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Journal articles on the topic 'New Linear-exponential distribution'

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Published: 5 June 2025

Last updated: 15 July 2025

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Binod, Kumar Sah. "New Linear-Exponential Distribution." APPLIED SCIENCE PERIODICAL XXIV, no. 2, May 2022 (2022): 1–16. https://doi.org/10.5281/zenodo.6559260.

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The proposed distribution is a continuous probability distribution with a single parameter. We named it ‘New Linear-exponential distribution’. We have been discussed about probability density function, probability distribution function and moment generating function. Moments about origin and hence, the first four moments about the mean of the proposed distribution have been obtained. Estimation of parameters have been discussed by the method of moments and that of the method of maximum likelihood. To test validity of the theoretical work, goodness of fit has been applied to some data-sets which were used earlier by others. It has been observed that the proposed distribution gives better fit to the most of data-sets than Lindley distribution and One-parameter Linear-exponential distribution.

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Binod, Kumar Sah. "Premium Linear-Exponential Distribution." APPLIED SCIENCE PERIODICAL XXIV, no. 3, August 2022 (2022): 1–16. https://doi.org/10.5281/zenodo.7093047.

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           In the present study, a one-parameter continuous probability distribution, which is a modified form of ‘New Linear-exponential distribution’ and ‘Modified Linear-exponential distribution’, has been proposed for statistical modeling of survival time data with better results. Several characteristics and descriptive measures of Statistics of the proposed distribution have been derived and discussed. To test validity of the theoretical work of this distribution, goodness of fit has been applied to some over-dispersed data-sets which were earlier used by other researchers. It is expected to be a better alternative of Lindley distribution and in some of the cases; it is expected to be a better alternative of New Linear-exponential distribution and Modified Linear-exponential distribution.

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Tian, Yuzhu, Maozai Tian, and Qianqian Zhu. "Transmuted Linear Exponential Distribution: A New Generalization of the Linear Exponential Distribution." Communications in Statistics - Simulation and Computation 43, no. 10 (2014): 2661–77. http://dx.doi.org/10.1080/03610918.2013.763978.

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Atem, Bol A. M., Suleman Nasiru, and Kwara Nantomah. "Topp–Leone Linear Exponential Distribution." Stochastics and Quality Control 33, no. 1 (2018): 31–43. http://dx.doi.org/10.1515/eqc-2017-0022.

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Abstract This article studies the properties of the Topp–Leone linear exponential distribution. The parameters of the new model are estimated using maximum likelihood estimation, and simulation studies are performed to examine the finite sample properties of the parameters. An application of the model is demonstrated using a real data set. Finally, a bivariate extension of the model is proposed.

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Radwan, Hossam, Mohamed Mahmoud, and Mohamed Ghazal. "Modified Generalized Linear Exponential Distribution: Properties and applications." Statistics, Optimization & Information Computing 12, no. 1 (2023): 231–55. http://dx.doi.org/10.19139/soic-2310-5070-1103.

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In this paper, we propose a new four-parameter lifetime distribution called modified generalized linear exponential distribution. The proposed distribution is a modification of the generalized linear exponential distribution. Several important lifetime distributions in reliability engineering and survival analysis are considered as special sub-models including modified Weibull, Weibull, linear exponential and generalized linear exponential distributions, among others. We study the mathematical and statistical properties of the proposed distribution including moments, moment generating function, modes, and quantile. We then examine hazard rate, mean residual life, and variance residual life functions of the distribution. A significant property of the new distribution is that it can have a bathtub-shaped, which is very flexible for modeling reliability data.The four unknown parameters of the proposed model are estimated by the maximum likelihood. Finally, two practical real data sets are applied to show that the proposed distribution provides a superior fit than the other sub-models and some well-known distributions.

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Tian, Yu-zhu, Mao-zai Tian, and Qian-qian Zhu. "A new generalized linear exponential distribution and its applications." Acta Mathematicae Applicatae Sinica, English Series 30, no. 4 (2014): 1049–62. http://dx.doi.org/10.1007/s10255-014-0442-4.

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Phani, Y., S. V. S. Girija, and A. V. Dattatreya Rao. "Arc Tan- Exponential Type Distribution Induced By Stereographic Projection / Bilinear Transformation On Modified Wrapped Exponential Distribution." Journal of Applied Mathematics, Statistics and Informatics 9, no. 1 (2013): 69–74. http://dx.doi.org/10.2478/jamsi-2013-0007.

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Abstract In this paper we make an attempt to construct a new three parameter linear model, we call this new model as Arc Tan-Exponential Type distribution, by applying Stereographic Projection or equivalently Bilinear transformation on Wrapped Exponential distribution, Probability density and cumulative distribution functions of this new model are presented and their graphs are plotted for various values of parameters.

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Poonia, Neeraj, and Sarita Azad. "A New Exponentiated Generalized Linear Exponential Distribution: Properties and Application." RMS: Research in Mathematics & Statistics 8, no. 1 (2021): 1953233. http://dx.doi.org/10.1080/27658449.2021.1953233.

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Althubyani, Faiza A., Ahmed M. T. Abd El-Bar, Mohamad A. Fawzy, and Ahmed M. Gemeay. "A New 3-Parameter Bounded Beta Distribution: Properties, Estimation, and Applications." Axioms 11, no. 10 (2022): 504. http://dx.doi.org/10.3390/axioms11100504.

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This study presents a new three-parameter beta distribution defined on the unit interval, which can have increasing, decreasing, left-skewed, right-skewed, approximately symmetric, bathtub, and upside-down bathtub shaped densities, and increasing, U, and bathtub shaped hazard rates. This model can define well-known distributions with various parameters and supports, such as Kumaraswamy, beta exponential, exponential, exponentiated exponential, uniform, the generalized beta of the first kind, and beta power distributions. We present a comprehensive account of the mathematical features of the new model. Maximum likelihood methods and a Bayesian method under squared error and linear exponential loss functions are presented; also, approximate confidence intervals are obtained. We present a simulation study to compare all the results. Two real-world data sets are analyzed to demonstrate the utility and adaptability of the proposed model.

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Sakthivel, K. M., and Alicia Mathew. "A Meaningful Construction of New Circular Distribution for Applications in Geomorphology." Indian Journal Of Science And Technology 18, no. 13 (2025): 1009–22. https://doi.org/10.17485/ijst/v18i13.4008.

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Objectives: This work introduces a novel circular probability distributionthe Double Truncated Wrapped Exponential (DTWE) distribution highlighting the importance of circular statistics with cyclical characteristics contrary to usual linear data. Methods: The DTWE distribution is developed using the principle of truncation on the wrapped exponential distribution, which satisfies the principles of circularity. The properties of the distribution, such as the trigonometric mean, skewness, and kurtosis, are derived to enhance interpretability. Parameter estimation is carried out using Maximum Likelihood Estimation, Least Squares, and Weighted Least Squares methods. The goodness-of-fit is carried out, which makes DTWE distribution comparable to other well-known circular probability models. Findings: The numerical results of the simulation study across sample sizes (𝑛 = 30, 50, 100, 1000) and parameter values (𝜃 = 0.5, 1, 2) demonstrate that the DTWE distribution achieves accurate and consistent parameter estimation. For 𝜃 = 0.5 and 𝑛 = 30, the key performance metrics, such as the bias, Mean Square Error (MSE), and standard deviation (SD) for MLE outperform the LS and WLS methods by approximately 20%. Similarly, for 𝜃 = 2 and 𝑛 = 1000, the MLE achieves greater consistency reducing the bias, MSE, and SD by more than 30%. Real world data analysis shows that the DTWE distribution captures the cyclical patterns in ecological and geological data perfectly and gives meaningful insights into directional behaviours. Novelty: This study introduces a novel truncationbased framework for constructing circular probability distributions. The new distribution provides a distinctive approach for evaluating the circular data in ecological and geological datasets. Keywords: Directional Statistics; Truncation; Exponential Distribution; Circular Distribution; MLE

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Okasha, Hassan M., and M. Kayid. "A new family of Marshall–Olkin extended generalized linear exponential distribution." Journal of Computational and Applied Mathematics 296 (April 2016): 576–92. http://dx.doi.org/10.1016/j.cam.2015.10.017.

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Al-Harbi, Khlood, Aisha Fayomi, Hanan Baaqeel, and Amany Alsuraihi. "A Novel Discrete Linear-Exponential Distribution for Modeling Physical and Medical Data." Symmetry 16, no. 9 (2024): 1123. http://dx.doi.org/10.3390/sym16091123.

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In real-life data, count data are considered more significant in different fields. In this article, a new form of the one-parameter discrete linear-exponential distribution is derived based on the survival function as a discretization technique. An extensive study of this distribution is conducted under its new form, including characteristic functions and statistical properties. It is shown that this distribution is appropriate for modeling over-dispersed count data. Moreover, its probability mass function is right-skewed with different shapes. The unknown model parameter is estimated using the maximum likelihood method, with more attention given to Bayesian estimation methods. The Bayesian estimator is computed based on three different loss functions: a square error loss function, a linear exponential loss function, and a generalized entropy loss function. The simulation study is implemented to examine the distribution’s behavior and compare the classical and Bayesian estimation methods, which indicated that the Bayesian method under the generalized entropy loss function with positive weight is the best for all sample sizes with the minimum mean squared errors. Finally, the discrete linear-exponential distribution proves its efficiency in fitting discrete physical and medical lifetime count data in real-life against other related distributions.

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Arshad, Mohd, Ashok Kumar Pathak, Qazi J. Azhad, and Mukti Khetan. "Modeling Bivariate Data Using Linear Exponential and Weibull Distributions as Marginals." Mathematica Slovaca 73, no. 4 (2023): 1075–96. http://dx.doi.org/10.1515/ms-2023-0079.

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ABSTRACT Modeling bivariate data with different marginals is an important problem and have numerous applications in diverse disciplines. This paper introduces a new family of bivariate generalized linear exponential Weibull distribution having generalized linear and exponentiated Weibull distributions as marginals. Some important quantities like conditional distributions, conditional moments, product moments and bivariate quantile functions are derived. Concepts of reliability and measures of dependence are also discussed. The methods of maximum likelihood and Bayesian estimation are considered to estimate model parameters. Monte Carlo simulation experiments are performed to demonstrate the performance of the estimators. Finally, a real data application is also discussed to demonstrate the usefulness of the proposed distribution in real-life situations.

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GAMARNIK, DAVID, TOMASZ NOWICKI, and GRZEGORZ ŚWIRSZCZ. "Invariant probability measures and dynamics of exponential linear type maps." Ergodic Theory and Dynamical Systems 28, no. 5 (2008): 1479–95. http://dx.doi.org/10.1017/s014338570700106x.

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AbstractWe consider the problem of the asymptotic size of the random maximum-weight matching of a sparse random graph, which we translate into dynamics of the operator in the space of distribution functions. A tight condition for the uniqueness of the globally attracting fixed point is provided, which extends the result of Karp and Sipser [Maximum matchings in sparse random graphs. 22nd Ann. Symp. on Foundations of Computer Science (Nashville, TN, 28–30 October, 1981). IEEE, New York, 1981, pp. 364–375] from deterministic weight distributions (Dirac measures μ) to general ones. Given a probability measure μ which corresponds to the weight distribution of a link of a random graph, we form a positive linear operator Φμ (convolution) on distribution functions and then analyze a family of its exponents, with parameter λ, which corresponds to the connectivity of a sparse random graph. The operator 𝕋 relates the distribution F on the subtrees to the distribution 𝕋F on the node of the tree by 𝕋F=exp (−λΦμF). We prove that for every probability measure μ and every λ<e, there exists a unique globally attracting fixed point of the operator; the probability measure corresponding to this fixed point can then be used to compute the expected maximum-weight matching on a sparse random graph. This result is called the e-cutoff phenomenon. For deterministic distributions and λ>e, there is no fixed point attractor. We further establish that the uniqueness of the invariant measure of the underlying operator is not a monotone property of the average connectivity; this parallels similar non-monotonicity results in the statistical physics context.

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Al-Marzouki, Sanaa, and Sharifah Alrajhi. "A New-Flexible Generated Family of Distributions Based on Half-Logistic Distribution." Journal of Computational and Theoretical Nanoscience 17, no. 11 (2020): 4813–18. http://dx.doi.org/10.1166/jctn.2020.9332.

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We proposed a new family of distributions from a half logistic model called the generalized odd half logistic family. We expressed its density function as a linear combination of exponentiated densities. We calculate some statistical properties as the moments, probability weighted moment, quantile and order statistics. Two new special models are mentioned. We study the estimation of the parameters for the odd generalized half logistic exponential and the odd generalized half logistic Rayleigh models by using maximum likelihood method. One real data set is assesed to illustrate the usefulness of the subject family.

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Yanev, George P. "Exponential and Hypoexponential Distributions: Some Characterizations." Mathematics 8, no. 12 (2020): 2207. http://dx.doi.org/10.3390/math8122207.

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The (general) hypoexponential distribution is the distribution of a sum of independent exponential random variables. We consider the particular case when the involved exponential variables have distinct rate parameters. We prove that the following converse result is true. If for some n≥2, X1,X2,…,Xn are independent copies of a random variable X with unknown distribution F and a specific linear combination of Xj’s has hypoexponential distribution, then F is exponential. Thus, we obtain new characterizations of the exponential distribution. As corollaries of the main results, we extend some previous characterizations established recently by Arnold and Villaseñor (2013) for a particular convolution of two random variables.

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K., M. Sakthivel, and Department of Statistics Bharathiar University Coimbatore Tamil Nadu India Professor. "A Meaningful Construction of New Circular Distribution for Applications in Geomorphology." Indian Journal of Science and Technology 18, no. 13 (2025): 1009–22. https://doi.org/10.17485/IJST/v18i13.4008.

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Abstract Objectives: This work introduces a novel circular probability distributionthe Double Truncated Wrapped Exponential (DTWE) distribution highlighting the importance of circular statistics with cyclical characteristics contrary to usual linear data. Methods: The DTWE distribution is developed using the principle of truncation on the wrapped exponential distribution, which satisfies the principles of circularity. The properties of the distribution, such as the trigonometric mean, skewness, and kurtosis, are derived to enhance interpretability. Parameter estimation is carried out using Maximum Likelihood Estimation, Least Squares, and Weighted Least Squares methods. The goodness-of-fit is carried out, which makes DTWE distribution comparable to other well-known circular probability models. Findings: The numerical results of the simulation study across sample sizes (𝑛 = 30, 50, 100, 1000) and parameter values (𝜃 = 0.5, 1, 2) demonstrate that the DTWE distribution achieves accurate and consistent parameter estimation. For 𝜃 = 0.5 and 𝑛 = 30, the key performance metrics, such as the bias, Mean Square Error (MSE), and standard deviation (SD) for MLE outperform the LS and WLS methods by approximately 20%. Similarly, for 𝜃 = 2 and 𝑛 = 1000, the MLE achieves greater consistency reducing the bias, MSE, and SD by more than 30%. Real world data analysis shows that the DTWE distribution captures the cyclical patterns in ecological and geological data perfectly and gives meaningful insights into directional behaviours. Novelty: This study introduces a novel truncationbased framework for constructing circular probability distributions. The new distribution provides a distinctive approach for evaluating the circular data in ecological and geological datasets. Keywords: Directional Statistics; Truncation; Exponential Distribution; Circular Distribution; MLE

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Povše, Aleš, Saša Skale, and Jelena Vojvodić-Tuma. "Evaluation of the Condition of the Bottom of the Tanks for Petroleum Products-Forecast of the Remaining Operating Life." Strojniški vestnik - Journal of Mechanical Engineering 70, no. 5-6 (2024): 282–92. http://dx.doi.org/10.5545/sv-jme.2023.682.

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Corrosion rate measurements in the tank enable us to develop a new model for predicting the remaining operating period of the tank. The empirical model for the study of bottom plate thicknesses and corrosion pits is more conservative than the standard linear model, as it considers the autocatalytic nature of the corrosion process. We used the double exponential distribution of maximum values (Gumbel's) to evaluate the maximum depth of pits, and the double exponential distribution of minimum values to evaluate the minimum values of the plate thickness. A comparison of the values of the parameters obtained using linear extrapolation and exponential models indicates the unreliability of linear extrapolation, since disregarding dynamic processes underestimates the actual rate of corrosion.

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Afify, Ahmed Z., Hassan M. Aljohani, Abdulaziz S. Alghamdi, Ahmed M. Gemeay, and Abdullah M. Sarg. "A New Two-Parameter Burr-Hatke Distribution: Properties and Bayesian and Non-Bayesian Inference with Applications." Journal of Mathematics 2021 (November 28, 2021): 1–16. http://dx.doi.org/10.1155/2021/1061083.

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This article introduces a two-parameter flexible extension of the Burr-Hatke distribution using the inverse-power transformation. The failure rate of the new distribution can be an increasing shape, a decreasing shape, or an upside-down bathtub shape. Some of its mathematical properties are calculated. Ten estimation methods, including classical and Bayesian techniques, are discussed to estimate the model parameters. The Bayes estimators for the unknown parameters, based on the squared error, general entropy, and linear exponential loss functions, are provided. The ranking and behavior of these methods are assessed by simulation results with their partial and overall ranks. Finally, the flexibility of the proposed distribution is illustrated empirically using two real-life datasets. The analyzed data shows that the introduced distribution provides a superior fit than some important competing distributions such as the Weibull, Fréchet, gamma, exponential, inverse log-logistic, inverse weighted Lindley, inverse Pareto, inverse Nakagami-M, and Burr-Hatke distributions.

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Jamal, Farrukh, Laba Handique, Abdul Hadi N. Ahmed, Sadaf Khan, Shakaiba Shafiq, and Waleed Marzouk. "The Generalized Odd Linear Exponential Family of Distributions with Applications to Reliability Theory." Mathematical and Computational Applications 27, no. 4 (2022): 55. http://dx.doi.org/10.3390/mca27040055.

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A new family of continuous distributions called the generalized odd linear exponential family is proposed. The probability density and cumulative distribution function are expressed as infinite linear mixtures of exponentiated-F distribution. Important statistical properties such as quantile function, moment generating function, distribution of order statistics, moments, mean deviations, asymptotes and the stress–strength model of the proposed family are investigated. The maximum likelihood estimation of the parameters is presented. Simulation is carried out for two of the mentioned sub-models to check the asymptotic behavior of the maximum likelihood estimates. Two real-life data sets are used to establish the credibility of the proposed model. This is achieved by conducting data fitting of two of its sub-models and then comparing the results with suitable competitive lifetime models to generate conclusive evidence.

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Miranda-Soberanis, V. F., and T. W. Yee. "New Link Functions for Distribution–Specific Quantile Regression Based on Vector Generalized Linear and Additive Models." Journal of Probability and Statistics 2019 (May 7, 2019): 1–11. http://dx.doi.org/10.1155/2019/3493628.

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In the usual quantile regression setting, the distribution of the response given the explanatory variables is unspecified. In this work, the distribution is specified and we introduce new link functions to directly model specified quantiles of seven 1–parameter continuous distributions. Using the vector generalized linear and additive model (VGLM/VGAM) framework, we transform certain prespecified quantiles to become linear or additive predictors. Our parametric quantile regression approach adopts VGLMs/VGAMs because they can handle multiple linear predictors and encompass many distributions beyond the exponential family. Coupled with the ability to fit smoothers, the underlying strong assumption of the distribution can be relaxed so as to offer a semiparametric–type analysis. By allowing multiple linear and additive predictors simultaneously, the quantile crossing problem can be avoided by enforcing parallelism constraint matrices. This article gives details of a software implementation called the VGAMextra package for R. Both the data and recently developed software used in this paper are freely downloadable from the internet.

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Fayomi, Aisha, Sadaf Khan, Muhammad Hussain Tahir, Ali Algarni, Farrukh Jamal, and Reman Abu-Shanab. "A new extended gumbel distribution: Properties and application." PLOS ONE 17, no. 5 (2022): e0267142. http://dx.doi.org/10.1371/journal.pone.0267142.

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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.

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Al-Duais, Fuad S. "Bayesian Analysis of Record Statistic from the Inverse Weibull Distribution under Balanced Loss Function." Mathematical Problems in Engineering 2021 (March 25, 2021): 1–9. http://dx.doi.org/10.1155/2021/6648462.

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The main contribution of this work is to develop a linear exponential loss function (LINEX) to estimate the scale parameter and reliability function of the inverse Weibull distribution (IWD) based on lower record values. We do this by merging a weight into LINEX to produce a new loss function called weighted linear exponential loss function (WLINEX). We then use WLINEX to derive the scale parameter and reliability function of the IWD. Subsequently, we discuss the balanced loss functions for three different types of loss function, which include squared error (SE), LINEX, and WLINEX. The majority of previous scholars determined the weighted balanced coefficients without mathematical justification. One of the main contributions of this work is to utilize nonlinear programming to obtain the optimal values of the weighted coefficients for balanced squared error (BSE), balanced linear exponential (BLINEX), and balanced weighted linear exponential (BWLINEX) loss functions. Furthermore, to examine the performance of the proposed methods—WLINEX and BWLINEX—we conduct a Monte Carlo simulation. The comparison is between the proposed methods and other methods including maximum likelihood estimation, SE loss function, LINEX, BSE, and BLINEX. The results of simulation show that the proposed models BWLINEX and WLINEX in this work have the best performance in estimating scale parameter and reliability, respectively, according to the smallest values of mean SE. This result means that the proposed approach is promising and can be applied in a real environment.

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Rannona, Kethamile, Broderick Oluyede, Fastel Chipepa, and Boikanyo Makubate. "Marshall-Olkin-Type II-Topp-Leone-G Family of Distributions: Model, Properties and Applications." Journal of Probability and Statistical Science 20, no. 1 (2022): 127–49. http://dx.doi.org/10.37119/jpss2022.v20i1.508.

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We develop a new family of distribution called the Marshall-Olkin-Type II-Topp-Leone-G (MO-TII-TL-G) family of distributions, which is a linear combination of the exponential-G family of distributions. The statistical properties of the new distributions are studied and its model parameters are obtainedusing the maximum likelihood method. A simulation study is carried out to determine the performance of the maximum likelihood estimates and lastly real data examples are provided to demonstrate the usefulness of the proposed model in comparison to other models.

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Roozegar, Rasool, and Ali Akbar Jafari. "ON BIVARIATE EXPONENTIATED EXTENDED WEIBULL FAMILY OF DISTRIBUTIONS." Ciência e Natura 38, no. 2 (2016): 564. http://dx.doi.org/10.5902/2179460x19496.

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In this paper, we introduce a new class of distributions by compounding the exponentiated extended Weibull family and power series family. This distribution contains several lifetime models such as the complementary extended Weibull-power series, generalized exponential-power series, generalized linear failure rate-power series, exponentiated Weibull-power series, generalized modifiedWeibull-power series, generalized Gompertz-power series and exponentiated extendedWeibull distributions as special cases. We obtain several properties of this new class of distributions such as Shannon entropy, mean residual life, hazard rate function, quantiles and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented.

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Shan, Chenghao, Weidong Zhou, Yefeng Yang, and Hanyu Shan. "A new robust Kalman filter with measurement loss based on mixing distribution." Transactions of the Institute of Measurement and Control 44, no. 8 (2021): 1699–707. http://dx.doi.org/10.1177/01423312211054942.

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A new robust Kalman filter (KF) based on mixing distribution is presented to address the filtering issue for a linear system with measurement loss (ML) and heavy-tailed measurement noise (HTMN) in this paper. A new Student’s t-inverse-Wishart-Gamma mixing distribution is derived to more rationally model the HTMN. By employing a discrete Bernoulli random variable (DBRV), the form of measurement likelihood function of double mixing distributions is converted from a weighted sum to an exponential product, and a hierarchical Gaussian state-space model (HGSSM) is therefore established. Finally, the system state, the intermediate random variables (IRVs) of the new STIWG distribution, and the DBRV are simultaneously estimated by utilizing the variational Bayesian (VB) method. Numerical example simulation experiment indicates that the proposed filter in this paper has superior performance than current algorithms in processing ML and HTMN.

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Muhammad, Mustapha, Rashad A. R. Bantan, Lixia Liu, et al. "A New Extended Cosine—G Distributions for Lifetime Studies." Mathematics 9, no. 21 (2021): 2758. http://dx.doi.org/10.3390/math9212758.

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In this article, we introduce a new extended cosine family of distributions. Some important mathematical and statistical properties are studied, including asymptotic results, a quantile function, series representation of the cumulative distribution and probability density functions, moments, moments of residual life, reliability parameter, and order statistics. Three special members of the family are proposed and discussed, namely, the extended cosine Weibull, extended cosine power, and extended cosine generalized half-logistic distributions. Maximum likelihood, least-square, percentile, and Bayes methods are considered for parameter estimation. Simulation studies are used to assess these methods and show their satisfactory performance. The stress–strength reliability underlying the extended cosine Weibull distribution is discussed. In particular, the stress–strength reliability parameter is estimated via a Bayes method using gamma prior under the square error loss, absolute error loss, maximum a posteriori, general entropy loss, and linear exponential loss functions. In the end, three real applications of the findings are provided for illustration; one of them concerns stress–strength data analyzed by the extended cosine Weibull distribution.

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Chaudhary, Arun Kumar, and Laxmi Prasad Sapkota. "New modified Inverted Weibull Distribution: Properties and Applications to COVID-19 Dataset of Nepal." Pravaha 27, no. 1 (2021): 1–12. http://dx.doi.org/10.3126/pravaha.v27i1.50603.

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We have developed a four parameters new modified inverted Weibull distribution and we named it exponentiated exponential inverted Weibull distribution. Linear representation of probability density function, reliability function, hazard function, moments about the origin and its generating function, mean residual life function, order statistics, two entropies namely Renyi and q-Entropy, and mean deviation for the proposed distribution are presented. For the parameter estimation the maximum likelihood, least-square, and Cramer-Von-Mises estimation methods are used. The application of the proposed distribution is analyzed using the deaths case of the COVID-19 dataset of Nepal from 1st April to 14th May 2021.

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Li, Hai-Jian, Jian-Hong Kang, Zhe-Jun Pan, Fu-Bao Zhou, Jin-Chang Deng, and Shuang-Jiang Zhu. "New Adsorption Models for Entirely Describing the Adsorption Isotherm and Heat of Methane in Heterogeneous Nanopore Structures of Coal." Journal of Nanoscience and Nanotechnology 21, no. 1 (2021): 212–24. http://dx.doi.org/10.1166/jnn.2021.18443.

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To understand the adsorption mechanism of methane in heterogeneous nanopore structures of coal, integral adsorption models based on linear, exponential, hyperbolic and quadratic energy distribution functions are established. The adsorption energy domain of the new models is assumed to be a finite interval. These new adsorption models can describe both the adsorption isotherm and the adsorption heat. A volumetric method of adsorption with a microcalorimetry system is used to measure the adsorption isotherms and integral heat, and then the parameters of the new models are obtained by fitting the experimental data. Since the adsorption heat can be different for different adsorption models, it is necessary to fit the adsorption isotherms and heat simultaneously. The fitting results of the adsorption isotherms and heat show that the new models are able to describe the experimental data better than the Langmuir model. By comparing the fitting results and the effective range of adsorption energy of the different adsorption models, it is shown that the exponential energy distribution function is the most reasonable model for methane adsorption in coals, which can be used to evaluate the energetic heterogeneity of nanopores in coal samples. The decreasing exponential energy distributions of three coal samples indicate that a larger adsorption energy corresponds to fewer adsorption sites in the coal samples. The proportion of high adsorption energy is related to the micro-nanopore volume in the coal samples.

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Nassar, Mazen, Refah Alotaibi, Hassan Okasha, and Liang Wang. "Bayesian Estimation Using Expected LINEX Loss Function: A Novel Approach with Applications." Mathematics 10, no. 3 (2022): 436. http://dx.doi.org/10.3390/math10030436.

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The loss function plays an important role in Bayesian analysis and decision theory. In this paper, a new Bayesian approach is introduced for parameter estimation under the asymmetric linear-exponential (LINEX) loss function. In order to provide a robust estimation and avoid making subjective choices, the proposed method assumes that the parameter of the LINEX loss function has a probability distribution. The Bayesian estimator is then obtained by taking the expectation of the common LINEX-based Bayesian estimator over the probability distribution. This alternative proposed method is applied to estimate the exponential parameter by considering three different distributions of the LINEX parameter, and the associated Bayes risks are also obtained in consequence. Extensive simulation studies are conducted in order to compare the performance of the proposed new estimators. In addition, three real data sets are analyzed to investigate the applicability of the proposed results. The results of the simulation and real data analysis show that the proposed estimation works satisfactorily and performs better than the conventional standard Bayesian approach in terms of minimum mean square error and Bayes risk.

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HANYU, Toshiki. "Mathematical model of reverberation decay in a rectangular room with uneven distribution of absorption." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 268, no. 3 (2023): 5987–95. http://dx.doi.org/10.3397/in_2023_0871.

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Based on traditional reverberation theories such as Sabine's or Eyring's theories, the reverberation energy decay becomes linear in logarithmic scale of energy. However, this is valid only in diffuse sound fields. Sound fields in actual rooms are non-diffuse sound field. It has been well known that nonlinear reverberation decay occurs in a rectangular room with uneven distribution of absorption. Hence, reverberation times measured in such rooms are not correspondent with those calculated by the theories. Reverberation theories for such rectangular rooms have been proposed by some researchers. However, these theories have not been sufficiently validated. In this study, a new mathematical model of reverberation decay in rectangular rooms with uneven distributions of absorption is proposed. Most of previous theories contained only exponential decays. The proposed model in this study contains not only exponential decays, but also power law decays.

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Nagy, M., Ehab M. Almetwally, Ahmed M. Gemeay, et al. "The New Novel Discrete Distribution with Application on COVID-19 Mortality Numbers in Kingdom of Saudi Arabia and Latvia." Complexity 2021 (December 24, 2021): 1–20. http://dx.doi.org/10.1155/2021/7192833.

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This paper aims to introduce a superior discrete statistical model for the coronavirus disease 2019 (COVID-19) mortality numbers in Saudi Arabia and Latvia. We introduced an optimal and superior statistical model to provide optimal modeling for the death numbers due to the COVID-19 infections. This new statistical model possesses three parameters. This model is formulated by combining both the exponential distribution and extended odd Weibull family to formulate the discrete extended odd Weibull exponential (DEOWE) distribution. We introduced some of statistical properties for the new distribution, such as linear representation and quantile function. The maximum likelihood estimation (MLE) method is applied to estimate the unknown parameters of the DEOWE distribution. Also, we have used three datasets as an application on the COVID-19 mortality data in Saudi Arabia and Latvia. These three real data examples were used for introducing the importance of our distribution for fitting and modeling this kind of discrete data. Also, we provide a graphical plot for the data to ensure our results.

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Alyami, Salem A., Ibrahim Elbatal, Naif Alotaibi, Ehab M. Almetwally, Hassan M. Okasha, and Mohammed Elgarhy. "Topp–Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data." Applied Sciences 12, no. 20 (2022): 10431. http://dx.doi.org/10.3390/app122010431.

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In this article, a four parameter lifetime model called the Topp–Leone modified Weibull distribution is proposed. The suggested distribution can be considered as an alternative to Kumaraswamy Weibull, generalized modified Weibull, extend odd Weibull Lomax, Weibull-Lomax, Marshall-Olkin alpha power extended Weibull and exponentiated generalized alpha power exponential distributions, etc. The suggested model includes the Topp-Leone Weibull, Topp-Leone Linear failure rate, Topp-Leone exponential and Topp-Leone Rayleigh distributions as a special case. Several characteristics of the new suggested model including quantile function, moments, moment generating function, central moments, mean, variance, coefficient of skewness, coefficient of kurtosis, incomplete moments, the mean residual life and the mean inactive time are derived. The probability density function of the Topp–Leone modified Weibull distribution can be right skewed and uni-modal shaped but, the hazard rate function may be decreasing, increasing, J-shaped, U-shaped and bathtub on its parameters. Three different methods of estimation as; maximum likelihood, maximum product spacing and Bayesian methods are used to estimate the model parameters. For illustrative reasons, applications of the Topp–Leone modified Weibull model to four real data sets related to medical and engineering sciences are provided and contrasted with the fit reached by several other well-known distributions.

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Khan, Sadaf, Oluwafemi Samson Balogun, Muhammad Hussain Tahir, Waleed Almutiry, and Amani Abdullah Alahmadi. "An Alternate Generalized Odd Generalized Exponential Family with Applications to Premium Data." Symmetry 13, no. 11 (2021): 2064. http://dx.doi.org/10.3390/sym13112064.

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In this article, we use Lehmann alternative-II to extend the odd generalized exponential family. The uniqueness of this family lies in the fact that this transformation has resulted in a multitude of inverted distribution families with important applications in actuarial field. We can characterize the density of the new family as a linear combination of generalised exponential distributions, which is useful for studying some of the family’s properties. Among the structural characteristics of this family that are being identified are explicit expressions for numerous types of moments, the quantile function, stress-strength reliability, generating function, Rényi entropy, stochastic ordering, and order statistics. The maximum likelihood methodology is often used to compute the new family’s parameters. To confirm that our results are converging with reduced mean square error and biases, we perform a simulation analysis of one of the special model, namely OGE2-Fréchet. Furthermore, its application using two actuarial data sets is achieved, favoring its superiority over other competitive models, especially in risk theory.

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Caporale, Guglielmo Maria, and Luis Alberiko Gil-Alana. "Exponential Time Trends in a Fractional Integration Model." Econometrics 12, no. 2 (2024): 15. http://dx.doi.org/10.3390/econometrics12020015.

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This paper introduces a new modelling approach that incorporates nonlinear, exponential deterministic terms into a fractional integration framework. The proposed model is based on a specific test on fractional integration that is more general than the standard methods, which allow for only linear trends.. Its limiting distribution is standard normal, and Monte Carlo simulations show that it performs well in finite samples. Three empirical examples confirm that the suggested specification captures the properties of the data adequately.

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Gbenga, Jemilohun Vincent, and Ipinyomi Reuben Adeyemi. "Alpha Power Poisson-G Distribution With an Application to Bur XII Distribution Lifetime Data." International Journal of Statistics and Probability 11, no. 2 (2022): 8. http://dx.doi.org/10.5539/ijsp.v11n2p8.

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We propose a new method of adding two shape parameters to a family of distributions for more flexibility and wider scope of applications called Alpha power Poisson-g distribution. A special case has been considered in details namely; one parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, moment generating function, mean and median deviation, Bonferroni and Lorenz curve, order statistics and expression of the Renyi entropies are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non-linear equations only. Further we consider an extension of the two-parameter Bur XII distribution also, mainly for data analysis purposes. Three data sets have been analyzed to show how the proposed models work in practice. We also carried out Monte Carlo simulation to further investigate the properties of the proposed method of estimation.

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Chui, Charles K., and Amos Ron. "On The Convolution of a Box Spline With a Compactly Supported Distribution: Linear Independence for the Integer Translates." Canadian Journal of Mathematics 43, no. 1 (1991): 19–33. http://dx.doi.org/10.4153/cjm-1991-002-7.

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AbstractThe problem of linear independence of the integer translates of μ * B, where μ is a compactly supported distribution and B is an exponential box spline, is considered in this paper. The main result relates the linear independence issue with the distribution of the zeros of the Fourier-Laplace transform, of μ on certain linear manifolds associated with B. The proof of our result makes an essential use of the necessary and sufficient condition derived in [12]. Several applications to specific situations are discussed. Particularly, it is shown that if the support of μ is small enough then linear independence is guaranteed provided that does not vanish at a certain finite set of critical points associated with B. Also, the results here provide a new proof of the linear independence condition for the translates of B itself.

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Jansen, Cees J. A. "On the calculation of the linear equivalence bias of jump controlled linear finite state machines." Tatra Mountains Mathematical Publications 45, no. 1 (2010): 51–63. http://dx.doi.org/10.2478/v10127-010-0005-x.

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ABSTRACT Jump controlled linear finite state machines were introduced several years ago as building blocks for stream ciphers that can efficiently be implemented in hardware and have intrinsically good side channel resistance. These constructions have found their way in concrete stream cipher designs. The bias in the distribution of linear relations of low degree in the key stream is important for the cryptographic strength of these stream ciphers. Recently, an algorithm was presented by the author to determine this bias. In this paper a new algorithm is introduced, that makes use of the properties of jump registers and has sub exponential order in the degree of the characteristic polynomial of the linear finite state machine.

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Linton, Oliver B. "EFFICIENT ESTIMATION OF GENERALIZED ADDITIVE NONPARAMETRIC REGRESSION MODELS." Econometric Theory 16, no. 4 (2000): 502–23. http://dx.doi.org/10.1017/s0266466600164023.

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We define new procedures for estimating generalized additive nonparametric regression models that are more efficient than the Linton and Härdle (1996, Biometrika 83, 529–540) integration-based method and achieve certain oracle bounds. We consider criterion functions based on the Linear exponential family, which includes many important special cases. We also consider the extension to multiple parameter models like the gamma distribution and to models for conditional heteroskedasticity.

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Hurtado, Paul, and Cameron Richards. "Time Is Of The Essence: Incorporating Phase-Type Distributed Delays And Dwell Times Into ODE Models." Mathematics in Applied Sciences and Engineering 9999, no. 9999 (2020): 1–14. http://dx.doi.org/10.5206/mase/10857.

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Ordinary differential equations models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models to yield much more general ODEs with phase-type distributed delays or dwell times.

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Al-Duais, Fuad S. "Bayesian Estimations under the Weighted LINEX Loss Function Based on Upper Record Values." Complexity 2021 (April 26, 2021): 1–7. http://dx.doi.org/10.1155/2021/9982916.

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The essential objective of this research is to develop a linear exponential (LINEX) loss function to estimate the parameters and reliability function of the Weibull distribution (WD) based on upper record values when both shape and scale parameters are unknown. We perform this by merging a weight into LINEX to produce a new loss function called the weighted linear exponential (WLINEX) loss function. Then, we utilized WLINEX to derive the parameters and reliability function of the WD. Next, we compared the performance of the proposed method (WLINEX) in this work with Bayesian estimation using the LINEX loss function, Bayesian estimation using the squared-error (SEL) loss function, and maximum likelihood estimation (MLE). The evaluation depended on the difference between the estimated parameters and the parameters of completed data. The results revealed that the proposed method is the best for estimating parameters and has good performance for estimating reliability.

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Kharazmi, Omid, Morad Alizadeh, Javier E. Contreras-Reyes, and Hossein Haghbin. "Arctan-Based Family of Distributions: Properties, Survival Regression, Bayesian Analysis and Applications." Axioms 11, no. 8 (2022): 399. http://dx.doi.org/10.3390/axioms11080399.

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In this paper, a new class of the continuous distributions is established via compounding the arctangent function with a generalized log-logistic class of distributions. Some structural properties of the suggested model such as distribution function, hazard function, quantile function, asymptotics and a useful expansion for the new class are given in a general setting. Two special cases of this new class are considered by employing Weibull and normal distributions as the parent distribution. Further, we derive a survival regression model based on a sub-model with Weibull parent distribution and then estimate the parameters of the proposed regression model making use of Bayesian and frequentist approaches. We consider seven loss functions, namely the squared error, modified squared error, weighted squared error, K-loss, linear exponential, general entropy, and precautionary loss functions for Bayesian discussion. Bayesian numerical results include a Bayes estimator, associated posterior risk, credible and highest posterior density intervals are provided. In order to explore the consistency property of the maximum likelihood estimators, a simulation study is presented via Monte Carlo procedure. The parameters of two sub-models are estimated with maximum likelihood and the usefulness of these sub-models and a proposed survival regression model is examined by means of three real datasets.

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Jiang, Yunlu, Jiangchuan Liu, Hang Zou, and Xiaowen Huang. "Model Selection for Exponential Power Mixture Regression Models." Entropy 26, no. 5 (2024): 422. http://dx.doi.org/10.3390/e26050422.

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Finite mixture of linear regression (FMLR) models are among the most exemplary statistical tools to deal with various heterogeneous data. In this paper, we introduce a new procedure to simultaneously determine the number of components and perform variable selection for the different regressions for FMLR models via an exponential power error distribution, which includes normal distributions and Laplace distributions as special cases. Under some regularity conditions, the consistency of order selection and the consistency of variable selection are established, and the asymptotic normality for the estimators of non-zero parameters is investigated. In addition, an efficient modified expectation-maximization (EM) algorithm and a majorization-maximization (MM) algorithm are proposed to implement the proposed optimization problem. Furthermore, we use the numerical simulations to demonstrate the finite sample performance of the proposed methodology. Finally, we apply the proposed approach to analyze a baseball salary data set. Results indicate that our proposed method obtains a smaller BIC value than the existing method.

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Emam, Walid, and Yusra Tashkandy. "The Weibull Claim Model: Bivariate Extension, Bayesian, and Maximum Likelihood Estimations." Mathematical Problems in Engineering 2022 (May 4, 2022): 1–10. http://dx.doi.org/10.1155/2022/8729529.

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Using a class of claim distributions, we introduce the Weibull claim distribution, which is a new extension of the Weibull distribution with three parameters. The maximum likelihood estimation method is used to estimate the three unknown parameters, and the asymptotic confidence intervals and bootstrap confidence intervals are constructed. In addition, we obtained the Bayesian estimates of the unknown parameters of the Weibull claim distribution under the squared error and linear exponential function (LINEX) and the general entropy loss function. Since the Bayes estimators cannot be obtained in closed form, we compute the approximate Bayes estimates via the Markov Chain Monte Carlo (MCMC) procedure. By analyzing the two data sets, the applicability and capabilities of the Weibull claim model are illustrated. The fatigue life of a particular type of Kevlar epoxy strand subjected to a fixed continuous load at a pressure level of 90% until the strand fails data set was analyzed.

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Elshahhat, Ahmed, Abdisalam Hassan Muse, Omer Mohamed Egeh, and Berihan R. Elemary. "Estimation for Parameters of Life of the Marshall-Olkin Generalized-Exponential Distribution Using Progressive Type-II Censored Data." Complexity 2022 (October 3, 2022): 1–36. http://dx.doi.org/10.1155/2022/8155929.

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A new three-parameter extension of the generalized-exponential distribution, which has various hazard rates that can be increasing, decreasing, bathtub, or inverted tub, known as the Marshall-Olkin generalized-exponential (MOGE) distribution has been considered. So, this article addresses the problem of estimating the unknown parameters and survival characteristics of the three-parameter MOGE lifetime distribution when the sample is obtained from progressive type-II censoring via maximum likelihood and Bayesian approaches. Making use of the s-normality of classical estimators, two types of approximate confidence intervals are constructed via the observed Fisher information matrix. Using gamma conjugate priors, the Bayes estimators against the squared-error and linear-exponential loss functions are derived. As expected, the Bayes estimates are not explicitly expressed, thus the Markov chain Monte Carlo techniques are implemented to approximate the Bayes point estimates and to construct the associated highest posterior density credible intervals. The performance of proposed estimators is evaluated via some numerical comparisons and some specific recommendations are also made. We further discuss the issue of determining the optimum progressive censoring plan among different competing censoring plans using three optimality criteria. Finally, two real-life datasets are analyzed to demonstrate how the proposed methods can be used in real-life scenarios.

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Oramulu, Dorathy O., Chinyere P. Igbokwe, Ifeanyi C. Anabike, Harrison O. Etaga, and Okechukwu J. Obulezi. "Simulation Study of the Bayesian and Non-Bayesian Estimation of a new Lifetime Distribution Parameters with Increasing Hazard Rate." Asian Research Journal of Mathematics 19, no. 9 (2023): 183–211. http://dx.doi.org/10.9734/arjom/2023/v19i9711.

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In this paper, a new distribution known as the Shifted Chris-Jerry (SHCJ) distribution is proposed. The proposition is motivated by the need to compare the efficiency of various classical estimation methods as well as the bayesian estimation using gamma prior at linear-exponential loss, squared error loss and generalized entropy loss functions. Some useful mathematical properties are derived. Single acceptance sampling plans (SASPs) are created for the distribution when the life test is truncated at a predetermined period. The median lifetime of the SHCJ distribution with pre-defined constants is taken as the truncation time. To guarantee that the specific life test is obtained at the defined risk to the user, the minimum sample size is required. For a particular consumer’s risk, the SHCJ distribution’s parameters, and the truncation time including numerical results are obtained. A simulation study is carried out for the bayesian and non-bayesian estimation of the parameters. Data on blood cancer patients is used to demonstrate the usefuleness of the proposed distribution.

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Spanos, Aris. "On Modeling Heteroskedasticity: The Student's t and Elliptical Linear Regression Models." Econometric Theory 10, no. 2 (1994): 286–315. http://dx.doi.org/10.1017/s0266466600008422.

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This paper proposes a new approach to modeling heteroskedastidty which enables the modeler to utilize information conveyed by data plots in making informed decisions on the form and structure of heteroskedasticity. It extends the well-known normal/linear/homoskedastic models to a family of non-normal/linear/heteroskedastic models. The non-normality is kept within the bounds of the elliptically symmetric family of multivariate distributions (and in particular the Student's t distribution) that lead to several forms of heteroskedasticity, including quadratic and exponential functions of the conditioning variables. The choice of the latter family is motivated by the fact that it enables us to model some of the main sources of heteroskedasticity: “thicktails,” individual heterogeneity, and nonlinear dependence. A common feature of the proposed class of regression models is that the weak exogeneity assumption is inappropriate. The estimation of these models, without the weak exogeneity assumption, is discussed, and the results are illustrated by using cross-section data on charitable contributions.

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Hassan, Amal S., Ehab M. Almetwally, Samia C. Gamoura, and Ahmed S. M. Metwally. "Inverse Exponentiated Lomax Power Series Distribution: Model, Estimation, and Application." Journal of Mathematics 2022 (June 20, 2022): 1–21. http://dx.doi.org/10.1155/2022/1998653.

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In this paper, we introduce the inverse exponentiated Lomax power series (IELoPS) class of distributions, obtained by compounding the inverse exponentiated Lomax and power series distributions. The IELoPS class contains some significant new flexible lifetime distributions that possess powerful physical explications applied in areas like industrial and biological studies. The IELoPS class comprises the inverse Lomax power series as a new subclass as well as several new flexible compounded lifetime distributions. For the proposed class, some characteristics and properties are derived such as hazard rate function, limiting behavior, quantile function, Lorenz and Bonferroni curves, mean residual life, mean inactivity time, and some measures of information. The methods of maximum likelihood and Bayesian estimations are used to estimate the model parameters of one optional model. The Bayesian estimators of parameters are discussed under squared error and linear exponential loss functions. The asymptotic confidence intervals, as well as Bayesian credible intervals, of parameters, are constructed. Simulations for a one-selective model, say inverse exponentiated Lomax Poisson (IELoP) distribution, are designed to assess and compare different estimates. Results of the study emphasized the merit of produced estimates. In addition, they appeared the superiority of Bayesian estimate under regarded priors compared to the corresponding maximum likelihood estimate. Finally, we examine medical and reliability data to demonstrate the applicability, flexibility, and usefulness of IELoP distribution. For the suggested two real data sets, the IELoP distribution fits better than Kumaraswamy–Weibull, Poisson–Lomax, Poisson inverse Lomax, Weibull–Lomax, Gumbel–Lomax, odd Burr–Weibull–Poisson, and power Lomax–Poisson distributions.

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Chinedu, Eberechukwu Q., Queensley C. Chukwudum, Najwan Alsadat, Okechukwu J. Obulezi, Ehab M. Almetwally, and Ahlam H. Tolba. "New Lifetime Distribution with Applications to Single Acceptance Sampling Plan and Scenarios of Increasing Hazard Rates." Symmetry 15, no. 10 (2023): 1881. http://dx.doi.org/10.3390/sym15101881.

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This article is an extension of the Chris-Jerry distribution (C-JD) in that a two-parameter Chris-Jerry distribution (TPCJD) is suggested and its characteristics are studied. Based on the determined domain of attraction and other major statistical properties, the proposed TPCJD seems to fit into the Gumbel domain. Additionally, it has been confirmed that the stress strength is reliable. The tail study suggests that the TPCJD’s substantial tail makes it suited for a range of applications. The study took into account the single acceptance sampling approach using both simulation and real-life situations. The parameters of the TPCJD were estimated by some classical and Bayesian approaches. The mean squared errors (MSE), linear-exponential, and generalized entropy loss functions were deployed to obtain the Bayesian estimators aided by the Markov chain Monte Carlo (MCMC) simulation. An analysis of lifetime data on two events justified the use of the proposed distribution after comparing the results with some standard lifetime models.

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Sengupta, Bhaskar. "Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue." Advances in Applied Probability 21, no. 1 (1989): 159–80. http://dx.doi.org/10.2307/1427202.

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This paper is concerned with a bivariate Markov process {Xt, Nt; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

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