In the following, we introduce new proportional hazard (P.H.) processes, which are derived by a marginal transformation applied to complementary power function distribution (CPFD) processes. Also, we introduce two new Pareto processes, which are derived from the proportional hazard family. We discuss the distributional features of such processes, explore inferential aspects, and include an example of applications of the new processes to real-life data.

KEYWORDS: proportional hazard process; power function distribution process; complementary power function distribution process; pareto process, stationarity; markovian property.

Trend renewal processes (TRPs) were introduced by Lindqvist (1993) as a timetransformed renewal process. After a brief introduction to TRPs with possibly some new results, independent ordinary renewal processes (ORPs) and TRPs are compared with respect to some stochastic orderings between the generating inter-arrival time random variables, like, the usual stochastic order, hazard rate order, likelihood ratio order and variability order, and on the basis of the trend function. Some illustrations are given.

KEYWORDS: Hazard rate ordering, Likelihood ratio ordering, Renewal process, Stochastic ordering, Trend renewal process, Variability ordering.

Recently count data with excess zeroes and ones have received much attention in the literature due to its empirical needs and applications. Here we consider a new class of zero-one inflated version of the logarithmic-with-zero series distribution through its probability mass function. We derive the probability generating function, mean and variance of the proposed class of distribution and obtain expressions for its r-th raw moment and r-th factorial moments. The parameters of the distribution are estimated by the method of maximum likelihood and certain likelihood ratio test procedures are suggested. Further, the distribution has been fitted to certain real life data sets and thereby shown that the proposed model gives better fit to the data sets compared to existing models.

KEYWORDS: probability generating function; maximum likelyhood estimato; zero-one inflated distributions; factorial moments.

Linnik distribution is heavy tailed compared to Laplace distribution and is used for modeling data sets in various fields especially in Finance and Economics. In this paper, a bivariate distribution related to geometric Pakes asymmetric Laplace and Linnik distribution is introduced and bivariate time series model corresponding to this distribution is developed.

KEYWORDS: Autoregressive process; Geometric marginal asymmetric Laplace and Linnik distribution; Geometric Pakes generalized asymmetric Linnik distribution; Geometric Stable distribution.

In this paper, we propose an extension to the existing reduced Kies distribution called the extended reduced Kies (ExRKD) distribution. We study the properties of the ExRKD and utilize it on real-world COVID-19 datasets to estimate parameters with the help of maximum likelihood estimation (MLE) procedures. Also perform simulation studies to evaluate the asymptotic behaviour of the MLEs, offering insights into their performance and reliability.

KEYWORDS: Beta Weibull distribution; COVID-19; Kies distribution; maximum likelihood estimation; model selection; simulation.