scholarly journals A Bimodal Extension of the Exponential Distribution with Applications in Risk Theory

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 679
Author(s):  
Jimmy Reyes ◽  
Emilio Gómez-Déniz ◽  
Héctor W. Gómez ◽  
Enrique Calderín-Ojeda
Keyword(s):  
Exponential Distribution ◽  
Hazard Rate ◽  
Rate Function ◽  
Estimation Methods ◽  
Inverse Gaussian ◽  
New Family ◽  
Tail Distribution ◽  
Zero Values ◽  
The Mean ◽  
Positive Support

There are some generalizations of the classical exponential distribution in the statistical literature that have proven to be helpful in numerous scenarios. Some of these distributions are the families of distributions that were proposed by Marshall and Olkin and Gupta. The disadvantage of these models is the impossibility of fitting data of a bimodal nature of incorporating covariates in the model in a simple way. Some empirical datasets with positive support, such as losses in insurance portfolios, show an excess of zero values and bimodality. For these cases, classical distributions, such as exponential, gamma, Weibull, or inverse Gaussian, to name a few, are unable to explain data of this nature. This paper attempts to fill this gap in the literature by introducing a family of distributions that can be unimodal or bimodal and nests the exponential distribution. Some of its more relevant properties, including moments, kurtosis, Fisher’s asymmetric coefficient, and several estimation methods, are illustrated. Different results that are related to finance and insurance, such as hazard rate function, limited expected value, and the integrated tail distribution, among other measures, are derived. Because of the simplicity of the mean of this distribution, a regression model is also derived. Finally, examples that are based on actuarial data are used to compare this new family with the exponential distribution.

scholarly journals [0, 1] truncated fréchet-gamma and inverted gam-ma distributions

2017 ◽  
Vol 5 (2) ◽  
pp. 151
Author(s):  
Salah Abid ◽  
Russul K. Abdulrazak
Keyword(s):  
Hazard Rate ◽  
Cumulative Distribution Function ◽  
Reliability Function ◽  
Rate Function ◽  
Cumulative Distribution ◽  
Design Practice ◽  
New Family ◽  
Special Cases ◽  
The Mean ◽  
Stress Models

In this paper, we introduce a new family of continuous distributions based on [0, 1]] truncated Fréchet distribution. [0, 1]] Truncated Fréchet Gamma ([0, 1]] TFG) and truncated Fréchet inverted Gamma ([0, 1]] TFIG) distributions are discussed as special cases. The cumulative distribution function, the rth moment, the mean, the variance, the skewness, the kurtosis, the mode, the median, the characteristic function, the reliability function and the hazard rate function are obtained for the distributions under consideration. It is well known that an item fails when a stress to which it is subjected exceeds the corresponding strength. In this sense, strength can be viewed as "resistance to failure." Good design practice is such that the strength is always greater than the expected stress. The safety factor can be defined in terms of strength and stress as strength/stress. So, the [0, 1]] TFG strength-stress and the [0, 1]] TFIG strength-stress models with different parameters will be derived here. The Shannon entropy and Relative entropy will be derived also.

scholarly journals Sharp bounds for exponential approximations under a hazard rate upper bound

2015 ◽  
Vol 52 (03) ◽  
pp. 841-850 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Exponential Distribution ◽  
Hazard Rate ◽  
Residual Life ◽  
Continuous Distribution ◽  
Rate Function ◽  
Mean Residual Life ◽  
Birth And Death Processes ◽  
Absolutely Continuous ◽  
Sharp Bounds ◽  
Life Distributions

Consider an absolutely continuous distribution on [0, ∞) with finite meanμand hazard rate functionh(t) ≤bfor allt. Forbμclose to 1, we would expectFto be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance betweenFand an exponential distribution with meanμ, as well as betweenFand an exponential distribution with failure rateb. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

scholarly journals A Mathematical WGED – Model Approach on Short-Term High in Density Exercise Training, Attenuated Acute Exercise – Induced Growth Hormone Response

2019 ◽  
Vol 8 (1) ◽  
pp. 1-5
Author(s):  
M. Kaliraja ◽  
K. Perarasan
Keyword(s):  
Growth Hormone ◽  
Exponential Distribution ◽  
Hazard Rate ◽  
Acute Exercise ◽  
Survival Function ◽  
Life Time ◽  
Rate Function ◽  
Hazard Rate Function ◽  
Short Term ◽  
Exercise Induced

In the current manuscript, we have demonstrated the recent generalization of Weibull-G exponential distribution (three-parameter) and it is a very familiar distribution as compared to other distribution.It has been found that Weibull-G exponential distribution (WGED) can be utilized pretty efficiently to evaluate the biological data in the position of gamma and log-normal Weibull distributions. It has two shape parameters and the three scale parameters namely, a, b, λ. Some of its statistical properties are acquired, which includes reserved hazard function, probability-density function, hazard-rate function and survival function. Our aim is to shore-up the results of life-time using three-parameter Weibull generalized exponential distribution. Hence, the corresponding probability functions, hazard-rate function, survival function as well as reserved hazard-rate function has been analyzed in the 3 weeks of high-intensity exercise training in short-term. The outcomes of the present study supporting the results of life-time data that the interim elevated intensity exercise activity attenuated an acute exercise induced growth hormone release.

scholarly journals The Availability of Systems with Bathtub Hazard Rate Function

2018 ◽  
Vol 15 ◽  
pp. 8162-8173 ◽  
Author(s):  
Dr. Mohamad Yousef Ashkar
Keyword(s):  
Exponential Distribution ◽  
Hazard Rate ◽  
Rate Function ◽  
Vital Role ◽  
Statistical Distribution ◽  
Repair Time ◽  
Hazard Rate Function ◽  
Normal Life

In our normal life we can see that the most realistic systems possess useful time governed by hazard rateof bathtub shaped. The hazard rate function, however, plays a vital role in the computation of theavailability function. The repair time, however, could be modeled as any statistical distribution. In thispaper I will investigate the nature of availability function and points of availability of systems with bathtubhazard function and exponential distribution repair time using Markovian method.

scholarly journals A New Compound Gamma and Lindley Distribution with Application to Failure Data

2019 ◽  
Vol 48 (3) ◽  
pp. 54-75
Author(s):  
Mousa Abdi ◽  
Akbar Asgharzadeh ◽  
Hassan S. Bakouch ◽  
Zahra Alipour
Keyword(s):  
Hazard Rate ◽  
Goodness Of Fit ◽  
Stochastic Order ◽  
Rate Function ◽  
Estimation Methods ◽  
Strength Parameter ◽  
Data Sets ◽  
Failure Data ◽  
Distribution Parameters ◽  
Stochastic Order Relations

 In this paper, we propose a new lifetime distribution by compounding the gamma and Lindley distributions. Construction of it can be interpreted in the viewpoint of the reliability analysis and Bayesian inference. Moreover, the distribution has decreasing and unimodal hazard rate shapes. Several properties of the distribution are obtained, involving characteristics of the (reverse) hazard rate function, quantiles, moments, extreme order statistics and some stochastic order relations. Estimating the distribution parameters is discussed by some estimation methods and their performance is evaluated by a simulation study. Also, estimation of the stress-strength parameter is investigated. Usefulness of the distribution among other models is illustrated by fitting two failure data sets and using some goodness-of-fit measures.

scholarly journals Modeling Extreme Values Utilizing an Asymmetric Probability Function

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1730
Author(s):  
Mohammed M. A. Almazah ◽  
Muqrin A. Almuqrin ◽  
Mohamed. S. Eliwa ◽  
Mahmoud El-Morshedy ◽  
Haitham M. Yousof
Keyword(s):  
Hazard Rate ◽  
Extreme Values ◽  
Residual Life ◽  
Real Data ◽  
Moment Generating Function ◽  
Rate Function ◽  
Estimation Methods ◽  
Model Parameters ◽  
Hazard Rate Function ◽  
Data Application

In this article, a new flexible probability density function with three parameters is proposed for modeling asymmetric data (positive and negative) with different types of kurtosis (mesokurtic, leptokurtic and platykurtic). Some of its statistical and reliability properties, including hazard rate function, moments, moment generating function, incomplete moments, mean deviations, moment of the residual life, moment of the reversed residual life, and order statistics are derived. Its hazard rate function can be either constant, increasing-constant, decreasing-constant, U shape, upside down shape or upside down-U shape. Seven classical estimation methods are considered to estimate the unknown model parameters. Monte Carlo simulation experiments are performed to compare the performance of the seven different estimation methods. Finally, a distinctive asymmetric real data application is analyzed for illustrating the flexibility of the new model.

scholarly journals Pseudo-Lindley Family of Distributions and its Properties

2021 ◽  
pp. 35-49
Author(s):  
U. U. Uwadi ◽  
E. E. Nwezza
Keyword(s):  
Hazard Rate ◽  
Maximum Likelihood Method ◽  
Quantile Function ◽  
Rate Function ◽  
Likelihood Method ◽  
Parameter Estimates ◽  
New Family ◽  
The Family ◽  
Renyi’S Entropy ◽  
Family Of Distributions

In this study, we proposed a family of distribution called the Pseudo Lindley family of distributions. The limiting behaviors of the density and hazard rate function of the new family are examined. Statistical properties of the proposed family of distributions derived include quantile function, moments, order statistics, and Renyi’s entropy. The maximum likelihood method was employed in obtaining the parameter estimates of the Pseudo Lindley family of distribution. Bivariate extension of the proposed family is discussed. Some special members of the family are obtained. The shape of the density function of special members could be unimodal, bathtub shaped, increasing and decreasing. 

scholarly journals Sharp bounds for exponential approximations under a hazard rate upper bound

2015 ◽  
Vol 52 (3) ◽  
pp. 841-850 ◽  
Author(s):  
Mark Brown
Keyword(s):  
Exponential Distribution ◽  
Hazard Rate ◽  
Residual Life ◽  
Continuous Distribution ◽  
Rate Function ◽  
Mean Residual Life ◽  
Birth And Death Processes ◽  
Absolutely Continuous ◽  
Sharp Bounds ◽  
Life Distributions

Consider an absolutely continuous distribution on [0, ∞) with finite mean μ and hazard rate function h(t) ≤ b for all t. For bμ close to 1, we would expect F to be approximately exponential. In this paper we obtain sharp bounds for the Kolmogorov distance between F and an exponential distribution with mean μ, as well as between F and an exponential distribution with failure rate b. We apply these bounds to several examples. Applications are presented to geometric convolutions, birth and death processes, first-passage times, and to decreasing mean residual life distributions.

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