A New Compound Family of Weibull Order Statistics and Left k-Truncated Power Series Distributions

This paper introduces a new family of distributions by compounding the left k-truncated power series distribution with the k-th order statistic of the Weibull distribution. The new family provides a flexible framework for modeling complex data, particularly in reliability engineering and survival analysis. We derive key functions, including the probability density function (PDF), cumulative distribution function (CDF), and hazard rate function (HRF). Several important special cases—such as the geometric, Poisson, binomial, and logarithmic power series distributions—are discussed. Fundamental properties of the family, including moments and quantiles, are explored. Parameter estimation is addressed using the maximum likelihood and the expectation-maximization methods. The paper concludes with potential applications and future research directions.