Mixture of Lindley and Lognormal Distributions: Properties, Estimation, and Application

Finite mixture models provide a flexible tool for handling heterogeneous data. This paper introduces a new mixture model which is the mixture of Lindley and lognormal distributions (MLLND). First, the model is formulated, and some of its statistical properties are studied. Next, maximum likelihood estimation of the parameters of the model is considered, and the performance of the estimators of the parameters of the proposed models is evaluated via simulation. Also, the flexibility of the proposed mixture distribution is demonstrated by showing its superiority to fit a well-known real data set of 128 bladder cancer patients compared to several mixture and nonmixture distributions. The Kolmogorov Smirnov test and some information criteria are used to compare the fitted models to the real dataset. Finally, the results are verified using several graphical methods.

IDENTIFICATION OF THE DISTRIBUTION FUNCTION AND CHARACTERIZATION OF THE NANO– AND MICROPOWDERS SIZES

В работе предлагаются два новых эффективных алгоритма, реализованных коротким программным кодом в MS Excel, предназначенных для идентификации и характеризации размеров нано– и микропорошков частиц в виде обобщенного гамма или логнормального распределений по данным опытных гистограмм. Предлагаемый метод представляет собой новый и достаточно общий подход к решению обратных задач идентификации параметров дифференциальных функций распределения по экспериментальным данным на основе на минимизации функционала, представляющего собой коэффициент детерминации.Алгоритм реализован формулами (менее 10) наиболее распространенного инструментария (электронных таблиц MS Excel без использования макросов), позволяющего исследователям, не обладающими навыками профессиональных программистов, простоту проверки и воспроизведения представленного материала, а также возможность модификации кода для решения более широкого круга задач. Текст статьи и комментарии на рабочих листах скриншотов представляют собой готовые инструкции по решению задач идентификация функций распределения и характеризации размеров нано– и микропорошков. The paper proposes two new efficient algorithms, implemented by a short program code in MS Excel, designed to identify and characterize the sizes of nano- and micropowders of particles in the form of generalized gamma or lognormal distributions according to experimental histograms. The proposed method is a new general approach to solving inverse problems of identifying the parameters of differential distribution functions from experimental data based on minimizing the functional that is the coefficient of determination.The algorithm is implemented with formulas (less than 10) of the most common tools (MS Excel spreadsheets without the use of macros), which allow researchers without the skills of professional programmers to easily check and reproduce the presented material, as well as the ability to modify the code to solve a wider range of problems. The text of the article and comments on the worksheets of screenshots represent ready-made instructions for solving problems of identification of distribution functions and characterization of the sizes of nano- and micropowders.

Life Cycle Inventory (LCI) Stochastic Approach Used for Rare Earth Elements (REEs), Considering Uncertainty

The purpose of the paper is to present the results of the stochastic modelling with uncertaintyperformed with the use of Monte Carlo (MC) simulation with 10,000 cycles and a confidence interval of95 %, as recommended. Analysed REEs were fitted by lognormal distributions by using the Crystal Ball®(CB) spreadsheet-based software after defining the geometric mean value (μg) and the standard deviation(σg), automatically calculated (matches) the lower, as well as, upper boundaries of lognormal distribution.The number of replications of a simulation affects the quality of the results. The principal output reportprovided by CB and presented in this study consists of the graphical representation in the form of thefrequency chart, percentiles summary, and statistics summary. Additional CB options provide a sensitivityanalysis with tornado diagrams. The data that was used for MC simulation of the LCI model includesavailable and published data concerning associated with the REEs. This paper discusses the results andshow that the adopted approach is applicable for any REEs used in the LCI studies under uncertainty. Theresults obtained from this study can be used as the first step in performing a full LCA analysis and helppractitioners as well as decision-makers in the environmental engineering and management.

A Generalized Stochastic Cost–Volume–Profit Model

Cost–volume–profit (CVP) analysis is a widely used decision tool across many business disciplines. The current literature on stochastic applications of the CVP model is limited in that the model is studied under the restrictive forms of the Gaussian and Lognormal distributions. In this paper we introduce the Mellin Transform as a methodology to generalize stochastic modeling of the CVP problem. We demonstrate the versatility of using the Mellin transform to model the CVP problem, and present a generalization of the CVP model when the contribution margin and sales volume are both defined by continuous random distributions.

Generalized Fractional Calculus for Gompertz-Type Models

This paper focuses on the construction of deterministic and stochastic extensions of the Gompertz curve by means of generalized fractional derivatives induced by complete Bernstein functions. Precisely, we first introduce a class of linear stochastic equations involving a generalized fractional integral and we study the properties of its solutions. This is done by proving the existence and uniqueness of Gaussian solutions of such equations via a fixed point argument and then by showing that, under suitable conditions, the expected value of the solution solves a generalized fractional linear equation. Regularity of the absolute p-moment functions is proved by using generalized Grönwall inequalities. Deterministic generalized fractional Gompertz curves are introduced by means of Caputo-type generalized fractional derivatives, possibly with respect to other functions. Their stochastic counterparts are then constructed by using the previously considered integral equations to define a rate process and a generalization of lognormal distributions to ensure that the median of the newly constructed process coincides with the deterministic curve.

Bayesian Analysis of Nonnegative Data Using Dependency-Extended Two-Part Models

This article is motivated by the challenge of analysing an agricultural field experiment with observations that are positive on a continuous scale or zero. Such data can be analysed using two-part models, where the distribution is a mixture of a positive distribution and a Bernoulli distribution. However, traditional two-part models do not include any dependencies between the two parts of the model. Since the probability of zero is anticipated to be high when the expected value of the positive part is low, and the other way around, this article introduces dependency-extended two-part models. In addition, these extensions allow for modelling the median instead of the mean, which has advantages when distributions are skewed. The motivating example is an incomplete block trial comparing ten treatments against weed. Gamma and lognormal distributions were used for the positive response, although any density on the support of real numbers can be accommodated. In a cross-validation study, the proposed new models were compared with each other and with a baseline model without dependencies. Model performance and sensitivity to choice of priors were investigated through simulation. A dependency-extended two-part model for the median of the lognormal distribution performed best with regard to mean square error in prediction. Supplementary materials accompanying this paper appear online.

Bus Queue Time Estimation Model for a Curbside Bus Stop Considering the Blocking Effect

Bus queue time estimation of a curbside bus stop is essential to evaluate the operation, reliability and performance of a bus system. Arriving buses and served buses on upstream berths form an overflow queue considering the no overtaking principle and limited overtaking principle. Therefore, the bus dwelling time at the C-th berth may directly influence the stop capacity. The bus queue delay is modeled as a function of bus dwell time at the C-th berth and the dwell time at every berth using different distributions (normal and lognormal distributions) and data fitting approaches. This study aims to estimate the queue time attributed to dwell time at the C-th berth. The results indicate that the queue time should be evaluated by bus dwell time and joint probability density to quantify the negative influence on queue delay. Therefore, in some cases, where decisions must be taken into consideration for more than two buses, the relationship between dwell time at a downstream berth and dwell time at the C-th berth must be considered.

Conceptually plausible Bayesian inference in interval timing

In a world that is uncertain and noisy, perception makes use of optimization procedures that rely on the statistical properties of previous experiences. A well-known example of this phenomenon is the central tendency effect observed in many psychophysical modalities. For example, in interval timing tasks, previous experiences influence the current percept, pulling behavioural responses towards the mean. In Bayesian observer models, these previous experiences are typically modelled by unimodal statistical distributions, referred to as the prior. Here, we critically assess the validity of the assumptions underlying these models and propose a model that allows for more flexible, yet conceptually more plausible, modelling of empirical distributions. By representing previous experiences as a mixture of lognormal distributions, this model can be parametrized to mimic different unimodal distributions and thus extends previous instantiations of Bayesian observer models. We fit the mixture lognormal model to published interval timing data of healthy young adults and a clinical population of aged mild cognitive impairment patients and age-matched controls, and demonstrate that this model better explains behavioural data and provides new insights into the mechanisms that underlie the behaviour of a memory-affected clinical population.