Testing Parameters of a Gamma Distribution for Small Samples

Group acceptance sampling plans for type-I heavy-tailed exponential distribution based on truncated life tests

The usefulness of a new heavy-tailed distribution is studied in this article. The type-I heavy-tailed exponential (TI-HTE) distribution studied here has been suggested in the literature but has not been studied anywhere other than now. Some of its properties, together with graphical representations, were considered. The study utilized the maximum likelihood method in the estimation of the parameters. The primary goal is to create group acceptance sampling plans (GASP) using the TI-HTE model to determine whether units from a production process should be accepted or rejected. Through simulation studies and real-world examples, the importance of the TI-HTE model in identifying extreme behaviors beyond typical distributions like exponential or heavy-tailed distributions is emphasized.

A Software Tool for Estimating Uncertainty of Bayesian Posterior Probability for Disease

The role of medical diagnosis is essential in patient care and healthcare. Established diagnostic practices typically rely on predetermined clinical criteria and numerical thresholds. In contrast, Bayesian inference provides an advanced framework that supports diagnosis via in-depth probabilistic analysis. This study's aim is to introduce a software tool dedicated to the quantification of uncertainty in Bayesian diagnosis, a field that has seen minimal exploration to date. The presented tool, a freely available specialized software program, utilizes uncertainty propagation techniques to estimate the sampling, measurement, and combined uncertainty of the posterior probability for disease. It features two primary modules and fifteen submodules, all designed to facilitate the estimation and graphical representation of the standard uncertainty of the posterior probability estimates for diseased and non-diseased population samples, incorporating parameters such as the mean and standard deviation of the test measurand, the size of the samples, and the standard measurement uncertainty inherent in screening and diagnostic tests. Our study showcases the practical application of the program by examining the fasting plasma glucose data sourced from the National Health and Nutrition Examination Survey. Parametric distribution models are explored to assess the uncertainty of Bayesian posterior probability for diabetes mellitus, using the oral glucose tolerance test as the reference diagnostic method.

Investigating surrogate safety measures at midblock pedestrian crossings using multivariate models with roadside camera data

This study aims to evaluate and compare Surrogate Safety Measures (SSMs) at five midblock Rectangular Rapid Flashing Beacons (RRFB) and two midblock Pedestrian Hybrid Beacons (PHB) sites in Florida using extensive video data collected over the study period of July to November 2021. Computer vision and data processing resulted in four pedestrian SSMs, namely spatial gap, temporal gap, relative time to collision (RTTC) and Post Encroachment Time (PET). An initial investigation of the SSMs using Mann-Whitney-Wilcoxon tests revealed significant differences in the SSM values across different treatment types and hours of the day. Additionally, univariate regression of spatial gap, and multivariate regression of temporal gap, RTTC and PET revealed significant differences of SSMs across RRFB and PHB sites. The study considered both linear and non-linear (gamma, inverse Gaussian and lognormal) regression models. After considering various traffic and operational parameters, the data were aggregated for each pedestrian-vehicle interaction on each lane to create a total of 395 observations. The SSMs included average spatial gap, temporal gap, RTTC and PET for each interaction of pedestrian and vehicle on each lane. The results indicated that non-linear models performed better than the linear models. Moreover, the presence of the PHB, weekday, signal activation, lane count, pedestrian speed, vehicle speed, land use mix, morning period and pedestrian starting position from the sidewalk have been found to be significant determinants of the SSMs. Results also suggest temporal SSMs increase at the PHB sites compared to the RRFB sites, indicating an improvement of traffic safety at PHB sites. However, the spatial gap decreased for PHB sites compared to the RRFB sites, which suggests that pedestrians tend to start to cross the RRFB sites when they perceive vehicles to be further away than at the PHB sites.

A New Distribution for Modeling Data with Increasing Hazard Rate: A Case of COVID-19 Pandemic and Vinyl Chloride Data

A novel lifetime distribution has been defined and examined in this study. The odd Lindley–Pareto (OLiP) distribution is the name we give to the new distribution. The new density function can be written as an odd Lindley-G distribution with Pareto amplification. The moment-generating function and characteristic function, entropy and asymptotic behavior, order statistics and moments, mode, variance, skewness, and kurtosis are some of the aspects of the OLiP distribution that are discovered. Seven non-Bayesian estimation techniques and Bayesian estimation utilizing Markov chain Monte Carlo were compared for performance. Additionally, when the lifetime test is truncated after a predetermined period, single acceptance sampling plans (SASPs) are created for the newly suggested, OLiP distribution. The median lifetime of the OLiP distribution with pre-specified factors is taken as the truncation time. To guarantee that the specific life test is obtained at the defined risk to the user, the minimum sample size is required. For a particular consumer’s risk, the OLiP distribution’s parameters, and the truncation time, numerical results are obtained. The new distribution is illustrated using mortality rates of COVID-19 patients in Canada and vinyl chloride data in (g/L) from ground-water monitoring wells that are located in clean-up-gradient areas.

Expected Bayesian estimation for exponential model based on simple step stress with Type-I hybrid censored data

The procedure of selecting the values of hyper-parameters for prior distributions in Bayesian estimate has produced many problems and has drawn the attention of many authors, therefore the expected Bayesian (E-Bayesian) estimation method to overcome these problems. These approaches are used based on the step-stress acceleration model under the Exponential Type-I hybrid censored data in this study. The values of the distribution parameters are derived. To compare the E-Bayesian estimates to the other estimates, a comparative study was conducted using the simulation research. Four different loss functions are used to generate the Bayesian and E-Bayesian estimators. In addition, three alternative hyper-parameter distributions were used in E-Bayesian estimation. Finally, a real-world data example is examined for demonstration and comparative purposes.

Inferences for Alpha Power Exponential Distribution Using Adaptive Progressively Type-II Hybrid Censored Data with Applications

One of the most important asymmetrical probability distributions that recently presented as an extension of the conventional exponential distribution is the alpha power exponential distribution. It may be compared to various asymmetrical well-known models, such as Weibull and gamma distributions. As a result, using an adaptive progressive Type-II hybrid censoring scheme, this paper investigates the estimation problems of the alpha power exponential distribution. Maximum likelihood and Bayesian methods are used to estimate unknown parameters, reliability, and hazard rate functions. Under the assumption of independent gamma priors and symmetric loss function, Bayesian estimators are examined. The Bayesian credible intervals and estimated confidence intervals of the relevant values are also calculated. The various estimating approaches are evaluated using a simulation study that considers various sample sizes and censoring schemes. Furthermore, numerous optimality criteria are examined, and the best progressive censoring schemes are offered. Finally, for an explanation, two real data sets from engineering and chemical fields are provided to show the applicability of the asymmetrical alpha power exponential distribution. The Bayesian method for estimating the parameters and reliability indices of the alpha power exponential distribution is recommended based on numerical results, especially when the number of observed data is small.

Analysis for Xgamma Parameters of Life under Type-II Adaptive Progressively Hybrid Censoring with Applications in Engineering and Chemistry

Censoring mechanisms are widely used in various life tests, such as medicine, engineering, biology, etc., as they save (overall) test time and cost. In this context, we consider the problem of estimating the unknown xgamma parameter and some survival characteristics, such as reliability and failure rate functions in the presence of adaptive type-II progressive hybrid censored data. For this purpose, the maximum likelihood and Bayesian inferential approaches are used. Using the observed Fisher information under s-normal approximation, different asymptotic confidence intervals for any function of the unknown parameter were constructed. Using the gamma flexible prior, Bayes estimators against the squared-error loss were developed. Two procedures of Bayesian approximations—Lindley’s approximation and Metropolis–Hastings algorithm—were used to carry out the Bayes estimates and to construct the associated credible intervals. An extensive simulation study was implemented to compare the performance of the different methods. To validate the proposed methodologies of inference—two practical studies using datasets that form engineering and chemical fields are discussed.

α-Power transformed transformed power function distribution with applications

By standard transformation of a random variable, we obtained a partially bounded one-parameter version of the bounded three-parameter power function distribution by Saran and Pandey (2004) which we called the Transformed Power Function (TPF) distribution and based on an alpha-power transformation method due to Mahdavi and Kundu (2017) we generalized the TPF distribution as the α-Power Transformed Transformed Power Function (αPTTPF) distribution. Some of the properties of the αPTTPF distribution are given, and we approached the parameter estimation by three methods, namely: maximum likelihood, ordinary least-squares, and weighted least-squares, but after comparing the results from a simulation study, we settled for the maximum likelihood. The new distribution is suitable for modeling data with either decreasing or upside-down bathtub hazard rates. Three real data-sets are used to demonstrate the usefulness of the new model.

A New Right-Skewed Upside Down Bathtub Shaped Heavy-tailed Distribution and its Applications

A one parameter right skewed, upside down bathtub type, heavy-tailed distribution is derived. Various statistical properties and maximum likelihood approaches for estimation purpose are studied. Five different real data sets with four different models are considered to illustrate the suitability of the proposed model.

A modified truncated distribution for modeling the heavy tail, engineering and environmental sciences data

Truncated models are imperative to efficiently analyze the finite data that we observe in almost all the real life situations. In this paper, a new truncated distribution having four parameters named Weibull-Truncated Exponential Distribution (W-TEXPD) is developed. The proposed model can be used as an alternative to the Exponential, standard Weibull and shifted Gamma-Weibull and three parameter Weibull distributions. The statistical characteristics including cumulative distribution function, hazard function, cumulative hazard function, central moments, skewness, kurtosis, percentile and entropy of the proposed model are derived. The maximum likelihood estimation method is employed to evaluate the unknown parameters of the W-TEXPD. A simulation study is also carried out to assess the performance of the model parameters. The proposed probability distribution is fitted on five data sets from different fields to demonstrate its vast application. A comparison of the proposed model with some extant models is given to justify the performance of the W-TEXPD.

Statistical Inference for Inverted Kumaraswamy Distribution Based on Dual Generalized Order Statistics

In this paper, the shape parameters, reliability and hazard rate functions of the inverted Kumaraswamy distribution are estimated using maximum likelihood and Bayesian methods based on dual generalized order statistics. The Bayes estimators are derived under the squared error loss function as a symmetric loss function and the linear-exponential loss function as an asymmetric loss function based on dual generalized order statistics. Confidence and credible intervals for the parameters, reliability and hazard rate functions are obtained. All results are specialized to lower record values, also a numerical study is presented to illustrate the theoretical procedures.

A New Parametric Life Distribution with Modified Bagdonavičius-Nikulin Goodness-of-Fit Test for Censored Validation, Properties, Applications, and Different Estimation Methods

In this paper, we first study a new two parameter lifetime distribution. This distribution includes "monotone" and "non-monotone" hazard rate functions which are useful in lifetime data analysis and reliability. Some of its mathematical properties including explicit expressions for the ordinary and incomplete moments, generating function, Renyi entropy, δ-entropy, order statistics and probability weighted moments are derived. Non-Bayesian estimation methods such as the maximum likelihood, Cramer-Von-Mises, percentile estimation, and L-moments are used for estimating the model parameters. The importance and flexibility of the new distribution are illustrated by means of two applications to real data sets. Using the approach of the Bagdonavicius-Nikulin goodness-of-fit test for the right censored validation, we then propose and apply a modified chi-square goodness-of-fit test for the Burr X Weibull model. The modified goodness-of-fit statistics test is applied for the right censored real data set. Based on the censored maximum likelihood estimators on initial data, the modified goodness-of-fit test recovers the loss in information while the grouped data follows the chi-square distribution. The elements of the modified criteria tests are derived. A real data application is for validation under the uncensored scheme.

Meta-Analysis of Rare Binary Adverse Event Data

We examine the use of fixed-effects and random-effects moment-based meta-analytic methods for analysis of binary adverse event data. Special attention is paid to the case of rare adverse events which are commonly encountered in routine practice. We study estimation of model parameters and between-study heterogeneity. In addition, we examine traditional approaches to hypothesis testing of the average treatment effect and detection of the heterogeneity of treatment effect across studies. We derive three new methods, simple (unweighted) average treatment effect estimator, a new heterogeneity estimator, and a parametric bootstrapping test for heterogeneity. We then study the statistical properties of both the traditional and new methods via simulation. We find that in general, moment-based estimators of combined treatment effects and heterogeneity are biased and the degree of bias is proportional to the rarity of the event under study. The new methods eliminate much, but not all of this bias. The various estimators and hypothesis testing methods are then compared and contrasted using an example dataset on treatment of stable coronary artery disease.