Inverse Exponential Distribution Research Articles

Statistical inference of entropy functions of generalized inverse exponential model under progressive type-II censoring test.

This article explores the estimation of Shannon entropy and Rényi entropy based on the generalized inverse exponential distribution under the condition of stepwise Type II truncated samples. Firstly, we analyze the maximum likelihood estimation and interval estimation of Shannon entropy and Rényi entropy for the generalized inverse exponential distribution. In this process, we use the bootstrap method to construct confidence intervals for Shannon entropy and Rényi entropy. Next, we select the gamma distribution as the prior distribution and apply the Lindley approximation algorithm to calculate `estimates of Shannon entropy and Rényi entropy under different loss functions including Linex loss function, entropy loss function, and DeGroot loss function respectively. Afterwards, simulation is used to calculate estimates and corresponding mean square errors of Shannon entropy and Rényi entropy in GIED model. The research results show that under DeGroot loss function, estimation accuracy of Shannon entropy and Rényi entropy for generalized inverse exponential distribution is relatively high, overall Bayesian estimation performs better than maximum likelihood estimation. Finally, we demonstrate effectiveness of our estimation method in practical applications using a set of real data.

Inverse unit exponential probability distribution: Classical and Bayesian inference with applications

This article examines the new inverse unit exponential distribution, utilizing both classical and Bayesian methodologies; it begins by presenting the general properties of the proposed model, highlighting characteristic features, such as the presence of a reverse-J or increasing and inverted bathtub-shaped hazard rate function. Furthermore, it explores various statistical properties of the suggested distribution. It employs 12 methods to estimate the associated parameters. A Monte Carlo simulation is conducted to evaluate the accuracy of the estimation procedure. Even for small samples, the results indicate that biases and mean square errors decrease as the sample size increases, demonstrating the robustness of the estimation method. The application of the suggested distribution to real datasets is then discussed. Empirical evidence, using model selection criteria and goodness-of-fit test statistics, supports the assertion that the suggested model outperforms several existing models considered in the study. This article extends its analysis to the Bayesian framework. Using the Hamiltonian Monte Carlo with the no U-turn sampling algorithm, 8000 real samples are generated. The convergence assessment reveals that the chains are convergent and the samples are independent. Subsequently, using the posterior samples, the parameters of the proposed model are estimated, and credible intervals and highest posterior density intervals are computed to quantify uncertainty. The applicability of the suggested model to real data under both classical and Bayesian methodologies provides insights into its statistical properties and performance compared to existing models.

Some Properties and Applications of the Gompertz Topp-Leone Inverse Exponential Distribution

The Gompertz Topp Leone Inverse Exponential (GoTLIE) distribution, which has three parameters, is a novel distribution that we introduce in this article. Inverse exponential (IE) distributions with various parameter values can be combined to form the new distribution. The moments, moment generating function, quantile function, reliability function, and hazard function are only a few of the crucial structural aspects of the new model that are derived. We utilize the approach of greatest likelihood to estimate the new distribution's parameters (MLE). To show how flexible the new distribution is, we utilized two real data used to demonstrate the GoTLIE distribution.

A new Topp-Leone Kumaraswamy Marshall-Olkin generated family of distributions with applications

We aim in this paper to propose a novel class of distributions that was created by merging the Topp-Leone distribution and the Generated families of Kumaraswamy and Marshall-Olkin. Its cumulative distribution function characterizes it and includes rational and polynomial functions. In particular, the following desirable properties of the new family are presented: Shannon entropy, order statistics, the quantile power series, and several associated measures and functions. Then, using a specific family member identified before, we create a parametric statistical model with the basic distribution being the inverse exponential distribution. Finally, a thorough investigation has been made to implement this new distribution with three data sets: the glass fibers data set, the glass Alumina data set and the hailing times data set. In comparison to six prominent competitors, the new model performs favorably on all statistical tests and criteria that were examined.

A novel family of distributions: Properties, inequality measures and applications to socio economic development indicators

In this paper, we focused on two families of distributions: the Topp–Leone Kumaraswamy family and a novel proposed family of distributions. Subsequently, we explore their composition, leading to a novel family of distributions exhibiting compelling features for data modeling. Specifically, we examine a special member of this novel family, employing the inverse exponential distribution as the cumulative density function. We establish the mathematical properties, investigate the moments and the stochastic properties, and propose a parameter estimation method based on the maximum likelihood of the new model. To assess the applicability of our model, we gather data related to development indicators in Benin Republic. Additionally, employing competing models, we analyze some real-life data and compare the results to the novel distribution. Model performance is evaluated in terms of fitting observed data, and we conduct an in-depth interpretation of the outcomes. This study makes a significant contribution by introducing a novel family of distributions tailored for modeling development indicators. The findings of this research may have substantial implications for statistical analysis and decision-making in the context of Benin’s economic and social development.

A novel extended inverse‐exponential distribution and its application to COVID‐19 data

AbstractThe aim of this article is to define a new flexible statistical model to examine the COVID‐19 data sets that cannot be modeled by the inverse exponential distribution. A novel extended distribution with one scale and three shape parameters is proposed using the generalized alpha power family of distributions to derive the generalized alpha power exponentiated inverse exponential distribution. Some important statistical properties of the new distribution such as the survival function, hazard function, quantile function, moment, Rényi entropy, and order statistics are all derived. The method of maximum likelihood estimation is used to estimate the parameters of the new distribution. The performance of the estimators are assessed through Monte Carlo simulation, which shows that the maximum likelihood method works well in estimating the parameters. The GAPEIEx distribution was applied to COVID‐19 data sets in order to access the flexibility and adaptability of the distribution, and it happens to perform better than its submodels and other well‐known distributions.

A versatile model for lifetime of a component under stress

In this study, a versatile model, called alpha-monotone inverse Weibull distribution (alphaIW), for lifetime of a component under stress is introduced by using the alpha-monotone concept. The alphaIW distribution is also expressed as a scale-mixture between the inverse Weibull distribution and uniform distribution on (0, 1). The alphaIW distribution includes alpha-monotone inverse exponential and alpha-monotone inverse Rayleigh distributions as submodels and converenges to the inverse Weibull, inverse exponential, and inverse Rayleigh distributions as limiting cases. Also, slash Weibull, slash Rayleigh, and slash exponential distribuitons can be obtained under certain variable transformation and parameter settings. The alphaIW distribution is characterized by its hazard rate function and characterizing conditions are provided as well. Maximum likelihood, maximum product of spacing, and least squares methods are used to estimate distribution parameters. A Monte-Carlo simulation study is conducted to compare the efficiencies of the considered estimation methods. In the application part, two practical data sets, Kevlar 49/epoxy and Kevlar 373/epoxy, are modeled via the alphaIW distribution. Modeling performance of the alphaIW distribution is compared with its rivals by means of some well-known goodness-of-fit statistics and results show that alphaIW distribution performs better modeling than them. Results of comparison also indicate that obtaining the alphaIW distribution by using the alpha-monotone concept is cost effective since the new shape parameter added to the distribution by using the alpha-monotone concept significantly increases the modeling capability of the IW distribution. As a result of this study, it is shown that the alphaIW distribution can be an alternative to the well-known and recently-introduced distributions for modeling purposes.

On the Distribution of the Product of Inverse Pareto and Exponential Random Variables

This article considers Inverse Pareto and Exponential distributions to create the distribution of the product. The researchers derived its properties, such as; survival functions and hazard functions, and used the model criterion such as Sum Square Error (SSE), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) in estimating the parameters for deriving the best joint distribution between monthly precipitation and temperature in the Philippines from 1974 to 2013. The results showed that considering the monthly precipitation and temperature data, the distribution of the product of Inverse Pareto and Exponential outperformed the other existing distribution of the product.

Discrete Odd Inverse Pareto Exponential Distribution: Properties, Estimation and Applications

In this paper, a new discrete distribution named as the discrete odd inverse Pareto exponential (DOIPEx) distribution is introduced for modeling count data. The proposed distribution can be viewed as a generalization of the discrete Marshall-Olkin exponential, discrete exponentiate exponential, and discrete exponential distributions. Some basic distributional properties, quantile function, hazard and reversed hazard functions, and order statistics are derived. Estimation of the parameters is illustrated using the maximum likelihood method (MLE), and three real data sets are discussed to demonstrate the usefulness and applicability of the DOIPEx distribution.

New Lomax-G family of distributions: Statistical properties and applications

This research article introduces a new family of distributions developed using the innovative beta-generated transformation technique. Among these distributions, the focus is on the inverse exponential power distribution, which exhibits unique reverse-J, inverted bathtub, or monotonically increasing hazard functions. This paper thoroughly investigates the distribution’s key characteristics and utilizes the maximum likelihood estimation method to determine its associated parameters. To assess the accuracy of the estimation procedure, the researchers conducted a simulation experiment, revealing diminishing biases and mean square errors with increasing sample sizes, even when working with small samples. Moreover, the practical applicability of the proposed distribution is demonstrated by analyzing real-world COVID-19 and medical datasets. The article establishes that the proposed model outperforms existing models by using model selection criteria and conducting goodness-of-fit test statistics. The potential applications of this research extend to various fields where modeling and analyzing hazard functions or survival data are crucial. Additionally, the study contributes to advancing probability theory and statistical inferences.

Bayesian Approach for estimating the unknown Scale parameter of Erlang Distribution Based on General Entropy Loss Function

We are used Bayes estimators for unknown scale parameter when shape Parameter is known of Erlang distribution. Assuming different informative priors for unknown scale parameter. We derived The posterior density with posterior mean and posterior variance using different informative priors for unknown scale parameter which are the inverse exponential distribution, the inverse chi-square distribution, the inverse Gamma distribution, and the standard Levy distribution as prior. And we derived Bayes estimators based on the general entropy loss function (GELF) is used the Simulation method to obtain the results. we generated different cases for the parameters of the Erlang model, for different sample sizes. The estimates have been compared in terms of their mean-squared error (MSE). We concluded that the best estimators of the scale parameterof the Erlang distribution, based on GELF for the shape parameter (c=1,2,3) under inverse gamma prior with for all samples sizes(n) where the true cases of the Erlang model are and according to the smallest values of MSE

The Cubic Rank Transmuted Inverse Exponential Distribution

This paper uses the recently proposed cubic rank transmutation technique to develop and propose a new distribution in the transmutation family called the Cubic Rank Transmuted Inverse Exponential distribution (CRTIE). Important mathematical properties of the new distribution including Quantile function, and Reliability analysis were derived. The estimates of the parameters of the new distribution were also derived.

Gamma Generalized Extended Inverse Exponential Distribution: A Novel Distribution in Modeling COVID-19 Cases in Nigerian Africa: A Personal Perspective

In this study, we developed a novel distribution called Gamma Inverse Exponential (GIE) distribution, which has proved to be a more flexible distribution in modeling COVID-19 case fatality in Nigeria. We studied some statistical properties of the new distribution, which include: moments, incomplete moments, quantile function, Renyi entropy, and mean deviation. A real-life data application to a number of reported cases of COVID-19 infection between March 2019 to 2021 shows that the GIE distribution has a better fit than some competing distributions in fitting the data. Time series analysis of the COVID-19 data is also considered.

Ribonucleic acid genome mutations induced by the Casimir effect

In this paper, we investigate the Casimir effect within a virus RNA, particularizing the study to the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Then, we discuss the possibility of occurring damage or mutation in its genome due to the presence of quantum vacuum fluctuations inside and around the RNA ribbon. For this, we consider the geometry and the nontrivial topology of the viral RNA as having a simple helical structure. We initially compute the non-thermal Casimir energy associated to that geometry, considering boundary conditions that constrain the zero point oscillations of a massless scalar field to the cylindrical cavity containing a helix pitch of RNA ribbon. Then we extend the obtained result to the electromagnetic field and, following, we calculate the probability of occurring damage or mutation in RNA by using the normalized inverse exponential distribution, which suppresses very low energies, and consider cutoff (threshold) energies corresponding to UV-A and UV-C rays, surely responsible by mutations. Then, by taking into account UV-A, we arrive at a mutation rate per base per infection cycle, which in the case of the SARS-CoV-2 is non-negligible. We find a maximum value of this mutation rate for an RNA ribbon radius, applying it for SARS-CoV-2, in particular. We also calculate a characteristic longitudinal oscillation frequency for the helix pitch value corresponding to the local minimum of the Casimir energy. Finally, we consider thermal fluctuations of classical and quantum nature and show that the corresponding probability of mutation is completely negligible for that virus. Therefore, we conclude that only the nontrivial topology and the geometric attributes of the RNA molecule contribute to the possible mutations caused by quantum vacuum fluctuations in the viral genome.

Analysis of ℝ=P[Y<X<Z] Using Ranked Set Sampling for a Generalized Inverse Exponential Model

In many real-world situations, systems frequently fail due to demanding operating conditions. In particular, when systems reach their lowest, highest, or both extremes operating conditions, they usually fail to accomplish their intended functions. This study considers estimating the stress–strength reliability, for a component with a strength (X) that is independent of the opposing lower bound stress (Y) and upper bound stress (Z). We assumed that the strength and stress random variables followed a generalized inverse exponential distribution with different shape parameters. Under ranked set sampling (RSS) and simple random sampling (SRS) designs, we obtained four reliability estimators using the maximum likelihood method. The first and second reliability estimators were deduced when the sample data of the strength and stress distributions used the sample design (RSS/SRS). The third reliability estimator was determined when the sample data for Y and Z were received from the RSS and the sample data for X were taken from the SRS. The fourth reliability estimator was derived when the sample data of Y and Z were selected from the SRS, while the sample data of X were taken from the RSS. The accuracy of the suggested estimators was compared using a comprehensive computer simulation. Lastly, three real data sets were used to determine the reliability.

Inverse Exponentiated Odd Lomax Exponential Distribution: Properties and Applications

Background: New family of distributions has important functions in generalization of distributions by modifying some existing distributions for getting more flexible irrespective to applied and practical view point. The Inverse Exponentiated Odd Lomax Exponential Distribution (IEOLE) having four parameters is suggested. Proposed model is based on T-X family of distribution which is the extended form of beta-generated distribution. Based on the LSE, MLE, and CVM methods, the parameters of the proposed distribution are estimated. Different model validation criteria and model comparisons are done by considering other existing models.
 Materials and methods: IEOLE is compound distribution derived by using theoretical concept of Odd Lomax Exponential distribution and T-X family of distribution. The parameters of the proposed distribution are estimated through the least square, Cramer–Von Mises and maximum likelihood methods. The applicability of the proposed model is evaluated using R programming on two real-life time data sets.
 Results: The statistical properties and different characteristics like the hazard rate function, the cumulative distribution function, quantile function, skewness, and kurtosis of the proposed model are discussed. Box plots, TTT plot, density fits etc. shows that the proposed model fits better to considered two real data sets. Different model validation criteria such as AIC, BIC, and CAIC are obtained and compared with some existing well-known probability distributions.
 Conclusion: This study presents new probability model called Inverse Exponentiated Odd Lomax Exponential distribution. The density curves of the model show that it is unimodal having right skewed. The suggested model has proven to be versatile for modeling real-world data due to its increasing-decreasing, right-skewed form. Also, the hazard rate function (HRF) graph is decreasing, decreasing-increasing or inverted bathtub shaped according to the value of the model constants. Different validation criteria show that the suggested model fits data well and the goodness-of-fit shows that proposed model has lesser test statistic value and higher p- value with respect to some existing models.

Inverse Exponentiated Odd Lomax Exponential Distribution: Properties and Applications

Mentioned in manuscript abstract.

Carr Memorial: Maximum Drawdown Derivatives to a Hitting Time

Drawdown is defined as the amount a portfolio has decreased from its running maximum. Drawdown has become ensconced in finance practice with some hedge funds shutting down portfolio managers who reach a certain drawdown limit. In this article, we show that, for every continuous local martingale that hits a given point <i>m</i> with probability 1, the running maximum of drawdown at the time of hitting <i>m</i> has the same inverse exponential distribution. We then derive prices and hedge ratios for binary calls on maximum absolute and relative drawdown maturing at the hitting time for <i>m</i>. We also derive prices for call spreads on maximum drawdown at the hitting time for <i>m</i>. These prices and hedge ratios are model independent across all continuous arbitrage-free stochastic processes that, with probability 1, either hit <i>m</i> or reach a drawdown equal to the strike price. This includes stochastic volatility models whose volatility is bounded away from 0 before hitting <i>m</i> or the strike. These results are both simpler and more general than prior work, which, while allowing for a fixed maturity, require infinite series representations, the use of complex derivatives to hedge and greater restrictions on the stochastic process. The key fact that facilitates our form of model independence is that the values of the derivatives at maturity are invariant to time changes.

On the Coronavirus Disease Death Rate Modeling Utilizing Generalized Exponential Kumaraswamy

This paper aims to model COVID-19 in different countries, including Iran, Canada, Italy, and Mexico. A novel five-parameter lifetime distribution termed the Odd generalized exponential Kumaraswamy-inverse exponential distribution (OEKIE) is presented by combining the Kumaraswamy-inverse exponential distribution with the odd generalized exponential generator. The theoretical features of the new distribution, as well as its reliability functions, moments, and order statistics are investigated. The novel distribution has numerous advantages, since its density has a variety of symmetric and asymmetric forms. Furthermore, The graphs of the hazard rate function exhibit various asymmetrical shapes such as decreasing, increasing, upside-down bathtub shapes, and inverted J-shapes making OEKIE suitable for modeling hazards behaviors more likely to be observed in practical settings like human mortality, and biological applications. The OEKIE’s parameters are estimated using the maximum likelihood approach. The effectiveness of OEKIE is demonstrated through both numerical study and applications to four COVID-19 mortality rate data sets. The OEKIE provides best fits to COVID-19 data compared to other extended forms of the Kumaraswamy and inverse exponential distributions which may attract wider applications in different fields.

Modeling COVID-19 mortality data in four countries using odd generalized exponential Kumaraswamy-Inverse exponential distribution

This study aims to introduce an optimum model to assess the COVID-19 death rate in Saudi Arabia, Canada, Italy, and Mexico. A novel five-parameter lifetime distribution termed the Odd generalized exponential Kumaraswamy-inverse exponential distribution is presented by combining the Kumaraswamy-inverse exponential distribution with the odd generalized exponential generator. The theoretical features of the new distribution, as well as its reliability functions, moments, and order statistics are investigated. The odd generalized exponential Kumaraswamy-inverse exponential distribution is of special importance since its density has a variety of symmetric and asymmetric forms. Furthermore, the graphs of the hazard rate function exhibit various asymmetrical shapes such as decreasing, increasing, and upside-down bathtub shapes, and inverted J-shapes making The Odd generalized exponential Kumaraswamy-inverse exponential distribution suitable for modeling hazards behaviors more likely to be observed in practical settings like human mortality, and biological applications. The proposed distribution parameters are estimated using the maximum likelihood approach and its effectiveness is demonstrated through both numerical study and applications to four COVID-19 mortality rate data sets. The Odd generalized exponential Kumaraswamy-inverse exponential distribution provides the best fit to COVID-19 data compared to other extended forms of the Kumaraswamy and inverse exponential distributions which may attract wider applications in different fields.