Confidence Intervals for the Ratio of the Coefficients of Variation of Inverse-Gamma Distributions

Herein, we present four methods for constructing confidence intervals for the ratio of the coefficients of variation of inverse-gamma distributions using the percentile bootstrap, fiducial quantities, and Bayesian methods based on the Jeffreys and uniform priors. We compared their performances using coverage probabilities and expected lengths via simulation studies. The results show that the confidence intervals constructed with the Bayesian method based on the uniform prior and fiducial quantities performed better than those constructed with the Bayesian method based on the Jeffreys prior and the percentile bootstrap. Rainfall data from Thailand was used to illustrate the efficacies of the proposed methods.

When to Use Different Tests for Power Analysis and Data Analysis for Mediation

Several options exist for conducting inference on indirect effects in mediation analysis. While methods which use bootstrapping are the preferred inferential approach for testing mediation, they are time consuming when the test must be performed many times for a power analysis. Alternatives which are more computationally efficient are not as robust, meaning accuracy of the inferences from these methods are more affected by nonnormal and heteroskedastic data (Biesanz et al., 2010). While previous research focused on how different sample sizes would be needed to achieve the same amount of power for different inferential approaches (Fritz & MacKinnon, 2007), we explore how similar power estimates are at the same sample size. We compare the power estimates from six tests using a Monte Carlo simulation study, varying the path coefficients and tests of the indirect effect. If tests produce similar power estimates, the more computationally efficient test could be used for power analysis and the more intensive test involving resampling can be used for data analysis. We found that when the assumptions of linear regression are met, three tests consistently perform similarly: the joint significance test, the Monte Carlo confidence interval, and the percentile bootstrap confidence interval. Based on these results, we recommend using the more computationally efficient joint significance test for power analysis then using the percentile bootstrap confidence interval for the data analysis.

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.

Stress-Strength Reliability for Exponentiated Inverted Weibull Distribution with Application on Breaking of Jute Fiber and Carbon Fibers

For the first time and by using an entire sample, we discussed the estimation of the unknown parameters θ 1 , θ 2 , and β and the system of stress-strength reliability R = P Y < X for exponentiated inverted Weibull (EIW) distributions with an equivalent scale parameter supported eight methods. We will use maximum likelihood method, maximum product of spacing estimation (MPSE), minimum spacing absolute-log distance estimation (MSALDE), least square estimation (LSE), weighted least square estimation (WLSE), method of Cramér-von Mises estimation (CME), and Anderson-Darling estimation (ADE) when X and Y are two independent a scaled exponentiated inverted Weibull (EIW) distribution. Percentile bootstrap and bias-corrected percentile bootstrap confidence intervals are introduced. To pick the better method of estimation, we used the Monte Carlo simulation study for comparing the efficiency of the various estimators suggested using mean square error and interval length criterion. From cases of samples, we discovered that the results of the maximum product of spacing method are more competitive than those of the other methods. A two real‐life data sets are represented demonstrating how the applicability of the methodologies proposed in real phenomena.

On the Admissibility of Simultaneous Bootstrap Confidence Intervals

Simultaneous confidence intervals are commonly used in joint inference of multiple parameters. When the underlying joint distribution of the estimates is unknown, nonparametric methods can be applied to provide distribution-free simultaneous confidence intervals. In this note, we propose new one-sided and two-sided nonparametric simultaneous confidence intervals based on the percentile bootstrap approach. The admissibility of the proposed intervals is established. The numerical results demonstrate that the proposed confidence intervals maintain the correct coverage probability for both normal and non-normal distributions. For smoothed bootstrap estimates, we extend Efron’s (2014) nonparametric delta method to construct nonparametric simultaneous confidence intervals. The methods are applied to construct simultaneous confidence intervals for LASSO regression estimates.

The Bayesian Confidence Interval for Coefficient of Variation of Zero-inflated Poisson Distribution with Application to Daily COVID-19 Deaths in Thailand

Coronavirus disease 2019 (COVID-19) has spread rapidly throughout the world and has caused millions of deaths. However, the number of daily COVID-19 deaths in Thailand has been low with most daily records showing zero deaths, thereby making them fit a Zero-Inflated Poisson (ZIP) distribution. Herein, confidence intervals for the Coefficient Of Variation (CV) of a ZIP distribution are derived using four methods: the standard bootstrap (SB), percentile bootstrap (PB), Markov Chain Monte Carlo (MCMC), and the Bayesian-based highest posterior density (HPD), for which using the variance of the CV is unnecessary. We applied the methods to both simulated data and data on the number of daily COVID-19 deaths in Thailand. Both sets of results show that the SB, MCMC, and HPD methods performed better than PB for most cases in terms of coverage probability and average length. Overall, the HPD method is recommended for constructing the confidence interval for the CV of a ZIP distribution. Doi: 10.28991/esj-2021-SPER-05 Full Text: PDF

The Percentile Bootstrap: A Primer With Step-by-Step Instructions in R

The percentile bootstrap is the Swiss Army knife of statistics: It is a nonparametric method based on data-driven simulations. It can be applied to many statistical problems, as a substitute to standard parametric approaches, or in situations for which parametric methods do not exist. In this Tutorial, we cover R code to implement the percentile bootstrap to make inferences about central tendency (e.g., means and trimmed means) and spread in a one-sample example and in an example comparing two independent groups. For each example, we explain how to derive a bootstrap distribution and how to get a confidence interval and a p value from that distribution. We also demonstrate how to run a simulation to assess the behavior of the bootstrap. For some purposes, such as making inferences about the mean, the bootstrap performs poorly. But for other purposes, it is the only known method that works well over a broad range of situations. More broadly, combining the percentile bootstrap with robust estimators (i.e., estimators that are not overly sensitive to outliers) can help users gain a deeper understanding of their data than they would using conventional methods.

Sex-Dependent Aggregation of Tinnitus in Swedish Families

Twin and adoption studies point towards a genetic contribution to tinnitus; however, how the genetic risk applies to different forms of tinnitus is poorly understood. Here, we perform a familial aggregation study and determine the relative recurrence risk for tinnitus in siblings (λs). Four different Swedish studies (N = 186,598) were used to estimate the prevalence of self-reported bilateral, unilateral, constant, and severe tinnitus in the general population and we defined whether these 4 different forms of tinnitus segregate in families from the Swedish Tinnitus Outreach Project (STOP, N = 2305). We implemented a percentile bootstrap approach to provide accurate estimates and confidence intervals for λs. We reveal a significant λs for all types of tinnitus, the highest found being 7.27 (95% CI (5.56–9.07)) for severe tinnitus, with a higher susceptibility in women (10.25; 95% CI (7.14–13.61)) than in men (5.03; 95% CI (3.22–7.01)), suggesting that severity may be the most genetically influenced trait in tinnitus in a sex-dependent manner. Our findings strongly support the notion that genetic factors impact on the development of tinnitus, more so for severe tinnitus. These findings highlight the importance of considering tinnitus severity and sex in the design of large genetic studies to optimize diagnostic approaches and ultimately improve therapeutic interventions.

Construction of Confidence Interval of the Parameter in Von Mises Distribution using Bootstrap Methods

Bootstrap method is a computer-based technique for making certain kind of statistical inferences which can simplify the often intricate calculations of traditional statistical theory. Recently, bootstrapping has been widely used for the parameter estimation of linear data. In this paper, we consider bootstrapping methods in the construction of the confidence interval of concentration parameter, for the von Mises distribution. The performances of confidence interval based on percentile bootstrap, bootstrap-t and calibration bootstrap are evaluated via simulation study. The numerical results found that confidence interval based on the calibration bootstrap is good in terms of coverage probability. Meanwhile, confidence interval based on the bootstrap-t method has a shorter expected length. The confidence intervals were illustrated using daily wind direction data recorded at maximum wind speed for four stations in Malaysia. From point estimates of the concentration parameter and the respective confidence interval, we note that the method works well for a wide range of values. The implication of the study is that confidence interval of the concentration parameter can be obtained using bootstrap as it provides good estimates. Keywords: bootstrap-t; calibration bootstrap; concentration parameter; percentile bootstrap; von Mises distribution

Classical Estimation of the Index Spmk and Its Confidence Intervals for Power Lindley Distributed Quality Characteristics

The process capability index (PCI) has been introduced as a tool to aid in the assessment of process performance. Usually, conventional PCIs perform well under normally distributed quality characteristics. However, when these PCIs are employed to evaluate nonnormally distributed process, they often provide inaccurate results. In this article, in order to estimate the PCI Spmk when the process follows power Lindley distribution, first, seven classical methods of estimation, namely, maximum likelihood method of estimation, ordinary and weighted least squares methods of estimation, Cramèr–von Mises method of estimation, maximum product of spacings method of estimation, Anderson–Darling, and right-tail Anderson–Darling methods of estimation, are considered and the performance of these estimation methods based on their mean squared error is compared. Next, three bootstrap confidence intervals (BCIs) of the PCI Spmk, namely, standard bootstrap, percentile bootstrap, and bias-corrected percentile bootstrap, are considered and compared in terms of their average width, coverage probability, and relative coverage. Besides, a new cost-effective PCI, namely, Spmkc is introduced by incorporating tolerance cost function in the index Spmk. To evaluate the performance of the methods of estimation and BCIs, a simulation study is carried out. Simulation results showed that the maximum likelihood method of estimation performs better than their counterparts in terms of mean squared error, while bias-corrected percentile bootstrap provides smaller confidence length (width) and higher relative coverage than standard bootstrap and percentile bootstrap across sample sizes. Finally, two real data examples are provided to investigate the performance of the proposed procedures.