##### A Note on α-stable and α-inverse Gaussian Laws

S. Satheesh

Asian Journal of Probability and Statistics, Page 29-34
DOI: 10.9734/ajpas/2022/v19i230465

In this article we obtain the first passage time distribution of α-stable Levy processes. We derive the moment estimators of the parameters of α-inverse Gaussian laws and also their asymptotic distribution.

##### Probability Risk Model of Claim Amount Affected by a Threshold Value

Qiao Li, Zhenhua Bao

Asian Journal of Probability and Statistics, Page 1-17
DOI: 10.9734/ajpas/2022/v19i230462

In this paper, we consider a new risk model of claim amount affected by a threshold value. The comparision between the claim interval and the threshold will affect the distribution of claims. The hypothesis of the model is presented and then we derive the roots of the Lundberg equation, and the expected discounted penalty function and its Laplace Transform. Besides, the Gerber- Shiu penalty function and some other functions are given when the initial surplus is zero and when they satifie some defective renewal equations. Some explicit expressions about the ruin probability are obtained too.

##### Dispersion of Count Data: A Case Study of Poisson Distribution and Its Limitations

Xhavit Bektashi, Shpëtim Rexhepi, Nora Limani–Bektashi

Asian Journal of Probability and Statistics, Page 18-28
DOI: 10.9734/ajpas/2022/v19i230464

Poisson distribution is one of the widely known distribution in the field of probability and statistics by statisticians. It has been widely applied in modeling of discrete observations including but not limited to the number of customers in a shop within a specified period, the number of accidents occurring within a specified time or the number of claims experienced by an insurance company within a specified period of time. Poisson regression model has been widely used in events where one response variable is influenced directly by other independent variables. One thing about Poisson model is that it is strict on the property of dispersion as it assumes that count data is equidispersed which is not the case in practice. By this assumption, the Poisson model states that the variance of the count data is equal to the mean which is not practically true. In most cases, the variance of real count data is always greater than the mean, a phenomenon described as over dispersion. This gives Poisson model a loss in its frequent use in modelling count observations. This paper seeks to study the concept of dispersion, how Poisson regression is applied and its possible limitations. A deep study of Poisson model is done, its properties up to the fourth moments outlined. A graphical representation of its probability density function is drawn from simulated data and its shape noted under different rates as it resumes symmetry as the rate increases. A histogram is also presented. An application to real data is done in R programing language and proof that Poisson regression is very poor on this analysis given. Finally, a counter distribution appropriate for taking care of over dispersion is analyzed and results compared. AIC is used to conclude that NB is better than Poisson regression model.

##### Comparison of the Autoregressive Vector VAR with the Dynamic Error Correction Vector DVECM for Modeling COVID-19 Deaths

Ahmed Razzaq Abed, Ayad Habeeb Shamil

Asian Journal of Probability and Statistics, Page 35-56
DOI: 10.9734/ajpas/2022/v19i230466

In this article, the Vector Auto-Regressive model and the Dynamic Error Correction Vector Model will be used in modeling data representing the number of deaths due to infection with the COVD-19 virus as a dependent variable and the variable platelet rate in the blood as an independent variable and finding the model equations that represent the relationship between the two variables using the two models and then estimating the equations that were obtained by estimating the two models using the least squares method, then choosing the best estimated equation from each model, and then, using the standard error of the regression and the coefficient of determination, selecting the best equations from the two models. The Dynamic Error Correction Vector Model is superior to the Vector Auto-Regressive model in assessing the link between corona virus mortality and the proportion of platelets in the blood, according to the analysis carried out using the E-Views application, and that there is a direct relationship through the equation for the Dynamic Error Correction Vector Model between the deaths of the corona virus and the proportion of platelets in the blood in the long term, which is logical as a result of the increase in the impact of the deaths of the Corona virus, the increase in the platelet rate, and thus the increase in the deaths of the corona virus.

##### The Equity-Based Modeling for Two-Player Gambler's Ruin Problem

Muhammad Asim Masood, Abid Hussain, Ehtasham ul Haq, Ishfaq Ahmad

Asian Journal of Probability and Statistics, Page 57-71
DOI: 10.9734/ajpas/2022/v19i230467

In the context of the classical two-player gambler's ruin problem, the winning probabilities and initial stakes are pre-decided. If a player (who is in financial crisis) starts with less amount than his/her opponent in the symmetric game, has more chances to be ruined. Besides, a player (based on previous record data) with more winning probability than his/her competitor, has fewer chances to be ruined. We observe that most of the time, usually a weaker player is not fully willing to make a contest with a strong player. To give a fair chance to fight back for a weaker player and to develop the audience's interest, equity-based modeling is required. In this research, we propose some new equity-based models for the game of two players. In this way, we advocate the weaker player (with less winning probability or less amount to start the game) is motivated to participate in the contest because of a fair chance to make a comeback. The working methodology of newly proposed schemes is executed by deriving general expressions of the ruin probabilities for mathematical evaluation along with observing the ruin times, and then findings are compared with the results of a classic two-player game. Hence, the prime objectives related to the study are achieved by taking diverse parametric settings in the favor of equity-based modeling.