Abstract
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.
Highlights
IntroductionA generalization of the Pearson differential equation has appeared in the literature: dfX (x) dx
A generalization of the Pearson differential equation has appeared in the literature: dfX (x) dx =a0 + a1x + a2x2 b0 + b1x + b2x2 + + · · amxm bnxn fX (x) (1)where m, n ≥ 1 are arbitrary integers, and the coefficients a and b are real numbers
It appears from the literature that not much attention has been paid to the study of the family of continuous pdf’s that can be generated as a solution to the generalized Pearson differential equation (1), except Dunning and Hanson (1977), Chaudhry and Ahmad (1993), and recently Shakil, Singh, and Kibria (2010)
Summary
A generalization of the Pearson differential equation has appeared in the literature: dfX (x) dx. It appears from the literature that not much attention has been paid to the study of the family of continuous pdf’s that can be generated as a solution to the generalized Pearson differential equation (1), except Dunning and Hanson (1977), Chaudhry and Ahmad (1993), and recently Shakil, Singh, and Kibria (2010). Some characteristics of our newly proposed distribution, including the expressions for the normalizing constant, pdf, cumulative distribution function (cdf), k-th moment, Shannon’s entropy and relationships to other probability distributions, are derived. The plots for the cdf and pdf of the new distribution, including the percentile points, for some selected values of parameters, have been provided. The derivations of the cdf, pdf, k-th moment, etc, in this paper involve some special functions, which are provided in the Appendix
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