The generalized Bilal (GB) distribution can be defined as the distribution of the median of three independent random variables drawn from the Weibull distribution. Its failure rate function can be monotonic (decreasing or increasing) or upside-down bathtub-shaped. In this study, we aim to reveal some important properties of the GB distribution that have not been considered before. The findings are both theoretical and practical. From the theoretical viewpoint, we present explicit expressions for both single and product moments of order statistics from the GB distribution. The L-moments are derived as well. From the practical viewpoint, the parameter estimations are accomplished using the maximum likelihood (ML) method, which is based on two different sampling schemes: simple random sampling (SRS) and ranked set sampling (RSS) schemes. Furthermore, the asymptotic confidence intervals for the SRS and RSS estimators are discussed. For the sake of comparison and illustration, a simulation study and a real data example are presented. Concluding remarks are given at the end.

We obtain explicit expressions for single and product moments of the order statistics of an omega distribution. We also discuss seven methods to estimate the omega parameters. Various simulation results are performed to compare the performance of the proposed estimators. Furthermore, the maximum likelihood method is adopted to estimate the omega parameters under the type II censoring scheme. The usefulness of the omega distribution is proven using a real data set.

Explicit algebraic expressions for both single and product moments of order statistics from the exponentiated moment exponential (EME) distribution are derived. By using these expressions, we have computed the means, variances and covariances of order statistics for $n=1(1)10$ and for arbitrarily chosen parameter values. Further, these moments are used to determine the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) of the location and scale parameters based on complete as well as Type-II right censored samples. Prediction of unobserved order statistics in Type-II right censored samples is also discussed. A simulation study and a real data example are presented for the sake of comparison and illustration. The concluding remarks are given at the end.

This study develops explicit algebraic expressions for the single and product moments of order statistics derived from the generalized Bilal (GB) distribution. These expressions facilitate the computation of means, variances and covariances of order statistics for sample sizes up to n = 10 with specified parameter values. The derived moments serve as the foundation for constructing the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the location and scale parameters applicable to both complete and type-II right censored samples. Additionally, the study explores the prediction of unobserved order statistics in type-II right censored samples. The theoretical results are validated through a simulation study, while a real data example highlights their practical utility. These findings establish a robust framework for statistical inference based on order statistics from the GB distribution.