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PloS One 2023In 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)...
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.
Topics: Likelihood Functions; Bayes Theorem; Computer Simulation; Statistical Distributions; Monte Carlo Method
PubMed: 36730300
DOI: 10.1371/journal.pone.0275430 -
PeerJ 2022The gamma distribution is commonly used to model environmental data. However, rainfall data often contain zero observations, which violates the assumption that all...
The gamma distribution is commonly used to model environmental data. However, rainfall data often contain zero observations, which violates the assumption that all observations must be positive in a gamma distribution, and so a gamma model with excess zeros treated as a binary random variable is required. Rainfall dispersion is important and interesting, the confidence intervals for the variance of a gamma distribution with excess zeros help to examine rainfall intensity, which may be high or low risk. Herein, we propose confidence intervals for the variance of a gamma distribution with excess zeros by using fiducial quantities and parametric bootstrapping, as well as Bayesian credible intervals and highest posterior density intervals based on the Jeffreys', uniform, or normal-gamma-beta prior. The performances of the proposed confidence interval were evaluated by establishing their coverage probabilities and average lengths via Monte Carlo simulations. The fiducial quantity confidence interval performed the best for a small probability of the sample containing zero observations () whereas the Bayesian credible interval based on the normal-gamma-beta prior performed the best for large . Rainfall data from the Kiew Lom Dam in Lampang province, Thailand, are used to illustrate the efficacies of the proposed methods in practice.
Topics: Bayes Theorem; Thailand; Probability; Statistical Distributions; Risk
PubMed: 36132216
DOI: 10.7717/peerj.14023 -
CPT: Pharmacometrics & Systems... May 2020Pharmacometric models using lognormal distributions have become commonplace in pharmacokinetic-pharmacodynamic investigations. The extent to which it can be interpreted...
Pharmacometric models using lognormal distributions have become commonplace in pharmacokinetic-pharmacodynamic investigations. The extent to which it can be interpreted by traditional description of variability through the normal distribution remains elusive. In this tutorial, the comparison is made using formal approximation methods. The quality of the resulting approximation was assessed by the similarity of prediction intervals (PIs) to true values, illustrated using 80% PIs. Approximated PIs were close to true values when lognormal standard deviation (omega) was smaller than about 0.25, depending mostly on the desired precision. With increasing omega values, the precision of approximation worsens and starts to deteriorate at omega values of about 1. With such high omega values, there is no resemblance between the lognormal and normal distribution anymore. To support dissemination and interpretation of these nonlinear properties, some additional statistics are discussed in the context of the three regions of behavior of the lognormal distribution.
Topics: Humans; Models, Biological; Models, Statistical; Normal Distribution; Pharmacokinetics
PubMed: 32198841
DOI: 10.1002/psp4.12507 -
PloS One 2021This paper studies the distribution of the firm size for the Colombian economy showing evidence against the Gibrat's law, which assumes a stable lognormal distribution....
This paper studies the distribution of the firm size for the Colombian economy showing evidence against the Gibrat's law, which assumes a stable lognormal distribution. On the contrary, we propose a lognormal expansion that captures deviations from the lognormal distribution with additional terms that allow a better fit at the upper distribution tail, which is overestimated according to the lognormal distribution. As a consequence, concentration indexes should be addressed consistently with the lognormal expansion. Through a dynamic panel data approach, we also show that firm growth is persistent and highly dependent on firm characteristics, including size, age, and leverage -these results neglect Gibrat's law for the Colombian case.
Topics: Statistical Distributions
PubMed: 34242353
DOI: 10.1371/journal.pone.0254487 -
Biostatistics (Oxford, England) Jul 2020We propose a novel model for hierarchical time-to-event data, for example, healthcare data in which patients are grouped by their healthcare provider. The most common...
We propose a novel model for hierarchical time-to-event data, for example, healthcare data in which patients are grouped by their healthcare provider. The most common model for this kind of data is the Cox proportional hazard model, with frailties that are common to patients in the same group and given a parametric distribution. We relax the parametric frailty assumption in this class of models by using a non-parametric discrete distribution. This improves the flexibility of the model by allowing very general frailty distributions and enables the data to be clustered into groups of healthcare providers with a similar frailty. A tailored Expectation-Maximization algorithm is proposed for estimating the model parameters, methods of model selection are compared, and the code is assessed in simulation studies. This model is particularly useful for administrative data in which there are a limited number of covariates available to explain the heterogeneity associated with the risk of the event. We apply the model to a clinical administrative database recording times to hospital readmission, and related covariates, for patients previously admitted once to hospital for heart failure, and we explore latent clustering structures among healthcare providers.
Topics: Algorithms; Cluster Analysis; Computer Simulation; Health Personnel; Humans; Patient Admission; Proportional Hazards Models; Statistical Distributions; Statistics, Nonparametric; Time Factors; Time-to-Treatment
PubMed: 30590499
DOI: 10.1093/biostatistics/kxy071 -
Anais Da Academia Brasileira de Ciencias 2023Poisson distribution is a popular discrete model used to describe counting information, from which traditional control charts involving count data, such as the c and u...
Poisson distribution is a popular discrete model used to describe counting information, from which traditional control charts involving count data, such as the c and u charts, have been established in the literature. However, several studies recognize the need for alternative control charts that allow for data overdispersion, which can be encountered in many fields, including ecology, healthcare, industry, and others. The Bell distribution, recently proposed by Castellares et al. (2018), is a particular solution of a multiple Poisson process able to accommodate overdispersed data. It can be used as an alternative to the usual Poisson (which, although not nested in the Bell family, is approached for small values of the Bell distribution) Poisson, negative binomial, and COM-Poisson distributions for modeling count data in several areas. In this paper, we consider the Bell distribution to introduce two new exciting, and useful statistical control charts for counting processes, which are capable of monitoring count data with overdispersion. The performance of the so-called Bell charts, namely Bell-c and Bell-u charts, is evaluated by the average run length in numerical simulation. Some artificial and real data sets are used to illustrate the applicability of the proposed control charts.
Topics: Computer Simulation; Poisson Distribution; Ecology; Models, Statistical
PubMed: 37283327
DOI: 10.1590/0001-3765202320200246 -
Biometrics Dec 2020This article concerns the problem of estimating a continuous distribution in a diseased or nondiseased population when only group-based test results on the disease...
This article concerns the problem of estimating a continuous distribution in a diseased or nondiseased population when only group-based test results on the disease status are available. The problem is challenging in that individual disease statuses are not observed and testing results are often subject to misclassification, with further complication that the misclassification may be differential as the group size and the number of the diseased individuals in the group vary. We propose a method to construct nonparametric estimation of the distribution and obtain its asymptotic properties. The performance of the distribution estimator is evaluated under various design considerations concerning group sizes and classification errors. The method is exemplified with data from the National Health and Nutrition Examination Survey study to estimate the distribution and diagnostic accuracy of C-reactive protein in blood samples in predicting chlamydia incidence.
Topics: Bias; Humans; Models, Statistical; Nutrition Surveys; Research Design; Statistical Distributions
PubMed: 32083733
DOI: 10.1111/biom.13236 -
PloS One 2013In experiments with many statistical tests there is need to balance type I and type II error rates while taking multiplicity into account. In the traditional approach,...
In experiments with many statistical tests there is need to balance type I and type II error rates while taking multiplicity into account. In the traditional approach, the nominal [Formula: see text]-level such as 0.05 is adjusted by the number of tests, [Formula: see text], i.e., as 0.05/[Formula: see text]. Assuming that some proportion of tests represent "true signals", that is, originate from a scenario where the null hypothesis is false, power depends on the number of true signals and the respective distribution of effect sizes. One way to define power is for it to be the probability of making at least one correct rejection at the assumed [Formula: see text]-level. We advocate an alternative way of establishing how "well-powered" a study is. In our approach, useful for studies with multiple tests, the ranking probability [Formula: see text] is controlled, defined as the probability of making at least [Formula: see text] correct rejections while rejecting hypotheses with [Formula: see text] smallest P-values. The two approaches are statistically related. Probability that the smallest P-value is a true signal (i.e., [Formula: see text]) is equal to the power at the level [Formula: see text], to an very good excellent approximation. Ranking probabilities are also related to the false discovery rate and to the Bayesian posterior probability of the null hypothesis. We study properties of our approach when the effect size distribution is replaced for convenience by a single "typical" value taken to be the mean of the underlying distribution. We conclude that its performance is often satisfactory under this simplification; however, substantial imprecision is to be expected when [Formula: see text] is very large and [Formula: see text] is small. Precision is largely restored when three values with the respective abundances are used instead of a single typical effect size value.
Topics: Humans; Probability; Research Design; Statistical Distributions
PubMed: 24376639
DOI: 10.1371/journal.pone.0083079 -
Optics Express May 2008An algorithm is reported for the design of a phase-only diffractive optical element (DOE) that reshapes a beam focused using a high numerical aperture (NA) lens. The...
An algorithm is reported for the design of a phase-only diffractive optical element (DOE) that reshapes a beam focused using a high numerical aperture (NA) lens. The vector diffraction integrals are used to relate the field distributions in the DOE plane and focal plane. The integrals are evaluated using the chirp-z transform and computed iteratively within the Method of Generalized Projections (MGP) to identify a solution that simultaneously satisfies the beam shaping and DOE constraints. The algorithm is applied to design a DOE that transforms a circularly apodized flat-top beam of wavelength lambda to a square irradiance pattern when focused using a 1.4-NA objective. A DOE profile is identified that generates a 50 lambda x 50 lambda square irradiance pattern having 7% uniformity error and 74.5% diffraction efficiency (fraction of focused power). The diffraction efficiency and uniformity decrease as the size of the focused profile is reduced toward the diffraction limited spot size. These observations can be understood as a manifestation of the uncertainty principle.
Topics: Algorithms; Computer Simulation; Equipment Design; Lasers; Light; Models, Statistical; Models, Theoretical; Normal Distribution; Optics and Photonics; Photochemistry; Reproducibility of Results
PubMed: 18545425
DOI: 10.1364/oe.16.007203 -
PloS One 2023Although many data sets are discrete and heavy tailed (for example, number of claims and claim amounts if recorded as rounded values), not many discrete heavy tailed...
Although many data sets are discrete and heavy tailed (for example, number of claims and claim amounts if recorded as rounded values), not many discrete heavy tailed distributions are available in the literature. In this paper, we discuss thirteen known discrete heavy tailed distributions, propose nine new discrete heavy tailed distributions and give expressions for their probability mass functions, cumulative distribution functions, hazard rate functions, reversed hazard rate functions, means, variances, moment generating functions, entropies and quantile functions. Tail behaviour and a measure of asymmetry are used to compare the known and new discrete heavy tailed distributions. The better fits of the discrete heavy tailed distributions over their continuous counterparts as assessed by probability plots are illustrated using three data sets. Finally, a simulated study is performed to assess the finite sample performance of the maximum likelihood estimators used in the data application section.
Topics: Likelihood Functions; Statistical Distributions
PubMed: 37146020
DOI: 10.1371/journal.pone.0285183