Generalized T – X Family of Distributions and Applications

Generalized T – X Family of Distributions

In this paper, we introduce a new generalization of univariate and bivariate modified discrete Weibull distribution. Various properties of the univariate generalized modified discrete Weibull distribution, such as survival function, probability mass function, hazard rate function, probability generating function, and moment generating function, are derived. The joint distribution function, joint probability mass function, marginal distributions, moment generating function, and conditional distribution of the proposed bivariate distribution are derived. Parameters of the distributions are estimated using maximum likelihood estimation. The use of these distributions is illustrated using real-life datasets.

In this paper, a new method of generating families of distributions using any p.d.f. as a generator which is called Transformed – Transformer ( T – X ) family of distributions is considered. Some members of this family including Gumbel – Uniform, Gumbel – Pareto, Gumbel – Frechet and their mathematical properties are explored. The method of maximum likelihood is used for estimating the distribution parameters. The flexibility of the new model is illustrated using real life data sets. Also industrial reliability test plans for acceptance or rejection of a lot of products submitted for inspection are developed. The acceptance sampling plan proposed here can save the test time in practical situations. Marshall – Olkin generalization of T – X family is also introduced. Stress-strength analysis is discussed. A simulation study is also conducted to study the behaviour of estimate of stress-strength reliability. Auto-regressive models are discussed based on Marshall – Olkin Gumbel uniform distribution and sample path properties are explored. Transmutation technique is also considered in T – X family. A real data is used to explain the flexibility of Gumbel Transmuted Uniform model, a member of Gumbel Transmuted X family.