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In this paper, a generalized bivariate Kummer-beta distribution is proposed. The name derives from the fact that its particular cases include univariate Kummer-beta distributions. This distribution generalizes a number of existing bivariate beta distributions, including Nadarajah's bivariate distributions, Libby and Novick's bivariate beta distribution and a central bivariate Kummer-beta distribution. Various properties associated with this newly introduced distribution are derived. The derived properties include product moments, marginal densities, marginal moments, conditional densities, conditional moments, Rényi entropy and Shannon entropy. Motivated by possible applications in economics, genetics, hydrology, meteorology, nuclear physics, and reliability, we also derive distributions of the product and the ratio of the components following the proposed distribution. Parameter estimation by maximum likelihood method is discussed by deriving expressions for score functions. Inference based on maximum likelihood estimation supposes that the maximum likelihood estimators have zero bias and zero mean squared errors. A simulation study is performed to check this for finite samples. [ABSTRACT FROM AUTHOR]

Prediction of satisfaction design, with binary endpoints, is an innovative strategy for phase II trials. We explain this hybrid frequentist-Bayesian strategy with an adept statistical plan and thorough findings, incorporating a description of study design features such as the sample size and the beta prior distribution, to simplify the Bayesian design. We also provide a set of tables and figures ranging from the stopping boundary for futility to the prediction of satisfaction, performance (type I error, power, and the probability of early termination PET), and sensitivity analysis for prediction of satisfaction. The statistical plan includes the operating characteristics through simulation study. Several trial examples from phase II lung cancer studies demonstrate the approach's practical use. The prediction of satisfaction design presents a flexible method in clinical study. This statistical study adds value to the application by broadening its scope. [ABSTRACT FROM AUTHOR]

This study aimed to propose a novel method for dynamic risk assessment using a Bayesian network (BN) based on fuzzy data to decrease uncertainty compared to traditional methods by integrating Interval Type-2 Fuzzy Sets (IT2FS) and Z-numbers. A bow-tie diagram was constructed by employing the System Hazard Identification, Prediction, and Prevention (SHIPP) approach, the Top Event Fault Tree, and the Barriers Failure Fault Tree. The experts then provided their opinions and confidence levels on the prior probabilities of the basic events, which were then quantified utilizing the IT2FS and combined using the Z-number to reduce the uncertainty of the prior probability. The posterior probability of the critical basic events (CBEs) was obtained using the beta distribution based on recorded data on their requirements and failure rates over five years. This information was then fed into the BN. Updating the BN allowed calculating the posterior probability of barrier failure and consequences. Spherical tanks were used as a case study to demonstrate and confirm the significant benefits of the methodology. The results indicated that the overall posterior probability of Consequences after the failure probability of barriers displayed an upward trend over the 5-year period. This rise in IT2FS-Z calculation outcomes exhibited a shallower slope compared to the IT2FS mode, attributed to the impact of experts' confidence levels in the IT2FS-Z mode. These differences became more evident by considering the 10−4 variance compared to the 10−5. This study offers industry managers a more comprehensive and reliable understanding of achieving the most effective accident prevention performance. [ABSTRACT FROM AUTHOR]

Even with the powerful statistical parameters derived from the Extreme Gradient Boost (XGB) algorithm, it would be advantageous to define the predicted accuracy to the level of a specific case, particularly when the model output is used to guide clinical decision-making. The probability density function (PDF) of the derived intracranial pressure predictions enables the computation of a definite integral around a point estimate, representing the event's probability within a range of values. Seven hold-out test cases used for the external validation of an XGB model underwent retinal vascular pulse and intracranial pressure measurement using modified photoplethysmography and lumbar puncture, respectively. The definite integral ±1 cm water from the median (DIICP) demonstrated a negative and highly significant correlation (-0.5213±0.17, p< 0.004) with the absolute difference between the measured and predicted median intracranial pressure (DiffICPmd). The concordance between the arterial and venous probability density functions was estimated using the two-sample Kolmogorov-Smirnov statistic, extending the distribution agreement across all data points. This parameter showed a statistically significant and positive correlation (0.4942±0.18, p< 0.001) with DiffICPmd. Two cautionary subset cases (Case 8 and Case 9), where disagreement was observed between measured and predicted intracranial pressure, were compared to the seven hold-out test cases. Arterial predictions from both cautionary subset cases converged on a uniform distribution in contrast to all other cases where distributions converged on either log-normal or closely related skewed distributions (gamma, logistic, beta). The mean±standard error of the arterial DIICP from cases 8 and 9 (3.83±0.56%) was lower compared to that of the hold-out test cases (14.14±1.07%) the between group difference was statistically significant (p<0.03). Although the sample size in this analysis was limited, these results support a dual and complementary analysis approach from independently derived retinal arterial and venous non-invasive intracranial pressure predictions. Results suggest that plotting the PDF and calculating the lower order moments, arterial DIICP, and the two sample Kolmogorov-Smirnov statistic may provide individualized predictive accuracy parameters. [ABSTRACT FROM AUTHOR]

In this work, a new flexible class, called the type-I extended-F family, is proposed. A special sub-model of the proposed class, called type-I extended-Weibull (TIEx-W) distribution, is explored in detail. Basic properties of the TIEx-W distribution are provided. The parameters of the TIEx-W distribution are obtained by eight classical methods of estimation. The performance of these estimators is explored using Monte Carlo simulation results for small and large samples. Besides, the Bayesian estimation of the model parameters under different loss functions for the real data set is also provided. The importance and flexibility of the TIEx-W model are illustrated by analyzing an insurance data. The real-life insurance data illustrates that the TIEx-W distribution provides better fit as compared to competing models such as Lindley–Weibull, exponentiated Weibull, Kumaraswamy–Weibull, α logarithmic transformed Weibull, and beta Weibull distributions, among others. [ABSTRACT FROM AUTHOR]

A common issue affecting DNA methylation analysis in tumor tissue is the presence of a substantial amount of non-tumor methylation signal derived from the surrounding microenvironment. Although approaches for quantifying and correcting for the infiltration component have been proposed previously, we believe these have not fully addressed the issue in a comprehensive and universally applicable way. We present a multi-population framework for adjusting DNA methylation beta values on the Illumina 450/850K platform using generic purity estimates to account for non-tumor signal. Our approach also provides an indirect estimate of the aggregate methylation state of the surrounding normal tissue. Using whole exome sequencing derived purity estimates and Illumina 450K methylation array data generated by The Cancer Genome Atlas project (TCGA), we provide a demonstration of this framework in breast cancer illustrating the effect of beta correction on the aggregate methylation beta value distribution, clustering accuracy, and global methylation profiles. [ABSTRACT FROM AUTHOR]

The concept of professional judgement underpins the way in which an occupational hygienist assesses an exposure problem. Despite the importance placed on professional judgement in the discipline, a method of assessment to characterise accuracy has not been available. In this paper, we assess the professional judgement of four occupational hygienists ('experts') when completing exposure assessments on a range of airborne contaminants across a number of job roles within a surface mining environment in the Pilbara region of Western Australia. The job roles assessed were project driller, mobile equipment operator, fixed plant maintainer, and drill and blast operator. The contaminants of interest were respirable crystalline silica, respirable dust, and inhalable dust. The novel approach of eliciting exposure estimates focusing on contaminant concentration and attribution of an exposure standard estimate was used. The majority of the elicited values were highly skewed; therefore, a scaled Beta distribution were fitted. These elicited fitted distributions were then compared to measured data distributions, the results of which had been collected as part of an occupational hygiene program assessing full-shift exposures to the same contaminants and job roles assessed by the experts. Our findings suggest that the participating experts within this study tended to overestimate exposures. In addition, the participating experts were more accurate at estimating percentage of an exposure standard than contaminant concentration. We demonstrate that this elicitation approach and the encoding methodology contained within can be applied to assess accuracy of exposure judgements which will impact on worker protection and occupational health outcomes. [ABSTRACT FROM AUTHOR]

Wheat is a staple crop that is critical for feeding a hungry and growing planet, but its nutritive value has declined as global temperatures have warmed. The price offered to producers depends not only on yield but also grain protein content (GPC), which are often negatively related at the field scale but can positively covary depending in part on management strategies, emphasizing the need to understand their variability within individual fields. We measured yield and GPC in a winter wheat field in Sun River, Montana, USA, and tested the ability of normalized difference vegetation index (NDVI) measurements from an unoccupied aerial vehicle (UAV) on spatial scales of ~10 cm and from Landsat on spatial scales of 30 m to predict them. Landsat observations were poorly related to yield and GPC measurements. A multiple linear model using information from four (three) UAV flyovers was selected as the most parsimonious and predicted 26% (40%) of the variability in wheat yield (GPC). We sought to understand the optimal spatial scale for interpreting UAV observations given that the ~ 10 cm pixels yielded more than 12 million measurements at far finer resolution than the 12 m scale of the harvester. The variance in NDVI observations was "averaged out" at larger pixel sizes but only ~ 20% of the total variance was averaged out at the spatial scale of the harvester on some measurement dates. Spatial averaging to the scale of the harvester also made little difference in the total information content of NDVI fit using Beta distributions as quantified using the Kullback-Leibler divergence. Radially-averaged power spectra of UAV-measured NDVI revealed relatively steep power-law relationships with exponentially less variance at finer spatial scales. Results suggest that larger pixels can reasonably capture the information content of within-field NDVI, but the 30 m Landsat scale is too coarse to describe some of the key features of the field, which are consistent with topography, historic management practices, and edaphic variability. Future research should seek to determine an 'optimum' spatial scale for NDVI observations that minimizes effort (and therefore cost) while maintaining the ability of producers to make management decisions that positively impact wheat yield and GPC. [ABSTRACT FROM AUTHOR]

In this work, a new family of distributions, which extends the Beta transmuted family, was obtained, called the Modified Beta Transmuted Family of distribution. This derived family has the Beta Family of Distribution and the Transmuted family of distribution as subfamilies. The Modified beta transmuted frechet, modified beta transmuted exponential, modified beta transmuted gompertz and modified beta transmuted lindley were obtained as special cases. The analytical expressions were studied for some statistical properties of the derived family of distribution which includes the moments, moments generating function and order statistics. The estimates of the parameters of the family were obtained using the maximum likelihood estimation method. Using the exponential distribution as a baseline for the family distribution, the resulting distribution (modified beta transmuted exponential distribution) was studied and its properties. The modified beta transmuted exponential distribution was applied to a real life time data to assess its flexibility in which the results shows a better fit when compared to some competitive models. [ABSTRACT FROM AUTHOR]

The distributive power of the arithmetic operators: multiplication, division, addition, and subtraction, gives the arithmetic optimization algorithm (AOA) its unique ability to find the global optimum for optimization problems used to test its performance. Several other mathematical operators exist with the same or better distributive properties, which can be exploited to enhance the performance of the newly proposed AOA. In this paper, we propose an improved version of the AOA called nAOA algorithm, which uses the high-density values that the natural logarithm and exponential operators can generate, to enhance the exploratory ability of the AOA. The addition and subtraction operators carry out the exploitation. The candidate solutions are initialized using the beta distribution, and the random variables and adaptations used in the algorithm have beta distribution. We test the performance of the proposed nAOA with 30 benchmark functions (20 classical and 10 composite test functions) and three engineering design benchmarks. The performance of nAOA is compared with the original AOA and nine other state-of-the-art algorithms. The nAOA shows efficient performance for the benchmark functions and was second only to GWO for the welded beam design (WBD), compression spring design (CSD), and pressure vessel design (PVD). [ABSTRACT FROM AUTHOR]

This paper provides a general framework for controlling quality characteristics related to control variables and limited to the intervals (0, 1], [0, 1), or [0, 1]. The proposed control chart is based on the inflated beta regression model considering a reparametrization of the inflated beta distribution indexed by the response mean, which is useful for modeling fractions and proportions. The contribution of the paper is twofold. First, we extend the inflated beta regression model by allowing a regression structure for the precision parameter. We also present closed-form expressions for the score vector and Fisher's information matrix. Second, based on the proposed regression model, we introduce a new model-based control chart. The control limits are obtained considering the estimates of the inflated beta regression model parameters. We conduct a Monte Carlo simulation study to evaluate the performance of the proposed regression model estimators, and the performance of the proposed control chart is evaluated in terms of run length distribution. Finally, we present and discuss an empirical application to show the applicability of the proposed regression control chart. [ABSTRACT FROM AUTHOR]

Goals and methods: A simulation study investigated how ceiling and floor effect (CFE) affect the performance of Welch’s t-test, F-test, Mann-Whitney test, Kruskal-Wallis test, Scheirer-Ray-Hare-test, trimmed t-test, Bayesian t-test, and the “two one-sided tests” equivalence testing procedure. The effect of CFE on the estimate of group difference and on its confidence interval, and on Cohen’s d and on its confidence interval was also evaluated. In addition, the parametric methods were applied to data transformed with log or logit function and the performance was evaluated. The notion of essential maximum from abstract measurement theory is used to formally define CFE and the principle of maximum entropy was used to derive probability distributions with essential maximum/minimum. These distributions allow the manipulation of the magnitude of CFE through a parameter. Beta, Gamma, Beta prime and Beta-binomial distributions were obtained in this way with the CFE parameter corresponding to the logarithm of the geometric mean. Wald distribution and ordered logistic regression were also included in the study due to their measure-theoretic connection to CFE, even though these models lack essential minimum/maximum. Performance in two-group, three-group and 2 × 2 factor design scenarios was investigated by fixing the group differences in terms of CFE parameter and by adjusting the base level of CFE. Results and conclusions: In general, bias and uncertainty increased with CFE. Most problematic were occasional instances of biased inference which became more certain and more biased as the magnitude of CFE increased. The bias affected the estimate of group difference, the estimate of Cohen’s d and the decisions of the equivalence testing methods. Statistical methods worked best with transformed data, albeit this depended on the match between the choice of transformation and the type of CFE. Log transform worked well with Gamma and Beta prime distribution while logit transform worked well with Beta distribution. Rank-based tests showed best performance with discrete data, but it was demonstrated that even there a model derived with measurement-theoretic principles may show superior performance. Trimmed t-test showed poor performance. In the factor design, CFE prevented the detection of main effects as well as the detection of interaction. Irrespective of CFE, F-test misidentified main effects and interactions on multiple occasions. Five different constellations of main effect and interactions were investigated for each probability distribution, and weaknesses of each statistical method were identified and reported. As part of the discussion, the use of generalized linear models based on abstract measurement theory is recommended to counter CFE. Furthermore, the necessity of measure validation/calibration studies to obtain the necessary knowledge of CFE to design and select an appropriate statistical tool, is stressed. [ABSTRACT FROM AUTHOR]

Employing historical records we are able to estimate the risk of premature death during the second plague pandemic, and identify the Black Death and pestis secunda epidemics. We show a novel method of calculating Bayesian credible intervals for a ratio of beta distributed random variables and use this to quantify uncertainty of relative risk estimates for these two epidemics which we consider in a 2 × 2 contingency table framework. [ABSTRACT FROM AUTHOR]

We apply a known algorithm for computing exactly inequalities between Beta distributions to assess whether a given position in a genome is differentially methylated across samples. We discuss the advantages brought by the adoption of this solution with respect to two approximations (Fisher's test and Z score). The same formalism presented here can be applied in a similar way to variant calling. [ABSTRACT FROM AUTHOR]

Recovery rate is essential to the estimation of the portfolio’s loss and economic capital. Neglecting the randomness of the distribution of recovery rate may underestimate the risk. The study introduces two kinds of models of distribution, Beta distribution estimation and kernel density distribution estimation, to simulate the distribution of recovery rates of corporate loans and bonds. As is known, models based on Beta distribution are common in daily usage, such as CreditMetrics by J.P. Morgan, Portfolio Manager by KMV and Losscalc by Moody’s. However, it has a fatal defect that it can’t fit the bimodal or multimodal distributions such as recovery rates of corporate loans and bonds as Moody’s new data show. In order to overcome this flaw, the kernel density estimation is introduced and we compare the simulation results by histogram, Beta distribution estimation and kernel density estimation to reach the conclusion that the Gaussian kernel density distribution really better imitates the distribution of the bimodal or multimodal data samples of corporate loans and bonds. Finally, a Chi-square test of the Gaussian kernel density estimation proves that it can fit the curve of recovery rates of loans and bonds. So using the kernel density distribution to precisely delineate the bimodal recovery rates of bonds is optimal in credit risk management. [ABSTRACT FROM AUTHOR]