In this study, we have derived the expression for Maximum Likelihood (ML) estimates and likelihood ratio test (LRT) for two Kumaraswamy populations under Joint Ranked Set Sampling (JRSS), Joint Modified Minimum Ranked Set Sampling (JMnRSS), Joint Modified Maximum Ranked Set Sampling (JMxRSS), and Joint Simple Random Sampling (JSRS). The performance of the ML estimates is evaluated using Root Mean Square Error (RMSE) and the bias criterion. LRT is conducted to test the equality of the first shape parameters of two Kumaraswamy populations when the other two shape parameters are known. The power of the LRT is determined to compare the test performance under the mentioned sampling schemes. This simulations for ML and LRT results are obtained using Monte Carlo simulations in R Studio. We found that all joint ranked sampling schemes demonstrate superior performance compared to JSRS in both ML estimation and LRT. However, the JRSS exhibited the best results in LRT when the other two shape parameters are known, while the JMnRSS excelled in ML estimation. Additionally, we provide an illustration using real-life lung cancer data to highlight these findings.
Traditionally, frailty models are built with the assumption that frailty influences the baseline hazard function in a multiplicative manner, known as a multiplicative frailty model. An alternate model available in the literature is the additive frailty model, in which the frailty variable is linearly related to the baseline hazard function. Although both models are popular, there is a need for a model that incorporates both multiplicative and additive models, especially in epidemiological data where neither the multiplicative nor the additive model adequately describes the data. This paper aims to address this gap by developing a family of models that include both multiplicative and additive frailty models under the inverse Gaussian frailty distribution. The paper also discusses the inference procedure for estimating model parameters using the MCMC method and applies the proposed model to real-life datasets.
T20 cricket is an exciting format characterized by explosive hitting and strategic play, engaging fans with each boundary and a wicket. A comprehensive dataset of T20 matches is analyzed to understand the factors affecting batsmen's performance in this highly dynamic format. Survival analysis approach is used to study the performance of the batsmen, measured in terms of 'number of runs' taken as the 'innings survival time'. In this context, dismissal of batsmen is taken as the 'event'. The dismissal may be due to getting Bowled, being Caught, LBW, Run out, Stumped or Hit wicket. These different forms of dismissal can be taken as 'competing risks' and this study specifically focuses on identifying factors associated with specific dismissals. In this process, Cumulative Incidence Function (CIF), Cause-Specific Hazard function (CSH), and Fine and Gray's Sub distribution Hazard function (SDH) were used. The results from this analysis offer insights into the game dynamics and aids in player's performance evaluation and strategic decision-making, such as, team composition, batting order and making the choice of 'batting first' or 'chasing'. Data for representative batsmen from the top ICC ranking of T20 game with specific inclusion and analysis was carried out using R Programming Language (R 4.5.0) with suitable packages.
This study proposes a novel statistical distribution called the power Gemeay distribution by modifying the Gemeay distribution. Various statistical properties like hazard rate function and its graphics, moments and related measures, order statistics, and incomplete moments are derived. Many estimation methods like the maximum likelihood method, Anderson-Darling estimation, right-tail Anderson-Darling estimation, and more are used to estimate the parameters. A simulation study is carried out to evaluate the efficiency of the estimation techniques. Based on the analysis of two real data sets, the novel distribution is more suitable than other existing models.
This essay explores the life and contributions of Andrei Nikolaevich Kolmogorov(1903-1987), one of the twentieth century's most influential mathematicians. Beginningwith the Borel-Kolmogorov paradox, we examine how Kolmogorov transformed probabilitytheory from a collection of informal methods into a rigorous mathematical framework. Wetrace his remarkable journey through the tumultuous Soviet era, his historic visit to theIndian Statistical Institute, Kolkata, and his profound contributions spanning probabilitytheory, turbulence, complexity theory, topology, and mathematical education. Kolmogorov'sideas continue to shape modern science, from stochastic modeling and statistical inference toturbulence theory and algorithmic information theory. This essay explores both the math-ematical contributions that reshaped probability theory and the historical context in whichthose ideas emerged
