Classes of Ordinary Differential Equations Obtained for the Probability Functions of Exponentiated Generalized Exponential Distribution

In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODE) for the probability density function, quantile function, survival function, inverse survival function, hazard function and reversed hazard function of the exponentiated generalized exponential distribution. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distribution. Solutions of these ODEs by using numerous available methods are new ways of understanding the nature of the probability functions that characterize the distribution. The method can be extended to other probability distributions and can serve an alternative to approximatio

Classes of Ordinary Differential Equations Obtained for the Probability Function of Exponentiated Pareto Distribution

In this paper, the differential calculus was used to obtain some classes of ordinary differential equations (ODEs) for the probability density function, quantile function, survival function, inverse survival function, hazard function and reversed hazard function of the exponentiated Pareto distribution. The stated necessary conditions required for the existence of the ODEs are consistent with the various parameters that defined the distribution. Solutions of these ODEs by using numerous available methods are new ways of understanding the nature of the probability functions that characterize the distribution. The method can be extended to other probability distributions and can serve as an alternative to approximation

A New Beta Power Generator for Continuous Random Variable: Features and Inference to Model Asymmetric Data

Statistical methodologies have broad applications in sports and other exercise sciences. These methods can be used to predict the winning probability of a team or individual in a match. Due to the applicability of the statistical methods in sports, this paper introduces a new method of obtaining statistical distributions. The new method is called a novel beta power-L family of distributions. Some mathematical characteristics of the new family are obtained. Based on the novel beta power-L family, a special model, namely, a novel beta power Weibull model is studied. Finally, the applicability/usefulness of the novel beta power Weibull distribution is shown by analyzing the time-to-even data taken from different football matches during 1964-2018. The data consist of seventy-eight observations and is representing the waiting time duration of the fastest goal scored ever in the history of football. The fitting results of the novel beta power Weibull distribution are compared with other models. Based on three model selection criteria, it is observed that the proposed novel beta power Weibull model provides a close fit to the waiting time data

Statistical modeling of skewed data using newly formed parametric distributions

Several newly formed continuous parametric distributions are introduced to analyze skewed data. Firstly, a two-parameter smooth continuous lognormal-Pareto composite distribution is introduced for modeling highly positively skewed data. The new density is a lognormal density up to an unknown threshold value and a Pareto density for the remainder. The resulting density is similar in shape to the lognormal density, yet its upper tail is larger than the lognormal density and the tail behavior is quite similar to the Pareto density. Parameter estimation methods and the goodness-of-fit criterion for the new distribution are presented. A large actuarial data set is analyzed to illustrate the better fit and applicability of the new distribution over other leading distributions. Secondly, the Odd Weibull family is introduced for modeling data with a wide variety of hazard functions. This three-parameter family is derived by considering the distributions of the odds of the Weibull and inverse Weibull families. As a result, the Odd Weibull family is not only useful for testing goodness-of-fit of the Weibull and inverse Weibull as submodels, but it is also convenient for modeling and fitting different data sets, especially in the presence of censoring and truncation. This newly formed family not only possesses all five major hazard shapes: constant, increasing, decreasing, bathtub-shaped and unimodal failure rates, but also has wide variety of density shapes. The model parameters for exact, grouped, censored and truncated data are estimated in two different ways due to the fact that the inverse transformation of the Odd Weibull family does not change its density function. Examples are provided based on survival, reliability, and environmental sciences data to illustrate the variety of density and hazard shapes by analyzing complete and incomplete data. Thirdly, the two-parameter logistic-sinh distribution is introduced for modeling highly negatively skewed data with extreme observations. The resulting family provides not only negatively skewed densities with thick tails, but also variety of monotonic density shapes. The advantages of using the proposed family are demonstrated and compared by illustrating well-known examples. Finally, the folded parametric families are introduced to model the positively skewed data with zero data values

Construction of multivariate distributions : a review of some recent results

The construction of multivariate distributions is an active field of research in theoretical and applied statistics. In this paper some recent developments in this field are reviewed. Specifically, we study and review the following set of methods: (a) Construction of multivariate distributions based on order statistics, (b) Methods based on mixtures, (c) Conditionally specified distributions, (d) Multivariate skew distributions, (e) Distributions based on the method of the variables in common and (f) Other methods, which include multivariate weighted distributions, vines and multivariate Zipf distributions

On the unit Burr-XII distribution with the quantile regression modeling and applications

In this paper, we modify the Burr-XII distribution through the inverse exponential scheme to obtain a new two-parameter distribution on the unit interval called the unit Burr-XII distribution. The basic statistical properties of the newly defined distribution are studied. Parameters estimation is dealt and different estimation methods are assessed through two simulation studies. A new quantile regression model based on the proposed distribution is introduced. Applications of the proposed distribution and its regression model to real data sets show that the proposed models have better modeling capabilities than competing models

Valuation and Risk Measurement of Guaranteed Annuity Options under Stochastic Environment

This thesis develops stochastic modelling frameworks for the accurate pricing and risk management of complex insurance products with option-embedded features. We propose stochastic models for the evolution of the two main risk factors, the interest rate and mortality rate, which could also have a correlation structure. For the valuation problem, a general framework is put forward where correlated interest and mortality rates are modelled as affine-diffusion processes. A new concept of endowment-risk-adjusted measure is introduced to facilitate the calculation of the GAO value. As a natural offshoot of addressing GAO valuation, we derive the convex-order upper and lower bounds of GAO values by employing the comonotonicity theory. As an alternative to affine structure, we construct a more flexible modelling framework that incorporate regime-switching dynamics of interest and mortality rates governed by a continuous-time Markov chain. The corresponding endowment-risk-adjusted measures are constructed and employed to obtain more efficient GAO pricing formulae. An extension of the previous modelling set-up is further developed by integrating the affine structure and regime-switching feature. Both interest and mortality risk factors follow correlated affine structure whilst their volatilities are modulated by a Markov chain process. The change of probability measure technique is again utilised to generate pricing expressions capable of significantly cutting down computing times. Finally, the risk management aspect of GAO is investigated by evaluating various risk measurement metrics. The bootstrap technique is used to quantify standard error for the estimates of risk measures under a stochastic modelling framework in which death is the only decrement