Four-parameter Distribution Research Articles

Exponentially Generated Modified Chen Distribution with Applications to Lifetime Dataset

In this paper, the exponentially generated system was used to modify a two-parameter Chen distribution to a four-parameter distribution with better performance. The property of complete probability distribution function was used to verify the completeness of the resulting distribution, which shows that the distribution is a proper probability distribution function. A simulation study involving varying sample sizes was used to ascertain the asymptotic property of the new distribution. Small and large sample sizes were considered which shows the closeness of the estimates to the true value as the sample size increases. Lifetime dataset were used for model comparison which shows the superiority of exponentially generated modify Chen distribution over some existing distributions. It is therefore recommended to use the four-parameter Chen distribution in place of the well-known two-parameter Chen distribution.

Efficient seismic fragility analysis of structures from dynamic reliability perspective

Seismic fragility analysis of structures quantitatively describes the relationship between the failure probabilities of structures and ground motion Intensity Measures (IMs). Generally, two steps are involved: (1). calculating the failure probabilities of the structure under consideration at a set of IM levels; (2). regressing the IM levels and the corresponding failure probabilities to obtain a bunch of fragility curves. In this paper, a new method is proposed for seismic fragility analysis of structures from a dynamic reliability perspective, where efficiency and accuracy are emphasized. First, the dynamic reliability of structures under fully non-stationary ground motions is presented, where the Fractional Exponential Moments-based Maximum Entropy Method (FEM-MEM) is applied to derive the extreme value distribution with high efficiency. In this method, only a total of 89 samples of fully non-stationary ground motions generated by the spectral representation method and consistent with the design response spectrum that are sufficient in the FEM-MEM for efficient dynamic reliability computations are required. Then, a four-parameter distribution model, which exhibits high flexibility, is used to recover the fragility curves. A two-step strategy is developed for parameter estimation based on four failure probabilities at four different IM levels. In addition, a strategy for selecting a small number of IM levels for all damage states is also suggested, which improves the computational efficiency. Two numerical examples, including a seven-story nonlinear shear frame structure and a five-story three-span frame structure designed according to Chinese codes, are investigated to verify the effectiveness of the proposed method, where the results obtained from the traditional lognormal distribution are also compared.

EVALUATION MODEL OF THE RECOVERY PROCESSES OF NON-MARKOVIAN SYSTEMS, CONSIDERING THE ELEMENTS UNRELIABILITY UNDER ARBITRARY DISTRIBUTION LAWS

The subject of the study is the reliability of recoverable non–Markovian systems, functioning of which is described by arbitrary distribution laws. The purpose of the article is to develop a mathematical model of the functioning of modern computer systems under arbitrary laws of the distribution of stay duration in each of the states, taking into account the recovery system and the provision of spare elements. The main task is to develop an adequate model of the system functioning process, taking into account the non-Markovian character of the processes occurring in the system, its possible large dimension, and the presence of a hierarchical recovery system. Based on this model, a method for calculating the density of the system recovery time distribution has been developed. At the same time, a universal four-parameter distribution is proposed to describe random processes occurring in the system. Using this approximation, the calculation of the desired parameter of the recovery flow is performed by solving the Volterra integral equation with a difference kernel.

Alpha power Topp-Leone Weibull distribution : Properties, Characterizations, Regression modeling and applications

This paper presents a new four-parameter distribution called the alpha power Topp-Leone Weibull distribution. It is more fexible than some existing well-known distributions due to its diferent shapes of the hazard and density functions. Fundamental mathematical properties of this distribution are obtained including the series expansions of density, quantile function, moments, generating function, conditional moments, mean deviations, Bonferroni and Lorenz curves, Rényi and Shanon entropies and order statistics. Characterizations of the new distribution based on two truncated moments and hazard function are investigated. Estimation of the model parameters is carried out by the maximum likelihood estimation method. A simulation study is conducted to evaluate the performance of the maximum likelihood estimators. A new log-alpha power Topp-Leone Weibull regression model is defined. Three real data sets are provided to examine the fexibility of the new models.

OPTIMIZATION OF RESOURCE DISTRIBUTION UNDER THE CONDITIONS OF FUZZY INITIAL DATA

The problem of resource distribution in several directions of its spending is considered for the case when the parameters of the distribution efficiency criterion are fuzzy numbers with given membership functions. The purpose of the study is the development of mathematical models and methods for solving the problem of resource allocation for practically the most important criteria, taking into account the fuzziness of the numerical values – of their parameters. An analysis of the well-known approach to solving the problem is carried out and its main shortcomings are identified, which motivate the continuation of research. A method for solving the stated problem is proposed, the computational implementation of which contains three stages. At the first stage, using the membership functions of the fuzzy parameters of the problem, the membership function of the criterion is formed. The function obtained in this case is approximated at the second stage using a four-parameter distribution. An important advantage of this distribution is the possibility, by varying the numerical values – of its parameters over a wide range, to change the mathematical expectation, variance, and asymmetry of the values – specified by this distribution, providing a high quality of approximation. Thus, the criterion for the effectiveness of the task is determined. At the third stage, a mathematical model of the optimization problem of the distribution of a limited resource is formulated. The following three options for constructing an optimality criterion are considered: maximizing the criterion with the maximum possible value of its membership function; maximization of the criterion, provided that the value of its membership function is not lower than the specified one; maximization of the criterion, provided that the value of the membership function of each of its terms is not lower than the specified one. Each of the resulting problems is a standard problem of mathematical programming and is solved by known methods. A possible direction for further research is discussed in order to improve the adequacy of the used analytical descriptions of the membership functions of the fuzzy parameters of the problem.

A New Probability Model Based on a Coherent System with Applications

The notion of a coherent system allows us to formalize how the random lifetime of the system is connected to the random lifetimes of its components. These connections are also generators of new pliant distributions, being those of various mixes of minimum and maximum of random variables. In this paper, a new four-parameter lifetime probability distribution is introduced by using the notion of a coherent system. Its structural properties are assessed and evaluated, including the analytical study of its main functions, stochastic dominance results, moments, and moment generating function. The proposed distribution, in particular, is proving to be efficient at fitting data with slight negative skewness and platykurtic as well as leptokurtic nature. This is illustrated by the analysis of three relevant real-life data sets, two in reliability and another in production, exhibiting the significance of the introduced model in comparison to various well-known models in statistical literature.

Seismic reliability evaluation of bridges under spatially varying ground motions using a four-parameter distribution

Large-span bridges are often subjected to spatially varying ground motion (SVGM) excitations. Meanwhile, bridge structures and ground motion excitations generally present a large degree of inherent uncertainties. Carrying out the seismic reliability analysis of bridges under SVGMs can provide a direct assessment for the safety probability of bridges during an earthquake event and is of paramount importance for the probabilistic performance analysis of bridges. To this end, this study proposes a new method for the seismic reliability evaluation of random nonlinear bridges subjected to SVGMs, which tackles the drawback of the current seismic reliability analysis method that can only be applied for uniform seismic excitations. To implement, the space correlated ground motions are first generated via a random function-based spectral representation method, based on which the uncertainty of ground motions is characterized by three elementary random variables. The first-passage probability of bridges under SVSGMs is then derived by the equivalent extreme value distribution (EVD), which is approximated by a flexible four-parameter distribution, i.e., shifted generalized lognormal distribution (SGLD). To estimate the parameters in the SGLD model with high efficiency and accuracy, a novel fractional moment-based method is proposed. A numerical example of a two-span continuous box-girder bridge subjected to space varying site conditions is investigated to verify and exemplify the proposed method. The results demonstrate that: 1) The proposed method is efficient and accurate for the seismic reliability of bridge under SVGM, even at low failure probability levels, and the computational burden of the proposed method is less than 2% of that of MCS; 2) The spatial variability is significantly influential on the seismic reliability of the bridge, ignoring the spatial variability of seismic excitation will significantly underestimate the failure probability of the bridge.

Normal-Power Function Distribution with Logistic Quantile Function: Properties and Application

Developing compound probability distributions is very important in the field of probability and statistics because there are different datasets from different fields with different features. These features range from high skewness, peakedness (kurtosis), bimodality, highly dispersed, and so on. Existing distributions might not easily fit well to these emerging data of interest. So, there is a need to develop more robust and flexible distributions that are positively skewed, negatively skewed, and bathup shape, to handle some of these features in the emerging data of interest. This paper, therefore, proposed a new four-parameter distribution called the Normal-Power{logistic} distribution. The proposed distribution was characterized by its density, distribution, survival, hazard, cumulative hazard, reversed hazard, and quantile functions. Properties such as the r-th moment, heavy tail property, stochastic ordering, mean inactive time were obtained. A useful transformation of the proposed distribution to normal distribution was shown to help generate its quantiles. The method of Maximum Likelihood Estimation (MLE) was used to estimate the model parameters. A simulation study was carried out to test the consistency of the maximum likelihood parameter estimates. The result of the simulation shows that the biases reduce as the sample size increases for different parameter values. The importance of the new distribution was proved empirically using a real-life dataset of gauge lengths of 10mm. The proposed distribution was compared with five other competing distributions, and the results show that the proposed Normal-Power{logistic} distribution (NPLD) performed favourably than the other five distributions using the AIC, CAIC, BIC, HQIC criteria.

Optimal Daily Trading of Battery Operations Using Arbitrage Spreads

An important revenue stream for electric battery operators is often arbitraging the hourly price spreads in the day-ahead auction. The optimal approach to this is challenging if risk is a consideration as this requires the estimation of density functions. Since the hourly prices are not normal and not independent, creating spread densities from the difference of separately estimated price densities is generally intractable. Thus, forecasts of all intraday hourly spreads were directly specified as an upper triangular matrix containing densities. The model was a flexible four-parameter distribution used to produce dynamic parameter estimates conditional upon exogenous factors, most importantly wind, solar and the day-ahead demand forecasts. These forecasts supported the optimal daily scheduling of a storage facility, operating on single and multiple cycles per day. The optimization is innovative in its use of spread trades rather than hourly prices, which this paper argues, is more attractive in reducing risk. In contrast to the conventional approach of trading the daily peak and trough, multiple trades are found to be profitable and opportunistic depending upon the weather forecasts.

Extended Exponentiated Chen Distribution: Mathematical Properties and Applications

In this paper, we introduce a new four-parameter distribution which is called Extended Exponentiated Chen (EE-C) distribution. Theoretical properties of this model including the hazard function, moments, conditional moments, mean residual life, mean past lifetime, coefficients of skewness and kurtosis, order statistics and asymptotic properties are derived and studied. The maximum likelihood estimation technique is used to estimate the parameters of this model. The estimation of the model parameters by Least squares, Weighted Least Squares, Crammer-von-Mises, Anderson-Darling and right-tailed Anderson-Darling methods are also briefly introduced and numerically investigated. Moreover, simulation schemes are derived. At the end, three applications of the model with two real data sets are presented for the illustration of the flexibility of the proposed distribution.

Transmuted Weibull Power Function Distribution: its Properties and Applications

In this paper, we introduce a new four-parameter distribution called the transmuted Weibull power function (TWPF) distribution which e5xtends the transmuted family proposed by Shaw and Buckley [1]. The hazard rate function of the TWPF distribution can be constant, increasing, decreasing, unimodal, upside down bathtub shaped or bathtub shape. Some mathematical properties are derived including quantile functions, expansion of density function, moments, moment generating function, residual life function, reversed residual life function, mean deviation, inequality measures. The estimation of the model parameters is carried out using the maximum likelihood method. The importance and flexibility of the proposed model are proved empirically using real data sets.

Inference for variance components in linear mixed-effect models with flexible random effect and error distributions.

In many biomedical investigations, parameters of interest, such as the intraclass correlation coefficient, are functions of higher-order moments reflecting finer distributional characteristics. One popular method to make inference for such parameters is through postulating a parametric random effects model. We relax the standard normality assumptions for both the random effects and errors through the use of the Fleishman distribution, a flexible four-parameter distribution which accounts for the third and fourth cumulants. We propose a Fleishman bootstrap method to construct confidence intervals for correlated data and develop a normality test for the random effect and error distributions. Recognizing that the intraclass correlation coefficient may be heavily influenced by a few extreme observations, we propose a modified, quantile-normalized intraclass correlation coefficient. We evaluate our methods in simulation studies and apply these methods to the Childhood Adenotonsillectomy Trial sleep electroencephalogram data in quantifying wave-frequency correlation among different channels.

A four-parameter negative binomial-Lindley distribution for modeling over and underdispersed count data with excess zeros

Count data often exhibits the property of dispersion and have large number of zeros. In order to take these properties into account, a new generalized negative binomial-Lindley distribution with four parameters is proposed, of which the two-parameter and three-parameter negative binomial-Lindley distributions are special cases. Several statistical properties of the proposed distribution are presented. The dispersion index for the proposed distribution is derived and based on the index, it is clear that the proposed distribution can adequately fit the data with properties of overdispersion or underdispersion depending on the choice of the parameters. The proposed distribution is fitted to three overdispersed datasets with large proportion of zeros. The best fitted model is selected based on the values of AIC, mean absolute error and root mean squared error. From the model fittings, it can be concluded that the proposed distribution outperforms Poisson and negative binomial distributions in fitting the count data with overdispersion and large number of zeros.

Transmuted Modified Inverse Weibull distribution: Properties and application

This paper examines the potential usefulness of the transmuted modified inverse Weibull distribution. This four-parameter distribution holds eleven life time distributions as special cases. Theoretical properties of the transmuted modified inverse Weibull distribution are studied; which includes the quantile, median, entropy, mean deviations, mean, geometric mean and harmonic mean. The estimation of parameters is obtained by using the method of maximumlikelihood. An application to real dataset is provided to show the better fit of the transmuted modified inverse Weibull distribution.

A New Extension of Lindley Geometric Distribution and its Applications

A new four-parameter distribution called the beta Lindley-geometric distribution is proposed. The hazard rate function of the new model can be constant, decreasing, increasing, upside down bathtub or bathtub failure rate shapes. Various structural properties including of the new distribution are derived. The estimation of the model parameters is performed by maximum likelihood method. The usefulness of the new distribution is illustrated using a real data set.

The Odd Gamma Weibull-Geometric Model: Theory and Applications

In this paper, we study a new four-parameter distribution called the odd gamma Weibull-geometric distribution. Having the qualities suggested by its name, the new distribution is a special member of the odd-gamma-G family of distributions, defined with the Weibull-geometric distribution as baseline, benefiting of their respective merits. Firstly, we present a comprehensive account of its mathematical properties, including shapes, asymptotes, quantile function, quantile density function, skewness, kurtosis, moments, moment generating function and stochastic ordering. Then, we focus our attention on the statistical inference of the corresponding model. The maximum likelihood estimation method is used to estimate the model parameters. The performance of this method is assessed by a Monte Carlo simulation study. An empirical illustration of the new distribution is presented by the analyses two real-life data sets. The results of the proposed model reveal to be better as compared to those of the useful beta-Weibull, gamma-Weibull and Weibull-geometric models.

Weight optimisation of functionally graded beams using modified differential evolution

In this article, an efficient numerical approach for weight optimisation of functionally graded (FG) beams in the presence of frequency constraints is presented. For the analysis purpose, a finite element (FE) solution based on the first order shear deformation theory (FSDT) is established to analyse the free vibration behaviour of FG beams. A four-parameter power law distribution and a five-parameter trigonometric distribution are used to describe the volume fraction of material constituents in the thickness direction. The goal is to tailor the thickness and material distribution for minimising the weight of FG beams while constraining the fundamental frequency to be greater than a prescribed value. The constrained optimisation problem is effectively solved by a novel differential evolution (DE) algorithm. The validity and efficiency of the proposed approach is demonstrated through two numerical examples corresponding to the four-parameter distribution and the five-parameter distribution.Keywords: FGM beam; lightweight design; frequency constraint; differential evolution.

The Weibull-Gamma Distribution: Properties and Applications.

A new member of the Weibull-generated (Weibull-G) family of distributions—namely the Weibull-gamma distribution—is proposed. This four-parameter distribution can provide great flexibility in modeling different data distribution shapes. Some special cases of the Weibull-gamma distribution are considered. Several properties of the new distribution are studied. The maximum likelihood method is applied to obtain an estimation of the parameters of the Weibull-gamma distribution. The usefulness of the proposed distribution is examined by means of five applications to real datasets.

A novel moment method using the log skew normal distribution for particle coagulation

A novel moment method is proposed to predict the detailed size distribution for particle coagulation. The method uses the log skew normal distribution (LSND) to approximate the actual size distribution. The LSND is a four-parameter distribution that generalizes the conventional log-normal distribution to allow for asymmetrical characteristics. The moment closure is achieved by applying the Gauss-Hermite quadrature for the integral terms in the moment equations. With this approach, no further treatments are needed for specific coagulation kernels. Then the method is validated by comparing it with other recognized numerical methods for Brownian coagulation in the continuum regime and the free-molecular regime. The results show that the present method well predicts the detailed size distribution over the entire coagulation time for both regimes. Moreover, the tails of the self-preserving size distribution are well reproduced by the present method. The asymptotic characteristics of Brownian coagulation are also analyzed based on the LSND. The results validate the LSND in analytically representing the self-preserving size distribution and predicting the asymptotic geometric standard deviation.

Extreme Response Prediction of Steel Risers Using a Four Parameter Distribution

Short-term extreme response estimates are required in many areas of ocean and offshore engineering, such as steel risers design. As in many cases, the response in non-Gaussian, a theoretical solution, is usually not readily available for this purpose. Hermite transformation and Weibull-based models, among others, are some alternatives that have been used in connection with sampled response time series. In this work, a new approach is investigated. Recently, a four-parameter distribution known as the shifted generalized lognormal distribution (SGLD) has been presented in the literature. One of its main advantages is that it covers regions of skewness–kurtosis not covered by other distributions of common use in engineering. In this paper, the performance of this distribution is evaluated in the extreme values' estimation of the utilization ratios of steel riser sections. Three alternatives for using SGLD are investigated in two case studies of different dynamic behavior. The first one is a steel-lazy wave riser (SLWR) connected to a turret-moored FPSO (floating, production, storage and offloading unit) in 914 m water depth, and the second is a SLWR connected to a spread-mooring FPSO in a water depth of 1400 m. The results obtained by the SGLD-based analysis, which considered several simulation lengths, are compared to those obtained by means of an extreme value distribution fitted to episodical extremes obtained from many distinct realizations. The results of a traditional Weibull-fitting approach to the response peaks and those obtained with a Hermite transformation-based model are also presented for comparison.