Weighted Poisson Distribution Research Articles

Constructing Optimal Quick Switching System with Hurdle Poisson Distribution

Objectives: Optimizing the sum of risks involved in the selection of acceptance sampling plans playing a vital role. This paper uses the Hurdle Poisson distribution to design an optimal quick switching system attribute plan for a given acceptable quality level (AQL) and limiting quality level (LQL) involving a minimum sum of risks. Methods: The sum of producer's and consumer's risks has been met for the specified AQL and LQL. The sum of these risks, as well as the acceptance and rejection numbers have been calculated using the Hurdle Poisson distribution. The operating characteristic function for the quick switching system attribute plan has also been derived using the Hurdle Poisson distribution. Findings: The producer and the consumer both represent the same party in the final inspection. As a result, the sum of these two risks should be minimized. In this paper, the sum of risks for various operating ratios are tabulated using the Hurdle Poisson distribution. These tabulated values are less than the sum of risks calculated using the Weighted Poisson distribution. Novelty: Reducing the sum of risks is the ultimate aim of the work. In this proposed paper, to attain the minimum sum of risks, the authors make an attempt to approach the Quick Switching System Sampling Plan, when the number of defectives in the submitted lots are very less. In other words, the probability of getting defective is very less. This indicates the quality of the lot selected for the inspection to ensure the protection for the consumer. And the plan is also designed in the way that the producer is also not get affected by rejecting a good lot by the consumer. This is the requirement of minimizing the risks. Keywords: Acceptable Quality Level (AQL), Limiting Quality Level (LQL), Minimum Risk Plan, Quick Switching System Sampling Attribute Plan, Operating Characteristic (OC) Function, Hurdle Poisson Distribution

Selection of Quick Switching Double Sampling System with MAPD AND MAAOQ Using Weighted Poisson Distribution

Selection of Quick Switching Double Sampling System with MAPD AND MAAOQ Using Weighted Poisson Distribution

Destructive cure models with proportional hazards lifetimes and associated likelihood inference

In survival analysis, cure models have gained much importance due to rapid advancements in medical sciences. More recently, a subset of cure models, called destructive cure models, have been studied extensively under competing risks scenario wherein initial competing risks undergo a destructive process. In this article, we study destructive cure models by assuming a flexible weighted Poisson distribution (exponentially weighted Poisson, length biased Poisson and negative binomial distributions) for the initial number of competing causes and lifetimes of the susceptible individuals being defined by proportional hazards. The expectation-maximization (EM) algorithm and profile likelihood approach are made use of to estimate the model parameters. An extensive simulation study is carried out under various parameter settings to examine the properties of the models, and accuracy and the robustness of the proposed estimation technique. Effects of model mis-specification on the parameter estimates are also discussed in detail. For further illustration of the proposed methodology, a real-life cutaneous melanoma data set is analyzed.

The Role of Random Vorticity Stretching in Tropical Depression Genesis

Abstract Tropical deep convection plays a key role in the tropical depression stage of tropical cyclogenesis by aggregating vorticity, but no existing theory can depict such a stochastic vorticity aggregation process. A vorticity probability distribution function (PDF) is proposed as a tool to predict the horizontal structure and wind speed of the tropical depression. The reason lies in the tendency for a vortex to adjust to an axisymmetric and monotonic vorticity structure. Assuming deep convection as independent and uniformly distributed vortex tube stretching events in the low–midtroposphere, repetitive vortex tube stretching will make the air column area shrink many times and significantly increase vorticity. A theory of the vorticity PDF is established by modeling the random stretching process as a Markov chain. The PDF turns out to be a weighted Poisson distribution, in good agreement with a randomly forced divergent barotropic model (weak temperature gradient model), and in rough agreement with a cloud-permitting simulation. The result shows that a stronger and sparser deep convective mode tends to produce more high-vorticity air columns, which leads to a more compact major vortex with a higher maximum wind. Based on the vorticity PDF theory, a parameterization of the eddy acceleration effect on the tangential flow is proposed.

A Weighted Poisson Distribution for Underdispersed Count Data

In this paper, we present a new weighted Poisson distribution for modeling underdispersed count data. Weighted Poisson distribution occurs naturally in contexts where the probability that a particular observation of Poisson variable enters the sample gets multiplied by some non-negative weight function. Suppose a realization y of Y a Poisson random variable enters the investigator’s record with probability proportional to w(y): Clearly, the recorded y is not an observation on Y, but on the random variable Yw, which is said to be the weighted version of Y. This distribution a two-parameter is from the exponential family, it includes and generalizes the Poisson distribution by weighting. It is a discrete distribution that is more flexible than other weighted Poisson distributions that have been proposed for modeling underdispersed count data, for example, the extended Poisson distribution (Dimitrov and Kolev, 2000). We present some moment properties and we estimate its parameters. One classical example is considered to compare the fits of this new distribution with the extended Poisson distribution.

On the estimation of destructive cure rate model: A new study with exponentially weighted Poisson competing risks

A new estimation method is proposed founded upon a nonlinear conjugate gradient‐type algorithm having an efficient line search technique for cure rate models with competing risks, which are subject to elimination. An extensive simulation study is carried out to compare the performance of the proposed algorithm with some existing algorithms, including other conjugate gradient‐type algorithms and the expectation maximization algorithm. For this purpose, it is assumed that the initial competing risks follow an exponentially weighted Poisson distribution. In particular, it is shown that that the proposed algorithm produces estimates that are more accurate and efficient (i.e., the bias and root mean square errors are smaller), specifically with respect to the parameters related to the cure rate. Although for the purpose of simulation study an exponentially weighted Poisson competing risks distribution is assumed, the proposed algorithm incorporates a generic framework that can accommodate any competing risks distribution. Finally, a real data application is provided.

A Note on the (Weighted) Bivariate Poisson Distribution

In the recent statistical literature, the univariate Poisson distribution has been generalized by many authors, among them: the univariate weighted Poisson distribution [13], the generalized univariate Poisson distribution [7], the bivariate Poisson distribution according to Holgate [11], the bivariate Poisson distribution according to Lakshminarayana, Pandit and Srinivasa Rao [15], the bivariate Poisson distribution according to Berkhout and Plug [4], the bivariate weighted Poisson distribution according to Elion et al. [8] and the generalized bivariate Poisson distribution according to Famoye [9]. In this paper, We highlight the weighted bivariate Poisson distribution and show that it is the synthesis of all the bivariate Poisson distributions which, under certain conditions, converge in distribution towards the bivariate Poisson distribution according to Berkhout and Plug [4] which can be considered like the standard distribution in N2 as is the univariate Poisson distribution in N.

A new method in designing group chain acceptance sampling plans (GChSP) for generalized exponential distribution

PurposeThis paper introduces group chain acceptance sampling plans (GChSP) for a truncated life test at preassumed time by using the minimum angle method. The proposed method is an approach, where both risks associated with acceptance sampling, namely consumers’ and producer’s risks, are considered. Currently, the GChSP only considers the consumer's risk (CR), which means the current plan only protects the consumer not the producer since it does not take into account the producer's risk (PR) at all.Design/methodology/approachThere are six phases involved when designing the GChSP, which are (1) identifying the design parameters, (2) implementing the operating procedures, (3) deriving the probability of lot acceptance, (4) deriving the probability of zero or one defective, (5) deriving the proportion defective and (6) measuring the performance.FindingsThe findings show that the optimal number of groups obtained satisfies both parties, i.e. consumer and producer, compared to the established GChSP, where the number of group calculated only satisfies the consumer not the producer.Research limitations/implicationsThere are three limitations identified for this paper. The first limitation is the distribution, in which this paper only proposes the GChSP for generalized exponential distribution. It can be extended to different distribution available in the literature. The second limitation is that the paper uses binomial distribution when deriving the probability of lot acceptance. Also, it can be derived by using different distributions such as weighted binomial distribution, Poisson distribution and weighted Poisson distribution. The final limitation is that the paper adopts the mean as a quality parameter. For the quality parameter, researchers have other options such as the median and the percentile.Practical implicationsThe proposed GChSP should provide an alternative for the industrial practitioners and for the inspection activity, as they have more options of the sampling plans before they finally decide to select one.Originality/valueThis is the first paper to propose the minimum angle method for the GChSP, where both risks, CR and PR, are considered. The GChSP has been developed since 2015, but all the researchers only considered the CR in their papers.

A Repetitive Sampling-based Control Chart for Multivariate Weighed Poisson Distribution with Two Different Indexes‎

A Repetitive Sampling-based Control Chart for Multivariate Weighted Poisson Distribution with Two Different Indices‎

The covariance structure of the bivariate weighted Poisson distribution and application to the Aleurodicus data

We meet in the literature the bivariate Poisson distribution put in evidence by Berkhout and Plug. From this distribution, Elion et al. put in evidence the bivariate weighted Poisson distribution like crossing of two univariate weighted Poisson distributions. The structure of the covariance of this bivariate weighted Poisson distribution has been put not again in evidence in the literature. Thus, in this paper, we remedy this hiatus. The overdispersion, underdispersion and the equidispersion will be valued with the help of the Fisher dispersion index for multivariate count distributions introduces by Kokonendji et al. An illustrative example based on the Aleurodicus data is presented. Keywords . Bivariate dispersion index; Moment generating function; Conditional law

The covariance structure of the bivariate weighted Poisson distribution and application to the Aleurodicus data

The covariance structure of the bivariate weighted Poisson distribution and application to the Aleurodicus data

Destructive cure rate models under proportional odds and associated likelihood inference

In this work, we introduce a flexible destructive cure rate model for lifetime data. We assume the number of competing causes of the event of interest to follow the Weighted Poisson distribution (including length-biased Poisson, negative binomial, and exponentially weighted Poisson [EWP]) and the lifetimes of the noncured individuals to follow a proportional odds survival model. The baseline odds distribution is considered to be either Weibull or log-logistic distribution. A damage distribution is introduced due to the fact that some of the competing causes may not remain active following treatment. The statistical inference is developed for this model under right-censored data. The maximum likelihood estimation of the model parameters is then developed with the full usage of expectation–maximization method. Model discrimination between destructive negative binomial cure rate model and destructive EWP cure rate model is carried out by likelihood-based criteria through AIC and BIC. An extensive simulation study is performed to evaluate the performance of all the models and inferential methods developed here. A real data example on cutaneous melanoma is analyzed for illustrative purpose.

A new demerit control chart for monitoring the quality of multivariate Poisson processes

ABSTRACTThis study aims to develop a new demerit control chart suitable for monitoring the quality of manufacturing processes with multiple characteristics subject to multivariate weighted Poisson distribution. Considering the correlation among different quality characteristics and their degrees of influence on the final product, we propose a new statistic for demerit scheme which gives different weights to different quality characteristics. Then, a new demerit control chart for multivariate weighted Poisson distribution (WMP chart) is developed accordingly. Moreover, a simulation study is conducted to evaluate the detecting performances of our proposed WMP chart and multivariate Poisson control chart (MP chart) using the out-of-control average run length (ARL) Finally, a numerical example with a two dimensional telecommunication data set is given to demonstrate the usefulness of our proposed WMP chart. Both the simulation results and numerical example show that the detecting ability of our proposed WMP chart outperforms that of the MP chart when a process shift occurs. Hopefully, the results of this research can provide a better alternative for detecting the mean shifts occurred in a multivariate Poisson processes.

Towards a Generalized Beer-Lambert Law

Anomalous deviations from the Beer-Lambert law have been observed for a long time in a wide range of application. Despite all the attempts, a reliable and accepted model has not been provided so far. In addition, in some cases the attenuation of radiation seems to follow a hyperbolic more than an exponential extinction law. Starting from a probabilistic interpretation of the Beer-Lambert law based on Poissonian distribution of extinction events, in this paper we consider deviations from the classical exponential extinction introducing a weighted version of the classical law. The generalized law is able to account for both sub or super-exponential extinction of radiation, and can be extended to the case of inhomogeneous media. Focusing on this case, we consider a generalized Beer-Lambert law based on an inhomogeneous weighted Poisson distribution involving a Mittag-Leffler function, and show how it can be directly related to hyperbolic decay laws observed in some applications particularly relevant to microbiology and pharmacology.

A weighted Poisson distribution and its application to cure rate models

ABSTRACTThe family of weighted Poisson distributions offers great flexibility in modeling discrete data due to its potential to capture over/under-dispersion by an appropriate selection of the weight function. In this paper, we introduce a flexible weighted Poisson distribution and further study its properties by using it in the context of cure rate modeling under a competing cause scenario. A special case of the new distribution is the COM-Poisson distribution which in turn encompasses the Bernoulli, Poisson, and geometric distributions; hence, many of the well-studied cure rate models may be seen as special cases of the proposed model. We focus on the estimation, through the maximum likelihood method, of the cured proportion and the properties of the failure time of the susceptibles/non cured individuals; a profile likelihood approach is also adopted for estimating the parameters of the weighted Poisson distribution. A Monte Carlo simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data set.

An EM type estimation procedure for the destructive exponentially weighted Poisson regression cure model under generalized gamma lifetime

ABSTRACTIn this paper, we assume the number of competing causes to follow an exponentially weighted Poisson distribution. By assuming the initial number of competing causes can undergo destruction and that the population of interest has a cure fraction, we develop the EM algorithm for the determination of the MLEs of the model parameters of such a general cure model. This model is more flexible than the promotion time cure model and also provides an interesting and realistic interpretation of the biological mechanism of the occurrence of an event of interest. Instead of assuming a particular parametric distribution for the lifetime, we assume the lifetime to belong to the wider class of generalized gamma distribution. This allows us to carry out a model discrimination to select a parsimonious lifetime distribution that provides the best fit to the data. Within the EM framework, a two-way profile likelihood approach is proposed to estimate the shape parameters. An extensive Monte Carlo simulation study is carried out to demonstrate the performance of the proposed estimation method. Model discrimination is carried out by means of the likelihood ratio test and information-based methods. Finally, a data on melanoma is analyzed for illustrative purpose.

Designing optimal double sampling plan using the weighted Poisson distribution

Purpose – The purpose of this paper is to present the optimal double sampling attribute plan using the weighted Poisson distribution. Design/methodology/approach – For the given AQL and LQL, sum of producer’s and consumer’s risks have been attained. Based on the weighted Poisson distribution, the sum of these risks has been optimized. Findings – In the final inspection, the producer and the consumer represent the same party. So, the sum these two risks should be minimized. In this paper, the sum of risks has been tabulated using the weighted Poisson distribution for different operating ratios. These tabulated values are comparatively less than the sum of risks derived using Poisson distribution. Originality/value – The sampling plan presented in this paper is particularly useful for testing the quality of finished products in shop floor situations.

Decomposition d'une loi de poisson ponderee en une combinaison convexe de lois duales

The probability distribution of a set of observation is most often dened as a convex combination of probability laws. To highlight this mixture of the laws, MCMC (Monte Carlo Markov Chain) which is an algorithm that generates a stationary Markov chain is often used; laws being considered as normal laws. In this paper, the observations are positive integer, so it is assumed that the mixture law is a Poisson weighted law and Blending laws are dual. The purpose of this work is to determine the dual laws by simple algebraic properties.Key words: Count Data; Exponetial Family; Weighted Poisson Distribution; Fisher Index; Overdispersion; Underdispersion; Dual Distribution; Convex Combination.

Selection of single sampling attribute plan for given AQL and LQL involving minimum risks using weighted Poisson distribution

PurposeThe purpose of this paper is to present the single sampling attribute plan for given acceptance quality level (AQL) and limiting quality level (LQL) involving minimum sum of risks using weighted Poisson distribution.Design/methodology/approachFor the given AQL and LQL, sum of producer's and consumer's risks have been attained. Based on weighted Poisson distribution, the sum of these risks has been arrived at, along with the acceptance number and the rejection number. Also, the operating characteristic function for the single sampling attribute sampling plan, using weighted Poisson distribution, has been derived.FindingsIn the final inspection, the producer and the consumer represent the same party. So, the sum these two risks should be minimized. In this paper, the sum of risks has been tabulated using weighted Poisson distribution for different operating ratios. These tabulated values are comparatively less than the sum of risks derived using Poisson distribution.Originality/valueThe sampling plan presented in this paper is particularly useful for testing the quality of finished products in shop floor situations.

Selection of modified quick switching systems for given acceptable and limiting quality levels with minimum risks using weighted poisson distribution

Since the first acceptance sampling plans have been developed 80 years ago, a number of selection principles have emerged. The majority of these principles are characterized by the fact that they look upon producer and consumer as two opposing parties. However, in many occasions, e.g., in final inspection, producer and consumer represent the same party and, therefore, the used sampling plan should not make an attempt to discriminate between their interests. In this case the interest is to avoid wrong decisions, i.e., reject product of sufficient quality and accept product of insufficient quality. Thus, the natural objective in these cases is to use overall risk for a wrong decision as optimization criteria. In this paper, a table and procedure are given for finding the Modified Quick Switching Systems QSS − m(n;CN, CT) involving minimum sum of producer's and consumer's risks for specified Acceptable Quality Level and Limiting Quality Level using weighted Poisson distribution.