses the estimation of $R= P(Y when $X$ and $Y$ are independent generalized exponential random variables with different scale parameters and the same shape parameter based on progressive type-II censoring samples. The maximum likelihood estimator (MLE) of the parameter $R$ is obtained. The approximate MLE (AMLE) of $R$ by the Taylor series expansion is proposed. The asymptotic distribution of $R$ is derived to construct the asymptotic MLE and AMLE confidence intervals. Also, the bootstrap-p and bootstrap-t confidence intervals are constructed. In addition, using the Metropolis-Hasting procedure, we derive the Bayes estimate and the corresponding credible interval. Finally, a Monte Carlo simulation and an analysis of two real datasets are presented to evaluate the various proposed methods.

A coverage probability of bootstrap-t confidence interval for the variance

We examine one-sided confidence intervals for the population variance, based on the ordinary t-statistics. We derive an unconditional coverage probability of the bootstrap-t interval for unknown variance. For that purpose, we find an Edgeworth expansion of the distribution of t-statistic to an order n-2. We can see that a number of simulation, B, has the influence on coverage probability of the confidence interval for the variance. If B equals sample size then coverage probability and its limit (when B ? ?) disagree at the level O(n-2). If we want that nominal coverage probability of the interval would be equal to ?, then coverage probability and its limit agree to order n-3/2 if B is of larger order than the square root of the sample size. We present a modeling application in insurance property, where the purpose of analysis is to measure variability of a data set.

Assessing the inter-rater agreement for ordinal data through weighted indexes.

Assessing the inter-rater agreement between observers, in the case of ordinal variables, is an important issue in both the statistical theory and biomedical applications. Typically, this problem has been dealt with the use of Cohen's weighted kappa, which is a modification of the original kappa statistic, proposed for nominal variables in the case of two observers. Fleiss (1971) put forth a generalization of kappa in the case of multiple observers, but both Cohen's and Fleiss' kappa could have a paradoxical behavior, which may lead to a difficult interpretation of their magnitude. In this paper, a modification of Fleiss' kappa, not affected by paradoxes, is proposed, and subsequently generalized to the case of ordinal variables. Monte Carlo simulations are used both to testing statistical hypotheses and to calculating percentile and bootstrap-t confidence intervals based on this statistic. The normal asymptotic distribution of the proposed statistic is demonstrated. Our results are applied to the classical Holmquist etal.'s (1967) dataset on the classification, by multiple observers, of carcinoma in situ of the uterine cervix. Finally, we generalize the use of s* to a bivariate case.

Reliability of stress–strength model for exponentiated Pareto distributions

ABSTRACTIn this paper, the reliability of a system is discussed when the strength of the system and the stress imposed on it are independent, non-identical exponentiated Pareto distributed random variables. Different point estimations and interval estimations are proposed. The point estimators obtained are maximum likelihood, uniformly minimum variance unbiased and Bayesian estimators. The interval estimations obtained are approximate, exact, bootstrap-p and bootstrap-t confidence intervals and Bayesian credible interval. Different methods and the corresponding confidence intervals are compared using Monte-carlo simulations.

Fullfact: an R package for the analysis of genetic and maternal variance components from full factorial mating designs

Full factorial breeding designs are useful for quantifying the amount of additive genetic, nonadditive genetic, and maternal variance that explain phenotypic traits. Such variance estimates are important for examining evolutionary potential. Traditionally, full factorial mating designs have been analyzed using a two‐way analysis of variance, which may produce negative variance values and is not suited for unbalanced designs. Mixed‐effects models do not produce negative variance values and are suited for unbalanced designs. However, extracting the variance components, calculating significance values, and estimating confidence intervals and/or power values for the components are not straightforward using traditional analytic methods. We introduce fullfact – an R package that addresses these issues and facilitates the analysis of full factorial mating designs with mixed‐effects models. Here, we summarize the functions of the fullfact package. The observed data functions extract the variance explained by random and fixed effects and provide their significance. We then calculate the additive genetic, nonadditive genetic, and maternal variance components explaining the phenotype. In particular, we integrate nonnormal error structures for estimating these components for nonnormal data types. The resampled data functions are used to produce bootstrap‐t confidence intervals, which can then be plotted using a simple function. We explore the fullfact package through a worked example. This package will facilitate the analyses of full factorial mating designs in R, especially for the analysis of binary, proportion, and/or count data types and for the ability to incorporate additional random and fixed effects and power analyses.

Confidence interval for the process capability index based on the bootstrap-t confidence interval for the standard deviation

This paper proposes a confidence interval for the process capability index based on the bootstrap-t confidence interval for the standard deviation. A Monte Carlo simulation study was conducted to compare the performance of the proposed confidence interval with the existing confidence interval based on the confidence interval for the standard deviation. Simulation results show that the proposed confidence interval performs well in terms of coverage probability in case of more skewed distributions. On the other hand, the existing confidence interval has a coverage probability close to the nominal level for symmetrical or less skewed distributions. The code to estimate the confidence interval in R language is provided.

Parameter and reliability estimation for an exponentiated half-logistic distribution under progressive type II censoring

In this article, we deal with a two-parameter exponentiated half-logistic distribution. We consider the estimation of unknown parameters, the associated reliability function and the hazard rate function under progressive Type II censoring. Maximum likelihood estimates (M LEs) are proposed for unknown quantities. Bayes estimates are derived with respect to squared error, linex and entropy loss functions. Approximate explicit expressions for all Bayes estimates are obtained using the Lindley method. We also use importance sampling scheme to compute the Bayes estimates. Markov Chain Monte Carlo samples are further used to produce credible intervals for the unknown parameters. Asymptotic confidence intervals are constructed using the normality property of the MLEs. For comparison purposes, bootstrap-p and bootstrap-t confidence intervals are also constructed. A comprehensive numerical study is performed to compare the proposed estimates. Finally, a real-life data set is analysed to illustrate the proposed methods of estimation.

Bootstrap Approach to Correlation Analysis of Two Mineral Components

Problem statement: In this article we considered pairs bootstrap thro ugh a truncated geometric bootstrap method for stationary time seri es data. Construction of valid inferential procedur es through the estimates of standard error, coefficien t of variation and other measures of statistical precision such as bootstrap confidence interval wer e considered. The method was used to confirm the correlation between Silicon Oxide (S iO2) and Aluminum Oxide (Al 2O3) from a geological data. A typical problem is that can these components exist together or they are mutually exclusive. Approach: We attempt to solve these problems through bootstr ap approach to correlation analysis and show that pair bootstrap method through truncated g eometric bootstrap method for stationary process revealed the correlation coefficient between Silico n Oxide (SiO 2) and Aluminum Oxide (Al 2O3) from the same geological field. Results: The computed measure of statistical precisions suc h as standard error, coefficient of variation and bootstrap-t con fidence interval revealed the correlation analysis of the bivariate stochastic processes of SiO 2 and Al 2O3 components from the same geological field. Conclusion: The correlation analysis of the bivariate stochasti c process of SiO 2 and Al 2O3 components through bootstrap method discussed in this study re vealed that the correlation coefficients are negati ve and bootstrap confidence intervals are negatively s kewed for all bootstrap replicates. This implies th at as one component increases, the other component decreases, which means that the two components are mutually exclusive and the abundance of one mineral prevents the other in the same oil reservoir of th e same geological field.

One-sided confidence intervals for population variances of skewed distributions

In this article we study the coverage accuracy of one-sided bootstrap-t confidence intervals for the population variances combined with Hall's and Johnson's transformation. We compare the coverage accuracy of all suggested intervals and intervals based on the Chi-square statistic for variances of positively skewed distributions. In addition, we describe and discuss an application of the presented methods for measuring and analyzing revenue variability within the food retail industry. The results show that both Hall's transformation and Johnson's transformation approaches yield good coverage accuracy of the lower endpoint confidence intervals, better than method based on the Chi-square statistic. For the upper endpoint confidence intervals Hall's bootstrap-t method yields the best coverage accuracy when compared with other methods.