A New Three-Parameter Inverse Weibull Distribution with Medical and Engineering Applications

The objective of this article is to provide a novel extension of the conventional inverse Weibull distribution that adds an extra shape parameter to increase its flexibility.This addition is beneficial in a variety of fields,including reliability,economics,engineering,biomedical science,biological research,environmental studies,and finance.For modeling real data,several expanded classes of distributions have been established.The modified alpha power transformed approach is used to implement the new model.The datamatches the new inverseWeibull distribution better than the inverse Weibull distribution and several other competing models.It appears to be a distribution designed to support decreasing or unimodal shaped distributions based on its parameters.Precise expressions for quantiles,moments,incomplete moments,moment generating function,characteristic generating function,and entropy expression are among the determined attributes of the new distribution.The point and interval estimates are studied using the maximum likelihood method.Simulation research is conducted to illustrate the correctness of the theoretical results.Three applications to medical and engineering data are utilized to illustrate the model’s flexibility.

A Novel Modified Alpha Power Transformed Weibull Distribution and Its Engineering Applications

This paper suggests a new modified version of the traditional Weibull distribution by adding a new shape parameter utilising the modified alpha power transformed technique.We refer to the new model as modified alpha power transformed Weibull distribution.The attractiveness and significance of the new distribution lie in its power to model monotone and non-monotone failure rate functions,which are quite familiar in environmental investigations.Its hazard rate function can be decreasing,increasing,bathtub and upside-down then bathtub shaped.Diverse structural properties of the proposed model are acquired including quantile function,moments,entropies,order statistics,residual life and reversed failure rate function.The parameters of the distribution were estimated using the maximum likelihood function.The maximum likelihood method is employed to estimate the model parameters and the approximate confidence intervals are also computed.Via a simulation study,the performance of the point and interval estimates are compared using different criteria.Employing real lifetime data sets,we verify that the offered model furnishes a better fit than some other lifetime models including Weibull,gamma and alpha powerWeibull models.

On a New Version of Weibull Model:Statistical Properties,Parameter Estimation and Applications

In this paper,we introduce a new four-parameter version of the traditional Weibull distribution.It is able to provide seven shapes of hazard rate,including constant,decreasing,increasing,unimodal,bathtub,unimodal then bathtub,and bathtub then unimodal shapes.Some basic characteristics of the proposedmodel are studied,including moments,entropies,mean deviations and order statistics,and its parameters are estimated using the maximum likelihood approach.Based on the asymptotic properties of the estimators,the approximate confidence intervals are also taken into consideration in addition to the point estimators.We examine the effectiveness of the maximum likelihood estimators of the model’s parameters through simulation research.Based on the simulation findings,it can be concluded that the provided estimators are consistent and that asymptotic normality is a good method to get the interval estimates.Three actual data sets for COVID-19,engineering and blood cancer are used to empirically demonstrate the new distribution’s usefulness inmodeling real-world data.The analysis demonstrates the proposed distribution’s ability in modeling many forms of data as opposed to some of its well-known sub-models,such as alpha powerWeibull distribution.

Modified DS np Chart Using Generalized Multiple Dependent State Sampling under Time Truncated Life Test

This study presents the design of a modified attributed control chart based on a double sampling(DS)np chart applied in combination with generalized multiple dependent state(GMDS)sampling to monitor the mean life of the product based on the time truncated life test employing theWeibull distribution.The control chart developed supports the examination of the mean lifespan variation for a particular product in the process of manufacturing.Three control limit levels are used:the warning control limit,inner control limit,and outer control limit.Together,they enhance the capability for variation detection.A genetic algorithm can be used for optimization during the in-control process,whereby the optimal parameters can be established for the proposed control chart.The control chart performance is assessed using the average run length,while the influence of the model parameters upon the control chart solution is assessed via sensitivity analysis based on an orthogonal experimental design withmultiple linear regression.A comparative study was conducted based on the out-of-control average run length,in which the developed control chart offered greater sensitivity in the detection of process shifts while making use of smaller samples on average than is the case for existing control charts.Finally,to exhibit the utility of the developed control chart,this paper presents its application using simulated data with parameters drawn from the real set of data.

Assessment of Upper Guinea’s Wind Energy Potential with a View to Identifying Exploitable Sites

As part of the national strategy to further develop the wind energy sector,the eight prefectures of Upper Guinea have been selected.Using meteorological data recorded over thirty years(1991-2021)at a height of 20 m,we assessed wind resources in terms of characteristic speeds,power and available energy.To this end,the Weibull distribution method was used and the following values were obtained:3.66 m/s for the average speed;1,102.83 W/m^(2)for the available power and 8,747.06 kWh/m^(2)/year for the annual available energy.

Estimating Weibull Parameters Using Least Squares and Multilayer Perceptron vs. Bayes Estimation

The Weibull distribution is regarded as among the finest in the family of failure distributions.One of the most commonly used parameters of the Weibull distribution(WD)is the ordinary least squares(OLS)technique,which is useful in reliability and lifetime modeling.In this study,we propose an approach based on the ordinary least squares and the multilayer perceptron(MLP)neural network called the OLSMLP that is based on the resilience of the OLS method.The MLP solves the problem of heteroscedasticity that distorts the estimation of the parameters of the WD due to the presence of outliers,and eases the difficulty of determining weights in case of the weighted least square(WLS).Another method is proposed by incorporating a weight into the general entropy(GE)loss function to estimate the parameters of the WD to obtain a modified loss function(WGE).Furthermore,a Monte Carlo simulation is performed to examine the performance of the proposed OLSMLP method in comparison with approximate Bayesian estimation(BLWGE)by using a weighted GE loss function.The results of the simulation showed that the two proposed methods produced good estimates even for small sample sizes.In addition,the techniques proposed here are typically the preferred options when estimating parameters compared with other available methods,in terms of the mean squared error and requirements related to time.

A Statistical Analysis of Wind Speed and Power Density Based on Weibull and Rayleigh Models of Jumla, Nepal

In the present study, wind speed data of Jumla, Nepal have been statistically analyzed. For this purpose, the daily averaged wind speed data for 10 year period (2004-2014: 2012 excluded) provided by Department of Hydrology and Meteorology (DHM) was analyzed to estimate wind power density. Wind speed as high as 18 m/s was recorded at height of 10 m. Annual mean wind speed was ascertained to be decreasing from 7.35 m/s in 2004 to 5.13 m/s in 2014 as a consequence of Global Climate Change. This is a subject of concern looking at government’s plan to harness wind energy. Monthly wind speed plot shows that the fastest wind speed is generally in month of June (Monsoon Season) and slowest in December/January (Winter Season). Results presented Weibull distribution to fit measured probability distribution better than the Rayleigh distribution for whole years in High altitude region of Nepal. Average value of wind power density based on mean and root mean cube seed approaches were 131.31 W/m2/year and 184.93 W/m2/year respectively indicating that Jumla stands in class III. Weibull distribution shows a good approximation for estimation of power density with maximum error of 3.68% when root mean cube speed is taken as reference.

Interval Estimation in a Two Parameter Weibull Distribution Based on Type-2 Censored Data

In this paper, we consider the construction of the approximate profile- likelihood confidence intervals for parameters of the 2-parameter Weibull distribution based on small type-2 censored samples. In previous research works, the traditional Wald method has been used to construct approximate confidence intervals for the 2-parameter Weibull distribution under type-2 censoring scheme. However, the Wald technique is based on normality assumption and thus may not produce accurate interval estimates for small samples. The profile-likelihood and Wald confidence intervals are constructed for the shape and scale parameters of the 2-parameter Weibull distribution based on simulated and real type-2 censored data, and are hence compared using confidence length and coverage probability.

Application of the Weibull Distribution to Estimate the Volume of Water Pumping by a Windmill

Given the steady increase in use of alternative energy sources to supply part of the energy demand of the world, and considering that Mexico has enough wind resources to produce all the electricity required by the country, in this work, it is carried out the development of a system of wind resource assessment to determine the volume of water that a windmill pumping system can provide in a specific area. To this end, it has developed a computer program for wind resource assessment by measurements of wind speed, direction, temperature, barometric pressure and relative humidity. Measuring equipments were mounted in a wind measurement tower of 10 m high, within the premises of the University of Quintana Roo which is located along the coast. Until this day, there are records over a period of five months of the variables of wind speed, wind direction, temperature, relative humidity and atmospheric pressure. These variables were used for statistical calculations using the Weibull distributions. To give a better use of the calculated data, they were applied to the evaluation of a Pumping System.

The Marshall-Olkin Right Truncated Fréchet-Inverted Weibull Distribution: Its Properties and Applications

In this paper, a new probability distribution is proposed by using Marshall and Olkin transformation. Some of its properties such as moments, moment generating function, order statistics and reliability functions are derived. The method of maximum likelihood is used to estimate the model parameters. The graphs of the reliability function and hazard rate function are plotted by taken some values of the parameters. Three real life applications are introduced to compare the behaviour of the new distribution with other distributions.

Weibull Probability Model for Fracture Strength of Aluminium (1101)-Alumina Particle Reinforced Metal Matrix Composite

In recent times, conventional materials are replaced by metal matrix composites （MMCs） due to their high specific strength and modulus. Strength reliability, one of the key factors restricting wider use of composite materials in various applications, is commonly characterized by Weibull strength distribution function. In the present work, statistical analysis of the strength data of 15% volume alumina particle （mean size 15 um） reinforced in aluminum alloy （1101 grade alloy） fabricated by stir casting method was carried out using Weibull probability model. Twelve tension tests were performed according to ASTM B577 standards and the test data, the corresponding Weibull distribution was obtained. Finally the reliability of the composite behavior in terms of its fracture strength was presented to ensure the reliability of composites for suitable applications. An important implication of the present study is that the Weibull distribution describes the experimentally measured strength data more appropriately.

A New Generalized Weibull Model:Classical and Bayesian Estimation

Statistical distributions play a prominent role in applied sciences,particularly in biomedical sciences.The medical data sets are generally skewed to the right,and skewed distributions can be used quite effectively to model such kind of data sets.In the present study,therefore,we propose a new family of distributions suitable for modeling right-skewed medical data sets.The proposed family may be called a new generalized-X family.A special sub-model of the proposed family called a new generalized-Weibull distribution is discussed in detail.The maximum likelihood estimators of the model parameters are obtained.A brief Monte Carlo simulation study is conducted to evaluate the performance of these estimators.Finally,the proposed model is applied to the remission times of the stomach cancer patient’s data.The comparison of the goodness of fit results of the proposed model is made with the other competing models such as Weibull,Kumaraswamy Weibull,and exponentiated Weibull distributions.Certain analytical measures such as Akaike information criterion,Bayesian information criterion,Anderson Darling statistic,and Kolmogorov-Smirnov test statistic are considered to show which distribution provides the best fit to data.Based on these measures,it is showed that the proposed distribution is a reasonable candidate for modeling data in medical sciences and other related fields.

Efficient Process Monitoring Under General Weibull Distribution

Product testing is a key ingredient in maintaining the quality of a pro-duction process.The production process is considered an efficient process if it is capable of quick identification of faulty products.The items produced by any production process are usually packed and acceptance or rejection of the pack depends upon its conformity to some specified quality level.Generally,the specified quality level is based upon the number of defective items found in the inspected number of items.Such decisions are based upon some rules and usually acceptance of the pack is based upon a fewer number of defective items in the pack.Such questions can be answered by using acceptance sampling plans.The acceptance sampling plans assume the fact that the quality level of the item follows some probability distribution.The sampling plans based upon some classical probability distributions are available but often it happens that the quality behavior of the product does not follow a simple probability model and hence the available sampling plans fail.In this paper,we have developed acceptance sampling plans when the product life follows a general Weibull distribution.The sampling plans have been constructed by considering the crisp and fuzzy behavior of the acceptance probability.These sampling plans have been constructed by assuming an infinite lot size.It has been found that the number of items required for inspection decreases with an increase in some parameters.

On Relations for Moments of Generalized Order Statistics for Lindley–Weibull Distribution

Moments of generalized order statistics appear in several areas of science and engineering.These moments are useful in studying properties of the random variables which are arranged in increasing order of importance,for example,time to failure of a computer system.The computation of these moments is sometimes very tedious and hence some algorithms are required.One algorithm is to use a recursive method of computation of these moments and is very useful as it provides the basis to compute higher moments of generalized order statistics from the corresponding lower-order moments.Generalized order statistics pro-vides several models of ordered data as a special case.The moments of general-ized order statistics also provide moments of order statistics and record values as a special case.In this research,the recurrence relations for single,product,inverse and ratio moments of generalized order statistics will be obtained for Lindley–Weibull distribution.These relations will be helpful for obtained moments of gen-eralized order statistics from Lindley–Weibull distribution recursively.Special cases of the recurrence relations will also be obtained.Some characterizations of the distribution will also be obtained by using moments of generalized order statistics.These relations for moments and characterizations can be used in differ-ent areas of computer sciences where data is arranged in increasing order.

Evaluation of the location parameter in Weibull distribution of CTOD testing results of high strength steel weldments

In this paper, the location parameter δ_(a) in Weibull distribution i.e. the lower limiting CTOD toughness of high strength steel weldments is evaluated from Gleeble simulated coarse grained specimens. The Charpy-V transition temperature obtained from these specimens is transformed to a fracture toughness K_(IC) value at a given temperature using Sanz' relation. Then, the location parameter a is δ_(a) evaluated. The predicted location parameter arrives at agreement with the lowest CTOD value as obtained from thick multilayer weldments. The improvement of the fracture toughness of multilayer weldments needs the increase of the lowering limiting CTOD.toughness i.e. the toughness of the coarse grained zone.

Bivariate Beta–Inverse Weibull Distribution:Theory and Applications

Probability distributions have been in use for modeling of random phenomenon in various areas of life.Generalization of probability distributions has been the area of interest of several authors in the recent years.Several situations arise where joint modeling of two random phenomenon is required.In such cases the bivariate distributions are needed.Development of the bivariate distributions necessitates certain conditions,in a field where few work has been performed.This paper deals with a bivariate beta-inverse Weibull distribution.The marginal and conditional distributions from the proposed distribution have been obtained.Expansions for the joint and conditional density functions for the proposed distribution have been obtained.The properties,including product,marginal and conditional moments,joint moment generating function and joint hazard rate function of the proposed bivariate distribution have been studied.Numerical study for the dependence function has been implemented to see the effect of various parameters on the dependence of variables.Estimation of the parameters of the proposed bivariate distribution has been done by using the maximum likelihood method of estimation.Simulation and real data application of the distribution are presented.

Parameter Estimation of Mixed Weibull Distributions Using Cuckoo Search

The lifetime data of products with multiple failure modes which are collected from life testing are often fitted by the mixed Weibull distributions. Since the mixed Weibull distributions contain no less than five parameters,the parameter estimation is difficult and inaccurate. In order to enhance the accuracy,a new method of parameter estimation based on Cuckoo search( CS) is proposed. An optimization model for the mixed Weibull distribution is formulated by minimizing the residual sum of squares. The optimal parameters are searched via CS algorithm. In the case study,the lifetime data come from the life testing of diesel injectors and are fitted by the twocomponent Weibull mixture. Regarding the maximum absolute error and the accumulative absolute error between estimated and observed values as the accuracy index of parameter estimation,the results of four parameter estimation methods that the graphic estimation method,the nonlinear least square method,the optimization method based on particle swarm optimization( PSO) and the proposed method are compared. The result shows that the proposed method is more efficient and more accurate than the other three methods.

Characterizing the Relationship between Weibull Location Parameter and the Minimal Observation in a Small Size of Sample Based on Stochastic Simulation

Three-parameter Weibull distribution is one of the preferable distribution models to describe product life. However, it is difficult to estimate its location parameter in the situation of a small size of sample. This paper presents a stochastic simulation method to estimate the Weibull location parameters according to a small size of sample of product life observations and a large amount of statistically simulated life date. Big data technique is applied to find the relationship between the minimal observation in a product life sample of size n (n ≥ 3) and the Weibull location parameter. An example is presented to demonstrate the applicability and the value of the big data based stochastic simulation method. Comparing with other methods, the stochastic simulation method can be applied to very small size of sample such as the sample size of three, and it is easy to apply.

An Algorithm to Determine the Truncated Weibull Parameters for Distribution of Throats and Pores in Random Network Models

In random network models, sizes for pores and throats are distributed according to a truncated Weibull distribution. As a result, parameters defining the shape of the distribution are critical for the characteristic of the network. In this paper, an algorithm to distribute pores and throats in random network was established to more representatively describe the topology of porous media. First, relations between Weibull parameters and the distribution of dimensionless throat sizes were studied and a series of standard curves were obtained. Then, by analyzing the capillary pressure curve of the core sample, frequency distribution histogram of throat sizes was obtained. All the sizes were transformed to dimensionless numbers ranged from 0 to 1. Curves of the core were compared to the standard curves, and truncated Weibull parameters could be determined according an inverse algorithm. Finally, aspect ratio and average length of throats were adjusted to simultaneously fit the porosity and the capillary pressure curves and the whole network was established. The predicted relative permeability curves were in good agreement with the experimental data of cores, indicating the validity of the algorithm.

A Class of Lindley and Weibull Distributions

In this paper, we introduce a class of Lindley and Weibull distributions (LW) that are useful for modeling lifetime data with a comprehensive mathematical treatment. The new class of generated distributions includes some well-known distributions, such as exponential, gamma, Weibull, Lindley, inverse gamma, inverse Weibull, inverse Lindley, and others. We provide closed-form expressions for the density, cumulative distribution, survival function, hazard rate function, moments, moments generating function, quantile, and stochastic orderings. Moreover, we discuss maximum likelihood estimation and the algorithm for computing the parameters estimates. Some sub models are discussed as an illustration with real data sets to show the flexibility of this class.