Estimation of parameters in discrete distributions based on different types of rank set sampling

Discrete distributions have important applications in many fields such as modeling of count data, reliability analysis, insurance mathematics and life cycle studies. In recent years, in order to overcome the flexibility and adaptability limitations of existing distributions, studies on the development of new discrete distributions have increased in the literature. In this context, new distributions obtained by combining record type structures based on continuous distributions with Poisson mixtures are attracting attention. In this thesis, two new Poisson transmuted record-type exponential and Poisson transmuted lower record-type exponential distributions, which are developed with the same approach and introduced to the literature, are proposed. In this study, the Poisson Ailamujia, Poisson-Lindley and Poisson new XLindley distributions, which have been proposed in recent years and are frequently preferred in modeling discrete data, are also included and comprehensive comparative analyses are made with these distributions. In the parameter estimation process, methods such as maximum likelihood, least squares, weighted least squares and Cramér-von Mises are used. The estimations are performed under both simple random sampling and ranked set sampling. The performance of these estimators is evaluated using Monte Carlo simulations in terms of criteria such as mean relative error, mean squared error, and bias. In the final section of this thesis, stress-strength reliability analyses are performed for the proposed distributions, and the estimators are evaluated using real data sets. The findings indicate that ranked set sampling is more flexible, consistent, and effective than simple random sampling in assessing the proposed discrete distributions and estimation strategies. This thesis contributes to the literature by introducing new discrete distributions and by comparatively analyzing the performance of various estimation methods under different sampling scenarios, supported by real-world data applications.