scholarly journals About the Stein equation for the generalized inverse Gaussian and Kummer distributions

2020 ◽  
Vol 24 ◽  
pp. 607-626
Author(s):  
Essomanda Konzou ◽  
Angelo Efoevi Koudou
Keyword(s):  
Differential Equation ◽  
Generalized Inverse ◽  
Inverse Gaussian ◽  
Generalized Inverse Gaussian Distribution ◽  
Stein Equation ◽  
Second Derivatives ◽  
Generalized Inverse Gaussian ◽  
A Minor ◽  
Derivatives Of

We observe that the density of the Kummer distribution satisfies a certain differential equation, leading to a Stein characterization of this distribution and to a solution of the related Stein equation. A bound is derived for the solution and for its first and second derivatives. To provide a bound for the solution we partly use the same framework as in Gaunt 2017 [Stein, ESAIM: PS 21 (2017) 303–316] in the case of the generalized inverse Gaussian distribution, which we revisit by correcting a minor error. We also bound the first and second derivatives of the Stein equation in the latter case.

scholarly journals A New Family of Distributions Based on the Generalized Pearson Differential Equation with Some Applications

2016 ◽  
Vol 39 (3) ◽  
Author(s):  
Mohammad Shakil ◽  
B.M. Golam Kibria ◽  
Jai Narain Singh
Keyword(s):  
Differential Equation ◽  
Generalized Inverse ◽  
Natural Generalization ◽  
Cumulative Distribution ◽  
Inverse Gaussian ◽  
Gaussian Distributions ◽  
New Family ◽  
Generalized Inverse Gaussian ◽  
New Distribution ◽  
Family Of Distributions

Recently, a generalization of the Pearson differential equation has appeared in the literature, from which a vast majority of continuous probability density functions (pdf’s) can be generated, known as the generalized Pearson system of continuous probability distributions. This paper derives a new family of distributions based on the generalized Pearson differential equation, which is a natural generalization of the generalized inverse Gaussian distribution. Some characteristics of the new distribution are obtained.Plots for the cumulative distribution function, pdf and hazard function, tableswith percentiles and with values of skewness and kurtosis are provided. Itis observed that the new distribution is skewed to the right and bears mostof the properties of skewed distributions. As a motivation, the statistical applicationsof the results to a problem of forestry have been provided. It is found that our newly proposed model fits better than gamma, log-normal and inverse Gaussian distributions. Since many researchers have studied the use of the generalized inverse Gaussian distributions in the fields of biomedicine, demography, environmental and ecological sciences, finance, lifetime data, reliability theory, traffic data, etc., we hope the findings of the paper will be useful for the practitioners in various fields of theoretical and applied sciences.

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