Shot-noise models deal with the cumulative output of a system whose input is subject to a random Poisson succession of equally distributed impulses or shots, each followed by some attenuation dynamics. With population dynamics in mind, we study the cases when the attenuation dynamics are either given by some ad hoc attenuation function or by some nonlinear ordinary dynamical system or by a (sub-)critical branching process. In the three cases, an interesting issue concerning extinction and idle periods is when the overlapping subpopulations can go extinct in finite time.

KEYWORDS: decay-surge shot-noise processes; linear vs nonlinear; extremal shot-nois; embedded Markov chain; (sub-)critical branching process with immigration; first time to (local) extinction.

MSC 2010 No.: 60G55, 60C05, 60J80, 92D25

Bivariate count models having one marginal and the other conditionals being of the Poissons form are called pseudo-Poisson distributions. Such models have simple, flexible dependence structures, possess fast computation algorithms, and generate a sufficiently large number of parametric families. It has been strongly argued that the pseudo-Poisson model will be the first choice to consider in modeling bivariate over-dispersed data with positive correlation and having one of the marginal equidispersed. Yet, before we start fitting, it is necessary to test whether the given data is compatible with the assumed pseudo-Poisson model. Hence, we derive and propose a few Goodness-of-Fit tests for the bivariate pseudo-Poisson distribution in the present note. Also, we emphasize two tests, a lesser-known test based on the Supremes of the absolute difference between the estimated probability generating function and its empirical counterpart. A new test has been proposed based on the difference between the estimated bivariate Fisher dispersion index and its empirical indices. However, we also consider the potential of applying the bivariate tests that depend on the generating function (like the Kocherlakota and Kocherlakota(K&K) and Mu˜noz and Gamero (M&G) tests) and the univariate Goodness-of-Fit tests (like the Chi-square test) to the pseudo-Poisson data. However, we analyze finite, large, and asymptotic properties for each of the tests considered. Nevertheless, we compare the power (bivariate classical Poisson and Conway-Maxwell bivariate Poisson as alternatives) of each of the tests suggested and also include examples of application to real-life data. In a nutshell, we are developing an R package that includes a test for the compatibility of the data with the bivariate pseudo-Poisson model.

KEYWORDS: Goodness-of-Fit test; Bivariate pseudo-Poisson; Marginal and Conditional distributions; Neyman Type A distribution; Thomas distribution

In this paper, we study Bayes estimators for the mean of the exponential distribution based on ranked set sample with unequal samples (RSSU) proposed and studied by Bhoj (2001). We obtain the Bayes estimates of the scale parameter using both the squared error loss (SEL) function and the linear exponential (LINEX) loss function. Under the assumption of gamma and Jeffreys prior distributions for the scale parameter, we obtain the Bayes estimators. We compare different estimators through simulations for illustration and compute the bias and mean squared error (MSE) of these estimators. We observe that the proposed estimators based on RSSU are more efficient than those based on SRS when the scale parameter follows the Jeffreys prior distribution compared to when it follows the gamma prior distribution.

KEYWORDS: Bayes estimation; Bias; Conjugate prior; Jeffreys prior; Mean squared error (MSE); Posterior distribution;, Ranked set sampling with unequal samples (RSSU).

Modeling data situations that incorporate multiple modes presents challenges for both normal and skew-normal distributions. To overcome this challenge, we present a new class of asymmetric normal distributions designed to accurately represent both asymmetry and multimodality in datasets. Moreover, we explore the locationscale extension of this novel model and provide insights into parameter estimation using the maximum likelihood estimation method. To illustrate the practical utility of the model, we analyze a real-world dataset. Additionally, we perform a concise simulation study to showcase the effectiveness of maximum likelihood estimators in parameter estimation.

KEYWORDS: Model selection; Multimodality; Maximum likelihood estimation; Application; Generalized likelihood ratio test; Simulation.

The study aimed to investigate whether there is a differential impact of age, lifestyle, BMI, and waist-to-height ratio on the risk of Type 2 diabetes mellitus in males and females in a cross-sectional study in Kolkata, West Bengal, India. A sample of size 428 was observed from the outpatient consultation department of Belle Vue Clinic in Kolkata. Interrelationships between the measures were explored using correlation heat maps. The predictive powers of models based on age, BMI, and lifestyle were compared to those based on age, waist-to-height ratio, and lifestyle using receiver operating characteristic curves based on logistic regression models, separately for males and females. The risk of diabetes was found to increase significantly with age in both males and females. Although exercise and BMI were found to have a significant impact on the risk of Type 2 diabetes in males, in females both turned out to be statistically insignificant. In both males and females, predictive models based on waist-to-height ratio were found to perform better than those based on BMI.

KEYWORDS: Type2 Diabetes Mellitus; BMI; Waist to height ratio (Whtr); Multicollinearity; Logistic Regression.