Tests of fit for the power function lognormal distribution

In this study, tests of fit for the power function lognormal distribution is considered. The probability plot, probability plot correlation coefficient, and goodness-of-fit tests-the Kolmogorov-Smirnov (KS), Cramér-von Mises (CvM), and Anderson-Darling (AD) tests are provided. Tables of critical values are presented by using simulation techniques, and the AD test outperforms KS and CvM tests based on power comparisons. Finally, to illustrate these test procedures, we fit this distribution to the data which represent the survival times of 121 breast cancer patients from one hospital.

Factor analyzing ordinal items requires substantive knowledge of response marginals

In the social sciences, measurement scales often consist of ordinal items and are commonly analyzed using factor analysis. Either data are treated as continuous, or a discretization framework is imposed in order to take the ordinal scale properly into account. Correlational analysis is central in both approaches, and we review recent theory on correlations obtained from ordinal data. To ensure appropriate estimation, the item distributions prior to discretization should be (approximately) known, or the thresholds should be known to be equally spaced. We refer to such knowledge as substantive because it may not be extracted from the data, but must be rooted in expert knowledge about the data-generating process. An illustrative case is presented where absence of substantive knowledge of the item distributions inevitably leads the analyst to conclude that a truly two-dimensional case is perfectly one-dimensional. Additional studies probe the extent to which violation of the standard assumption of underlying normality leads to bias in correlations and factor models. As a remedy, we propose an adjusted polychoric estimator for ordinal factor analysis that takes substantive knowledge into account. Also, we demonstrate how to use the adjusted estimator in sensitivity analysis when the continuous item distributions are known only approximately. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Statistical distribution of micro and macro pores in acrylic bone cement- effect of amount of antibiotic content

Aseptic loosening due to mechanical failure of bone cement is considered to be a leading cause of revision of joint replacement systems. Detailed quantified information on the number, size and distribution pattern of pores can help to obtain a deeper understanding of the bone cement's fatigue behavior. The objective of this study was to provide statistical descriptions for the pore distribution characteristics of laboratory bone cement specimens with different amounts of antibiotic contents. For four groups of bone cement (Palacos) specimens, containing 0.3, 0.6, 1.2 and 2.4 wt/wt% of telavancin antibiotic, seven samples per group were micro computed tomography scanned (38.97 μm voxel size). The images were first preprocessed in Mimics and then analyzed in Dragonfly, with the level of threshold being set such that single-pixel pores become visible. The normalized pore volume data of the specimens were then used to extract the logarithmic histograms of the pore densities for antibiotic groups, as well as their three-parameter Weibull probability density functions. Statistical comparison of the pore distribution data of the antibiotic groups using the Mann-Whitney non-parametric test revealed a significantly larger porosity (p < 0.05) in groups with larger added antibiotic contents (2.4 and 0.6 wt/wt% vs 0.3 wt/wt%). Further analysis revealed that this effect was associated with the significantly larger frequency of micropores of 0.1-0.5 mm diameter (p < 0.05) in groups with larger antibiotic content (2.4 wt/wt% vs and 0.6 and 0.3 wt/wt%), implying that the elution of the added antibiotic produces micropores in this diameter range mainly. Based on this observation and the fatigue test results in the literature, it was suggested that micropore clusters have a detrimental effect on the mechanical properties of bone cement and play a major role in initiating fatigue cracks in highly antibiotic added specimens.

Comparing the variances of several treatments with that of a control treatment: Theory and applications

A common and important problem in medicine, economics and environmental studies is the comparison of the variances of several treatments with that of a control treatment. Among the existing methods, Spurrier's optimal test based on multivariate F distribution has exact type I error rates. However, it requires equal sample sizes among the treatment groups. To extend the application scope, in this paper, we propose a new efficient test for comparing several variances with a control using the marginal inferential model (MIM). Simulation studies show that the MIM test guarantees the exact type I error rate whether the sample size is equal or unequal. Moreover, the power of the MIM test is competitive with that of Spurrier's optimal test. Finally, two real examples are used to demonstrate the application of the proposed method.

Multibridge: an R package to evaluate informed hypotheses in binomial and multinomial models

The multibridge R package allows a Bayesian evaluation of informed hypotheses [Formula: see text] applied to frequency data from an independent binomial or multinomial distribution. multibridge uses bridge sampling to efficiently compute Bayes factors for the following hypotheses concerning the latent category proportions 𝜃: (a) hypotheses that postulate equality constraints (e.g., 𝜃1 = 𝜃2 = 𝜃3); (b) hypotheses that postulate inequality constraints (e.g., 𝜃1 < 𝜃2 < 𝜃3 or 𝜃1 > 𝜃2 > 𝜃3); (c) hypotheses that postulate combinations of inequality constraints and equality constraints (e.g., 𝜃1 < 𝜃2 = 𝜃3); and (d) hypotheses that postulate combinations of (a)-(c) (e.g., 𝜃1 < (𝜃2 = 𝜃3),𝜃4). Any informed hypothesis [Formula: see text] may be compared against the encompassing hypothesis [Formula: see text] that all category proportions vary freely, or against the null hypothesis [Formula: see text] that all category proportions are equal. multibridge facilitates the fast and accurate comparison of large models with many constraints and models for which relatively little posterior mass falls in the restricted parameter space. This paper describes the underlying methodology and illustrates the use of multibridge through fully reproducible examples.

Modeling COVID-19 contact-tracing using the ratio regression capture-recapture approach

Contact-tracing is one of the most effective tools in infectious disease outbreak control. A capture-recapture approach based upon ratio regression is suggested to estimate the completeness of case detection. Ratio regression has been recently developed as flexible tool for count data modeling and has proved to be successful in the capture-recapture setting. The methodology is applied here to Covid-19 contact tracing data from Thailand. A simple weighted straight line approach is used which includes the Poisson and geometric distribution as special cases. For the case study data of contact tracing for Thailand, a completeness of 83% could be found with a 95% confidence interval of 74%-93%.

The unit ratio-extended Weibull family and the dropout rate in Brazilian undergraduate courses

We propose a new family of distributions, so-called the unit ratio-extended Weibull family ([Formula: see text]). It is derived from ratio transformation in an extended Weibull random variable. The use of this transformation is a novelty of the work since it has been less explored than the exponential and has not yet been studied within the extended Weibull class. Moreover, we offer a valuable alternative to model double-bounded variables on the unit interval. Five [Formula: see text] special models are studied in detail, namely the: i) unit ratio-Gompertz; ii) unit ratio-Burr XII; iii) unit ratio-Lomax; v) unit ratio-Rayleigh, and vi) unit ratio-Weibull distributions. We propose a quantile-parameterization for the new family. The maximum likelihood estimators (MLEs) are presented. A Monte Carlo study is performed to evaluate the behavior of the MLEs of unit ratio-Gompertz and unit ratio-Rayleigh distributions. This last model has closed-form and approximately unbiased MLE for small sample sizes. Further, the [Formula: see text] submodels are adjusted to the dropout rate in Brazilian undergraduate courses. We focus on the areas of civil engineering, economics, computer sciences, and control engineering. The applications show that the new family is suitable for modeling educational data and may provide effective alternatives compared to other usual unit models, such as the Beta, Kumaraswamy, and unit gamma distributions. They can also outperform some recent contributions in the unit distribution literature. Thus, the [Formula: see text] family can provide competitive alternatives when those models are unsuitable.

Half logistic-truncated exponential distribution: Characteristics and applications

Gul and Mohsin 2021 developed a new modified form of renowned "Half logistic" distribution introduced by Balakrishnan (1991) and named it half logistic-truncated exponential distribution (HL-TEXPD). Some mathematical characteristics are studied, including hazard function, Pth percentile, moment generating function and Shannon entropy. Simulation study is performed to examine the behaviour of parameter estimates. The proposed model is fitted on three real data sets to check its efficacy. Additionally, TTT (total time on test) plot is drawn to study the failure rate of the three data sets. The results verdict that HL-TEXPD can be efficiently utilized in the field of engineering and medical sciences based on the data sets under study contrary to the classical and baseline models.

Confidence intervals for ratio of means of delta-lognormal distributions based on left-censored data with application to rainfall data in Thailand

Thailand is a country that is prone to both floods and droughts, and these natural disasters have significant impacts on the country's people, economy, and environment. Estimating rainfall is an important part of flood and drought prevention. Rainfall data typically contains both zero and positive observations, and the distribution of rainfall often follows the delta-lognormal distribution. However, it is important to note that rainfall data can be censored, meaning that some values may be missing or truncated. The interval estimator for the ratio of means will be useful when comparing the means of two samples. The purpose of this article was to compare the performance of several approaches for statistically analyzing left-censored data. The performance of the confidence intervals was evaluated using the coverage probability and average length, which were assessed through Monte Carlo simulation. The approaches examined included several variations of the generalized confidence interval, the Bayesian, the parametric bootstrap, and the method of variance estimates recovery approaches. For (ξ1, ξ2) = (0.10,0.10), simulations showed that the Bayesian approach would be a suitable choice for constructing the credible interval for the ratio of means of delta-lognormal distributions based on left-censored data. For (ξ1, ξ2) = (0.10,0.25), the parametric bootstrap approach was a strong alternative for constructing the confidence interval. However, the generalized confidence interval approach can be considered to construct the confidence when the sample sizes are increase. Practical applications demonstrating the use of these techniques on rainfall data showed that the confidence interval based on the generalized confidence interval approach covered the ratio of population means and had the smallest length. The proposed approaches' effectiveness was illustrated using daily rainfall datasets from the provinces of Chiang Rai and Chiang Mai in Thailand.

Tight bounds for the median of a gamma distribution

The median of a standard gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we prove the tightest upper and lower bounds of the form 2-1/k(A + k): an upper bound with A = e-γ (with γ being the Euler-Mascheroni constant) and a lower bound with [Formula: see text]. These bounds are valid over the entire domain of k > 0, staying between 48 and 55 percentile. We derive and prove several other new tight bounds in support of the proofs.

Ubiquitous lognormal distribution of neuron densities in mammalian cerebral cortex

Numbers of neurons and their spatial variation are fundamental organizational features of the brain. Despite the large corpus of cytoarchitectonic data available in the literature, the statistical distributions of neuron densities within and across brain areas remain largely uncharacterized. Here, we show that neuron densities are compatible with a lognormal distribution across cortical areas in several mammalian species, and find that this also holds true within cortical areas. A minimal model of noisy cell division, in combination with distributed proliferation times, can account for the coexistence of lognormal distributions within and across cortical areas. Our findings uncover a new organizational principle of cortical cytoarchitecture: the ubiquitous lognormal distribution of neuron densities, which adds to a long list of lognormal variables in the brain.

A model of behavioural response to risk accurately predicts the statistical distribution of COVID-19 infection and reproduction numbers

One clear aspect of behaviour in the COVID-19 pandemic has been people's focus on, and response to, reported or observed infection numbers in their community. We describe a simple model of infectious disease spread in a pandemic situation where people's behaviour is influenced by the current risk of infection and where this behavioural response acts homeostatically to return infection risk to a certain preferred level. This homeostatic response is active until approximate herd immunity is reached: in this domain the model predicts that the reproduction rate R will be centred around a median of 1, that proportional change in infection numbers will follow the standard Cauchy distribution with location and scale parameters 0 and 1, and that high infection numbers will follow a power-law frequency distribution with exponent 2. To test these predictions we used worldwide COVID-19 data from 1st February 2020 to 30th June 2022 to calculate [Formula: see text] confidence interval estimates across countries for these R, location, scale and exponent parameters. The resulting median R estimate was [Formula: see text] (predicted value 1) the proportional change location estimate was [Formula: see text] (predicted value 0), the proportional change scale estimate was [Formula: see text] (predicted value 1), and the frequency distribution exponent estimate was [Formula: see text] (predicted value 2); in each case the observed estimate agreed with model predictions.

General two-parameter distribution: Statistical properties, estimation, and application on COVID-19

In this paper, we introduced a novel general two-parameter statistical distribution which can be presented as a mix of both exponential and gamma distributions. Some statistical properties of the general model were derived mathematically. Many estimation methods studied the estimation of the proposed model parameters. A new statistical model was presented as a particular case of the general two-parameter model, which is used to study the performance of the different estimation methods with the randomly generated data sets. Finally, the COVID-19 data set was used to show the superiority of the particular case for fitting real-world data sets over other compared well-known models.

Classical and Bayesian estimation for type-I extended-F family with an actuarial application

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.

Estimation of population parameters using sample extremes from nonconstant sample sizes

We examine the accuracy and precision of parameter estimates for both the exponential and normal distributions when using only a collection of sample extremes. That is, we consider a collection of random variables, where each of the random variables is either the minimum or maximum of a sample of nj independent, identically distributed random variables drawn from a normal or exponential distribution with unknown parameters. Previous work derived estimators for the population parameters assuming the nj sample sizes are constant. Since sample sizes are often not constant in applications, we derive new unbiased estimators that take into account the varying sample sizes. We also perform simulations to assess how the previously derived estimators perform when the constant sample size is simply replaced with the average sample size. We explore how varying the mean, standard deviation, and probability distribution of the sample sizes affects the estimation error. Overall, our results demonstrate that using the average sample size in place of the constant sample size still results in reliable estimates for the population parameters, especially when the average sample size is large. Our estimation framework is applied to a biological example involving plant pollination.

Generalized Marshall-Olkin exponentiated exponential distribution: Properties and applications

In this study, we propose a generalized Marshall-Olkin exponentiated exponential distribution as a submodel of the family of generalized Marshall-Olkin distribution. Some statistical properties of the proposed distribution are examined such as moments, the moment-generating function, incomplete moment, and Lorenz and Bonferroni curves. We give five estimators for the unknown parameters of the proposed distribution based on maximum likelihood, least squares, weighted least squares, and the Anderson-Darling and Cramer-von Mises methods of estimation. To investigate the finite sample properties of the estimators, a comprehensive Monte Carlo simulation study is conducted for the models with three sets of randomly selected parameter values. Finally, four different real data applications are presented to demonstrate the usefulness of the proposed distribution in real life.

Estimation of the stress-strength reliability for the inverse Weibull distribution under adaptive type-II progressive hybrid censoring

In this article, we compare the maximum likelihood estimate (MLE) and the maximum product of spacing estimate (MPSE) of a stress-strength reliability model, θ = P(Y < X), under adaptive progressive type-II progressive hybrid censoring, when X and Y are independent random variables taken from the inverse Weibull distribution (IWD) with the same shape parameter and different scale parameters. The performance of both estimators is compared, through a comprehensive computer simulation based on two criteria, namely bias and mean squared error (MSE). To demonstrate the effectiveness of our proposed methods, we used two examples of real-life data based on Breakdown Times of an Insulated Fluid by (Nelson, 2003) and Head and Neck Cancer Data by (Efron, 1988). It is concluded that the MPSE method outperformed the MLE method in terms of bias and MSE values.

System resilience distribution identification and analysis based on performance processes after disruptions

Resilience is a system’s ability to withstand a disruption and return to a normal state quickly. It is a random variable due to the randomness of both the disruption and resilience behavior of a system. The distribution characteristics of resilience are the basis for resilience design and analysis, such as test sample size determination and assessment model selection. In this paper, we propose a systematic resilience distribution identification and analysis (RDIA) method based on a system’s performance processes after disruptions. Typical performance degradation/recovery processes have linear, exponential, and trigonometric functions, and they have three key parameters: the maximum performance degradation, the degradation duration, and the recovery duration. Using the Monte Carlo method, these three key parameters are first sampled according to their corresponding probability density functions. Combining the sample results with the given performance function type, the system performance curves after disruptions can be obtained. Then the sample resilience is computed using a deterministic resilience measure and the resilience distribution can be determined through candidate distribution identification, parameter estimation, and a goodness-of-fit test. Finally, we apply our RDIA method to systems with typical performance processes, and both the orthogonal experiment method and the control variable method are used to investigate the resilience distribution laws. The results show that the resilience of these systems follows the Weibull distribution. An end-to-end communication system is also used to explain how to apply this method with simulation or test data in practice.

A new asymmetric extended family: Properties and estimation methods with actuarial applications

In the present work, a class of distributions, called new extended family of heavy-tailed distributions is introduced. The special sub-models of the introduced family provide unimodal, bimodal, symmetric, and asymmetric density shapes. A special sub-model of the new family, called the new extended heavy-tailed Weibull (NEHTW) distribution, is studied in more detail. The NEHTW parameters have been estimated via eight classical estimation procedures. The performance of these methods have been explored using detailed simulation results which have been ordered, using partial and overall ranks, to determine the best estimation method. Two important risk measures are derived for the NEHTW distribution. To prove the usefulness of the two actuarial measures in financial sciences, a simulation study is conducted. Finally, the flexibility and importance of the NEHTW model are illustrated empirically using two real-life insurance data sets. Based on our study, we observe that the NEHTW distribution may be a good candidate for modeling financial and actuarial sciences data.

Beta distribution misspecification tests with application to Covid-19 mortality rates in the United States

The beta distribution is routinely used to model variables that assume values in the standard unit interval, (0, 1). Several alternative laws have, nonetheless, been proposed in the literature, such as the Kumaraswamy and simplex distributions. A natural and empirically motivated question is: does the beta law provide an adequate representation for a given dataset? We test the null hypothesis that the beta model is correctly specified against the alternative hypothesis that it does not provide an adequate data fit. Our tests are based on the information matrix equality, which only holds when the model is correctly specified. They are thus sensitive to model misspecification. Simulation evidence shows that the tests perform well, especially when coupled with bootstrap resampling. We model state and county Covid-19 mortality rates in the United States. The misspecification tests indicate that the beta law successfully represents Covid-19 death rates when they are computed using either data from prior to the start of the vaccination campaign or data collected when such a campaign was under way. In the latter case, the beta law is only accepted when the negative impact of vaccination reach on death rates is moderate. The beta model is rejected under data heterogeneity, i.e., when mortality rates are computed using information gathered during both time periods.

A reduced distribution of the modified Weibull distribution and its applications to medical and engineering data

In this work, we suggest a reduced distribution with two parameters of the modified Weibull distribution to avoid some estimation difficulties. The hazard rate function of the reduced distribution exhibits decreasing, increasing or bathtub shape. The suggested reduced distribution can be applied to many problems of modelling lifetime data. Some statistical properties of the proposed distribution have been discussed. The maximum likelihood is employed to estimate the model parameters. The Fisher information matrix is derived and then applied to construct confidence intervals for parameters. A simulation is conducted to illustrate the performance of maximum likelihood estimation. Four sets of real data are tested to prove the proposed distribution advantages. According to the statistical criteria, the proposed distribution fits the tested data better than some well-known two-and three-parameter distributions.

A new statistical approach for modeling the bladder cancer and leukemia patients data sets: Case studies in the medical sector

Statistical methods are frequently used in numerous healthcare and other related sectors. One of the possible applications of the statistical methods is to provide the best description of the data sets in the healthcare sector. Keeping in view the applicability of statistical methods in the medical sector, numerous models have been introduced. In this paper, we also introduce a novel statistical method called, a new modified-G family of distributions. Several mathematical properties of the new modified-G family are derived. Based on the new modified-G method, a new updated version of the Weibull model called, a new modified-Weibull distribution is introduced. Furthermore, the estimators of the parameters of the new modified-G distributions are also obtained. Finally, the applicability of the new modified-Weibull distribution is illustrated by analyzing two medical sets. Using certain analytical tools, it is observed that the new modified-Weibull distribution is the best choice to deal with the medical data sets.

Testing for Benford’s Law in very small samples: Simulation study and a new test proposal

Benford’s Law defines a statistical distribution for the first and higher order digits in many datasets. Under very general condition, numbers are expected to naturally conform to the theorized digits pattern. On the other side, any deviation from the Benford distribution could identify an exogenous modification of the expected pattern, due to data manipulation or even fraud. Many statistical tests are available for assessing the Benford conformity of a sample. However, in some practical applications, the limited number of data to analyze may raise questions concerning their reliability. The first aim of this article is then to analyze and compare the behavior of Benford conformity testing procedures applied to very small samples through an extensive Monte Carlo experiment. Simulations will consider a thorough choice of compliance tests and a very heterogeneous selection of alternative distributions. Secondly, we will use the simulation results for defining a new testing procedure, based on the combination of three tests, that guarantees suitable levels of power in each alternative scenario. Finally, a practical application is provided, demonstrating how a sounding testing Benford compliance test for very small samples is important and profitable in anti-fraud investigations.

An empirical data analysis of “price runs” in daily financial indices: Dynamically assessing market geometric distributional behavior

In financial time series there are time periods in which market indices values or assets prices increase or decrease monotonically. We call those events “price runs”, “elementary uninterrupted trends” or just “uninterrupted trends”. In this paper we study the distribution of the duration of uninterrupted trends for the daily indices DJIA, NASDAQ, IPC and Nikkei 225 during the period of time from 10/30/1978 to 08/07/2020 and we compare the simple geometric statistical model with p = 1 2 consistent with the EMH to the empirical data. By a fitting procedure, it is found that the geometric distribution with parameter p = 1 2 provides a good model for uninterrupted trends of short and medium duration for the more mature markets; however, longest duration events still need to be statistically characterized. Estimated values of the parameter p were also obtained and confirmed by calculating the mean value of p fluctuations from empirical data. Additionally, the observed trend duration distributions for the different studied markets are compared over time by means of the Anderson-Darling (AD) test, to the expected geometric distribution with parameter p = 1 2 and to a geometric distribution with a free parameter p, making possible to assess and compare different market geometric behavior for different dates as well as to measure the fraction of time runs duration from studied markets are consistent with the geometric distribution with p = 1 2 and in parametric free way.

Confidence interval estimation of the common mean of several gamma populations

Gamma distributions are widely used in applied fields due to its flexibility of accommodating right-skewed data. Although inference methods for a single gamma mean have been well studied, research on the common mean of several gamma populations are sparse. This paper addresses the problem of confidence interval estimation of the common mean of several gamma populations using the concept of generalized inference and the method of variance estimates recovery (MOVER). Simulation studies demonstrate that several proposed approaches can provide confidence intervals with satisfying coverage probabilities even at small sample sizes. The proposed methods are illustrated using two examples.

Validity of Lognormal Distribution in Analyzing Laboratory Test Values

We compared the distribution of laboratory test values with several parametric statistical distributions to show that a lognormal distribution can represent the distribution of laboratory test values. Then, we estimated the distributions of laboratory test values of four datasets including only three published values: two endpoints of reference interval (RI) and one median.

Exponentiated Odd Lomax Exponential distribution with application to COVID-19 death cases of Nepal

This study suggested a new four-parameter Exponentiated Odd Lomax Exponential (EOLE) distribution by compounding an exponentiated odd function with Lomax distribution as a generator. The proposed model is unimodal and positively skewed whereas the hazard rate function is monotonically increasing and inverted bathtubs. Some important properties of the new distribution are derived such as quintile function and median; asymptotic properties and mode; moments; mean residual life, mean path time; mean deviation; order statistics; and Bonferroni & Lorenz curve. The value of the parameters is obtained from the maximum likelihood estimation, least-square estimation, and Cramér-Von-Mises methods. Here, a simulation study and two real data sets, “the number of deaths per day due to COVID-19 of the first wave in Nepal" and ‘‘failure stresses (In Gpa) of single carbon fibers of lengths 50 mm", have been applied to validate the different theoretical findings. The finding of an order of COVID-19 deaths in 153 days in Nepal obey the proposed distribution, it has a significantly positive relationship between the predictive test positive rate and the predictive number of deaths per day. Therefore, the intended model is an alternative model for survival data and lifetime data analysis.

A new extended gumbel distribution: Properties and application

A robust generalisation of the Gumbel distribution is proposed in this article. This family of distributions is based on the T-X paradigm. From a list of special distributions that have evolved as a result of this family, three separate models are also mentioned in this article. A linear combination of generalised exponential distributions can be used to characterise the density of a new family, which is critical in assessing some of the family’s properties. The statistical features of this family are determined, including exact formulations for the quantile function, ordinary and incomplete moments, generating function, and order statistics. The model parameters are estimated using the maximum likelihood method. Further, one of the unique models has been systematically studied. Along with conventional skewness measures, MacGillivray skewness is also used to quantify the skewness measure. The new probability distribution also enables us to determine certain critical risk indicators, both numerically and graphically. We use a simulated assessment of the suggested distribution, as well as apply three real-world data sets in modelling the proposed model, in order to ensure its authenticity and superiority.

Contagious statistical distributions: k-connections and applications in infectious disease environments

Contagious statistical distributions are a valuable resource for managing contagion by means of k–connected chains of distributions. Binomial, hypergeometric, Pólya, uniform distributions with the same values for all parameters except sample size n are known to be strongly associated. This paper describes how the relationship can be obtained via factorial moments, simplifying the process by including novel elements. We describe the properties of these distributions and provide examples of their real–world application, and then define a chain of k–connected distributions, which generalises the relationship among samples of any size for a given population and the Pólya urn model.

Bayesian estimation for the mean of delta-gamma distributions with application to rainfall data in Thailand

Precipitation and flood forecasting are difficult due to rainfall variability. The mean of a delta-gamma distribution can be used to analyze rainfall data for predicting future rainfall, thereby reducing the risks of future disasters due to excessive or too little rainfall. In this study, we construct credible and highest posterior density (HPD) intervals for the mean and the difference between the means of delta-gamma distributions by using Bayesian methods based on Jeffrey's rule and uniform priors along with a confidence interval based on fiducial quantities. The results of a simulation study indicate that the Bayesian HPD interval based on Jeffrey's rule prior performed well in terms of coverage probability and provided the shortest expected length. Rainfall data from Chiang Mai province, Thailand, are also used to illustrate the efficacies of the proposed methods.

Bayesian interval estimations for the mean of delta-three parameter lognormal distribution with application to heavy rainfall data

Flash flooding is caused by heavy rainfall that frequently occurs during a tropical storm, and the Thai population has been subjected to this problem for a long time. The key to solving this problem by planning and taking action to protect the population and infrastructure is the motivation behind this study. The average weekly rainfall in northern Thailand during Tropical Storm Wipha are approximated using interval estimations for the mean of a delta-three parameter lognormal distribution. Our proposed methods are Bayesian confidence intervals-based noninformative (NI) priors (equal-tailed and highest posterior density (HPD) intervals based on NI1 and NI2 priors). Our numerical evaluation shows that the HPD-NI1 prior was closer to the nominal confidence level and possessed the narrowest expected length when the variance was small-to-medium for a large threshold. The efficacy of the methods was illustrated by applying them to weekly natural rainfall data in northern Thailand to examine their abilities to indicate flooding occurrence.

Confidence intervals for rainfall dispersions using the ratio of two coefficients of variation of lognormal distributions with excess zeros

Rainfall fluctuation is directly affected by the Earth’s climate change. It can be described using the coefficient of variation (CV). Similarly, the ratio of CVs can be used to compare the rainfall variation between two regions. The ratio of CVs has been widely used in statistical inference in a number of applications. Meanwhile, the confidence interval constructed with this statistic is also of interest. In this paper, confidence intervals for the ratio of two independent CVs of lognormal distributions with excess zeros using the fiducial generalized confidence interval (FGCI), Bayesian methods based on the left-invariant Jeffreys, Jeffreys rule, and uniform priors, and the Wald and Fieller log-likelihood methods are proposed. The results of a simulation study reveal that the highest posterior density (HPD) Bayesian using the Jeffreys rule prior method performed the best in terms of the coverage probability and the average length for almost all cases of small sample size and a large sample size together with a large variance and a small proportion of non-zero values. The performance of the statistic is demonstrated on two rainfall datasets from the central and southern regions in Thailand.

On the estimation of the fill rate for the continuous (s, S) inventory system for the lost sales context

In the continuous review reorder point, base-stock (s, S) policy, the replenishment order is launched when the inventory position reaches the reorder point, s. It is commonly assumed that the inventory position is exactly equal to the reorder point at the moment the order is launched, when actually it could be lower at that moment. This implies neglecting the possible undershoots at the reorder point, which has a direct impact on the calculation of the expected shortages per replenishment cycle. This article presents a method for an exact calculation of the fill rate (fraction of demand that is immediately satisfied from shelf) which takes explicit account of the existence of undershoots and is applicable to any discrete demand distribution function in a context of lost sales. This method is based on the determination of the stock probability vector at the moment the replenishment order is launched. Furthermore, neglecting the undershoots is shown to lead to an overestimation of the fill rate, particularly when we move farther away from the unitary demand assumption. From a practical point of view, this behaviour involves underestimating the base-stock level, S, when a target fill rate is set for its determination. The method proposed in this paper overcomes these shortcomings.

A new generalized family of distributions based on combining Marshal-Olkin transformation with T-X family

Data analysis in real life often relies mainly on statistical probability distributions. However, data arising from different fields such as environmental, financial, biomedical sciences and other areas may not fit the classical distributions. Therefore, the need arises for developing new distributions that would capture high degree of skewness and kurtosis and enhance the goodness-of-fit in empirical distribution. In this paper, we introduce a novel family of distributions which can extend some popular classes of distributions to include different new versions of the baseline distributions. The proposed family of distributions is referred as the Marshall-Olkin Weibull generated family. The proposed family of distributions is a combination of Marshall-Olkin transformation and the Weibull generated family. Two special members of the proposed family are investigated. A variety of shapes for the densities and hazard rate are presented of the considered sub-models. Some of the main mathematical properties of this family are derived. The estimation for the parameters is obtained via the maximum likelihood method. Moreover, the performance of the estimators for the considered members is examined through simulation studies in terms of bias and root mean square error. Besides, based on the new generated family, the log Marshall-Olkin Weibull-Weibull regression model for censored data is proposed. Finally, COVID-19 data and three lifetime data sets are used to demonstrate the importance of the newly proposed family. Through such an applications, it is shown that this family of distributions provides a better fit when compared with other competitive distributions.

Bayesian computation for the common coefficient of variation of delta-lognormal distributions with application to common rainfall dispersion in Thailand

Rainfall fluctuation makes precipitation and flood prediction difficult. The coefficient of variation can be used to measure rainfall dispersion to produce information for predicting future rainfall, thereby mitigating future disasters. Rainfall data usually consist of positive and true zero values that correspond to a delta-lognormal distribution. Therefore, the coefficient of variation of delta-lognormal distribution is appropriate to measure the rainfall dispersion more than lognormal distribution. In particular, the measurement of the dispersion of precipitation from several areas can be determined by measuring the common coefficient of variation in the rainfall from those areas together. Herein, we compose confidence intervals for the common coefficient of variation of delta-lognormal distributions by employing the fiducial generalized confidence interval, equal-tailed Bayesian credible intervals incorporating the independent Jeffreys or uniform priors, and the method of variance estimates recovery. A combination of the coverage probabilities and expected lengths of the proposed methods obtained via a Monte Carlo simulation study were used to compare their performances. The results show that the equal-tailed Bayesian based on the independent Jeffreys prior was suitable. In addition, it can be used the equal-tailed Bayesian based on the uniform prior as an alternative. The efficacies of the proposed confidence intervals are demonstrated via applying them to analyze daily rainfall datasets from Nan, Thailand.

Decision tree accelerated CTU partition algorithm for intra prediction in versatile video coding

Versatile video coding (VVC) achieves enormous improvement over the advanced high efficiency video coding (HEVC) standard due to the adoption of the quadtree with nested multi-type tree (QTMT) partition structure and other coding tools. However, the computational complexity increases dramatically as well. To tackle this problem, we propose a decision tree accelerated coding tree units (CTU) partition algorithm for intra prediction in VVC. Firstly, specially designated image features are extracted to characterize the coding unit (CU) complexity. Then, the trained decision tree is employed to predict the partition results. Finally, based on our newly designed intra prediction framework, the partition process is early terminated or redundant partition modes are screened out. The experimental results show that the proposed algorithm could achieve around 52% encoding time reduction for various test video sequences on average with only 1.75% Bjontegaard delta bit rate increase compared with the reference test model VTM9.0 of VVC.

Sample size calculation for two-arm trials with time-to-event endpoint for nonproportional hazards using the concept of Relative Time when inference is built on comparing Weibull distributions

Sample size calculations for two-arm clinical trials with a time-to-event endpoint have traditionally used the assumption of proportional hazards (PH) or the assumption of exponentially distributed survival times. Available software provides methods for sample size calculation using a nonparametric logrank test, Schoenfeld's formula for Cox PH model, or parametric calculations specific to the exponential distribution. In cases where the PH assumption is not valid, the first-choice method is to compute sample size assuming a piecewise linear survival curve (Lakatos approach) for both the control and treatment arms with judiciously chosen cut-points. Recent advances in literature have used the assumption of Weibull distributed times for single-arm trials, and, newer methods have emerged that allow sample size calculations for two-arm trials using the assumption of proportional time (PT) while considering non-PH. These methods, however, always assume an instantaneous effect of treatment relative to control requiring that the effect size be defined by a single number whose magnitude is preserved throughout the trial duration. Here, we consider the scenarios where the hypothesized benefit of treatment relative to control may not be constant giving rise to the notion of Relative Time (RT). By assuming that survival times for control and treatment arm come from two different Weibull distributions with different location and shape parameters, we develop the methodology for sample size calculation for specific cases of both non-PH and non-PT. Simulations are conducted to assess the operation characteristics of the proposed method and a practical example is discussed.

Firm size and economic concentration: An analysis from a lognormal expansion

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.

Threshold effect of government subsidy, corporate social responsibility and brand value using the data of China's top 500 most valuable brands

An increasing number of firms have begun to attach importance to corporate social responsibility (CSR) to obtain sustainable strategic advantages in the competitive market. On the basis of nonlinear perspective, panel data of A-share listed companies in the ranking list of China’s Top 500 Most Valuable Brands in 2012–2018 and Hansen panel threshold regression technology were adopted. With government subsidy and CSR being threshold variables, the internal mechanism about the influence of government subsidy and CSR on brand value was explored. Results show that the following. (1) CSR has a significantly inverted U-type threshold effect on brand value. (2) Government subsidy facilitates CSR with diminishing marginal utility. (3) When a difference exists in the strength of government subsidy, the influence of CSR on brand value presents a significant N-type law. Furthermore, threshold regression method was used to innovatively explore the complex nonlinear relationship among government subsidy, CSR, and brand value. This relationship has a significantly practical significance for listed firms for weighing the business decisions regarding the input of CSR and brand value, as well as subsidy policies for enterprises by the government.

Utilization of a Mixture Cure Rate Model based on the Generalized Modified Weibull Distribution for the Analysis of Leukemia Patients

OBJECTIVE

Cure rate models are survival models, commonly applied to model survival data with a cured fraction. In the existence of a cure rate, if the distribution of survival times for susceptible patients is specified, researchers usually prefer cure models to parametric models. Different distributions can be assumed for the survival times, for instance, generalized modified Weibull (GMW), exponentiated Weibull (EW), and log-beta Weibull. The purpose of this study is to select the best distribution for uncured patients' survival times by comparing the mixture cure models based on the GMW distribution and its particular cases.

MATERIALS AND METHODS

A data set of 91 patients with high-risk acute lymphoblastic leukemia (ALL) followed for five years from 1982 to 1987 was chosen for fitting the mixture cure model. We used the maximum likelihood estimation technique via R software 3.6.2 to obtain the estimates for parameters of the proposed model in the existence of cure rate, censored data, and covariates. For the best model choice, the Akaike information criterion (AIC) was implemented.

RESULTS

After comparing different parametric models fitted to the data, including or excluding cure fraction, without covariates, the smallest AIC values were obtained by the EW and the GMW distributions, (953.31/969.35) and (955.84/975.99), respectively. Besides, assuming a mixture cure model based on GMW with covariates, an estimated ratio between cure fractions for allogeneic and autologous bone marrow transplant groups (and its 95% confidence intervals) were 1.42972 (95% CI: 1.18614 - 1.72955).

CONCLUSION

The results of this study reveal that the EW and the GMW distributions are the best choices for the survival times of Leukemia patients.
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Acceptance sampling plans under two-parameter Quasi Shanker distribution assuring mean life with an application to manufacturing data

The acceptance sampling plan (ASP) is a statistical tool used in industry for quality control to determine the quality of products by selecting a specified number for testing in order to accept or reject the lot. The main objective is to develop a new ASP based on truncated life tests assuming that the lifetime follows the two parameters Quasi Shanker distribution, since this distribution showed its superiority in providing a better model for some applications than the exponential distribution. The ASP steps are carried out to find the minimum sample sizes needed to assert the certain life mean that are calculated under a given customer's risk. The operating characteristic values of the sampling plan and the producer risk values are obtained. The efficiency of the suggested plans is analyzed based on real data that is fitted to the Quasi Shanker distribution. For various values of the Quasi Shanker distribution parameters, numerical examples are presented for illustrative purposes. The results indicate that the suggested ASP provides smaller sample sizes than other competitors considered in this study. The suggested ASP has been found to provide a substantial sampling economy in terms of reducing the sample. Hence, it is recommended that the ASP can be used in industry and for future research works as double and group ASP.

New generalized-X family: Modeling the reliability engineering applications

As is already known, statistical models are very important for modeling data in applied fields, particularly in engineering, medicine, and many other disciplines. In this paper, we propose a new family to introduce new distributions suitable for modeling reliability engineering data. We called our proposed family a new generalized-X family of distributions. For the practical illustration, we introduced a new special sub-model, called the new generalized-Weibull distribution, to describe the new family's significance. For the proposed family, we introduced some mathematical reliability properties. The maximum likelihood estimators for the parameters of the new generalized-X distributions are derived. For assessing the performance of these estimators, a comprehensive Monte Carlo simulation study is carried out. To assess the efficiency of the proposed model, the new generalized-Weibull model is applied to the coating machine failure time data. Finally, Bayesian analysis and performance of Gibbs sampling for the coating machine failure time data are also carried out. Furthermore, the measures such as Gelman-Rubin, Geweke and Raftery-Lewis are used to track algorithm convergence.

On the Burr XII-moment exponential distribution

In this study, a new flexible lifetime model called Burr XII moment exponential (BXII-ME) distribution is introduced. We derive some of its mathematical properties including the ordinary moments, conditional moments, reliability measures and characterizations. We employ different estimation methods such as the maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling methods for estimating the model parameters. We perform simulation studies on the basis of the graphical results to see the performance of the above estimators of the BXII-ME distribution. We verify the potentiality of the BXII-ME model via monthly actual taxes revenue and fatigue life applications.

Control of tumor growth distributions through kinetic methods

The mathematical modeling of tumor growth has a long history, and has been mathematically formulated in several different ways. Here we tackle the problem in the case of a continuous distribution using mathematical tools from statistical physics. To this extent, we introduce a novel kinetic model of growth which highlights the role of microscopic transitions in determining a variety of equilibrium distributions. At variance with other approaches, the mesoscopic description in terms of elementary interactions allows to design precise microscopic feedback control therapies, able to influence the natural tumor growth and to mitigate the risk factors involved in big sized tumors. We further show that under a suitable scaling both the free and controlled growth models correspond to Fokker-Planck type equations for the growth distribution with variable coefficients of diffusion and drift, whose steady solutions in the free case are given by a class of generalized Gamma densities which can be characterized by fat tails. In this scaling the feedback control produces an explicit modification of the drift operator, which is shown to strongly modify the emerging distribution for the tumor size. In particular, the size distributions in presence of therapies manifest slim tails in all growth models, which corresponds to a marked mitigation of the risk factors. Numerical results confirming the theoretical analysis are also presented.

Nonparametric estimation of distributions and diagnostic accuracy based on group-tested results with differential misclassification

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.

Benchmarking of public health surveillance of COVID-19 in Colombia: First semester

Introduction: Public health surveillance together with good sanitary decisions is essential for the proper management of the SARS-CoV-2 pandemic. Objective: To compare the performance of Colombian departments based on the quality of the data and to build the national ranking. Materials and methods: We analyzed the accumulated cases published between March 6 and September 1, 2020, by the Instituto Nacional de Salud. To achieve comparability, the analyses considered the day the first case was diagnosed as the first analysis date for each department. The fulfillment of Benford’s law was assessed with p-values in the log-likelihood ratio or chi-square tests. The analysis was completed with the lethality observed in each department and then the performance ranking was established. Results: Bogotá and Valle del Cauca had optimal public health surveillance performance all along. The data suggest that Antioquia, Nariño, and Tolima had good containment and adequate public health surveillance after the economic opening beginning on June 1, 2020. Conclusion: We obtained the ranking of the departments regarding the quality of public health surveillance data. The best five departments can be case studies to identify the elements associated with good performance.

Compound distributions for financial returns

In this paper, we propose six Student’s t based compound distributions where the scale parameter is randomized using functional forms of the half normal, Fréchet, Lomax, Burr III, inverse gamma and generalized gamma distributions. For each of the proposed distribution, we give expressions for the probability density function, cumulative distribution function, moments and characteristic function. GARCH models with innovations taken to follow the compound distributions are fitted to the data using the method of maximum likelihood. For the sample data considered, we see that all but two of the proposed distributions perform better than two popular distributions. Finally, we perform a simulation study to examine the accuracy of the best performing model.

A novel family of lifetime distribution with applications to real and simulated data

The paper investigates a new scheme for generating lifetime probability distributions. The scheme is called Exponential- H family of distribution. The paper presents an application of this family by using the Weibull distribution, the new distribution is then called New Flexible Exponential distribution or in short NFE. Various statistical properties are derived, such as quantile function, order statistics, moments, etc. Two real-life data sets and a simulation study have been performed so that to assure the flexibility of the proposed model. It has been declared that the proposed distribution offers nice results than Exponential, Weibull Exponential, and Exponentiated Exponential distribution.

Development of a Synthetic Population Model for Assessing Excess Risk for Cardiovascular Disease Death

Importance

Evaluating the association of social determinants of health with chronic diseases at the population level requires access to individual-level factors associated with disease, which are rarely available for large populations. Synthetic populations are a possible alternative for this purpose.

Objective

To construct and validate a synthetic population that statistically mimics the characteristics and spatial disease distribution of a real population, using real and synthetic data.

Design, Setting, and Participants

This population-based decision analytical model used data for Allegheny County, Pennsylvania, collected from January 2015 to December 2016, to build a semisynthetic population based on the synthetic population used by the modeling and simulation platform FRED (A Framework for Reconstructing Epidemiological Dynamics). Disease status was assigned to this population using health insurer claims data from the 3 major insurance providers in the county or from the National Health and Nutrition Examination Survey. Biological, social, and other variables were also obtained from the National Health Interview Survey, Allegheny County, and public databases. Data analysis was performed from November 2016 to February 2020.

Exposures

Risk of cardiovascular disease (CVD) death.

Main Outcomes and Measures

Difference between expected and observed CVD death risk. A validated risk equation was used to estimate CVD death risk.

Results

The synthetic population comprised 1 188 112 individuals with demographic characteristics similar to those of the 2010 census population in the same county. In the synthetic population, the mean (SD) age was 40.6 (23.3) years, and 622 997 were female individuals (52.4%). Mean (SD) observed 4-year rate of excess CVD death risk at the census tract level was -40 (523) per 100 000 persons. The correlation of social determinant data with difference between expected and observed CVD death risk indicated that income- and education-based social determinants were associated with risk. Estimating improved social determinants of health and biological factors associated with disease did not entirely remove the excess in CVD death rates. That is, a 20% improvement in the most significant determinants still resulted in 105 census tracts with excess CVD death risk, which represented 24% of the county population.

Conclusions and Relevance

The results of this study suggest that creating a geographically explicit synthetic population from real and synthetic data is feasible and that synthetic populations are useful for modeling disease in large populations and for estimating the outcome of interventions.