When conducting reliability studies, the progressive first-failure censoring (PFFC) method is useful in situations in which the units of the life testing experiment are separated into groups consisting of k units each with the intention of seeing only the first failure in each group. Using progressive first-failure censored samples, the statistical inference for the parameters, reliability, and hazard functions of the extended Rayleigh distribution (ERD) are investigated in this study. The asymptotic normality theory of maximum likelihood estimates (MLEs) is used in order to acquire the maximum likelihood estimates (MLEs) together with the asymptotic confidence intervals (Asym. CIs). Bayesian estimates (BEs) of the parameters and the reliability functions under different loss functions may be produced by using independent gamma informative priors and non-informative priors. The Markov chain Monte Carlo (MCMC) approach is used so that Bayesian computations are performed with ease. In addition, the MCMC method is used in order to create credible intervals (Cred. CIs) for the parameters, which may be used for either informative or non-informative priors. Additionally, computations for the reliability functions are carried out. A Monte Carlo simulation study is carried out in order to provide a comparison of the behaviour of the different estimations that were created for this work. At last, an actual data set is dissected for the purpose of providing an example.

In this paper, we establish several recurrence relations satisfiedby the single and product moment generating function of progressive first failure censoring from an right truncated exponential distribution. These relations may then be used, for example, several recurrence relations satisfiedby the single and product moment generating function of progressive first failure censoring from an right truncated exponential distribution and to compute all the means, variances and covariances of right truncated exponential distribution progressive first failure censoring for all sample sizes n and all censoring schemes (R_1,R_2,…,R_m), m < n. The results presented in the paper generalize the results given by Aggarwala and Balakrishnan [1] for the single moments and product moments of order statistics from the exponential and right truncated exponential distributions. To further generalize these results, we consider also the right truncated exponential distribution. Recurrence relations for the single and product moment generating function are established for progressive first failure censoring from the right truncated exponential distribution. Further, the results are specialized to the progressively type-II right censored order statistics (PTIIRCOS).

In this article, we establish recurrence relations (RR) for single and product moment generating function (MGF) based on based on progressive first failure censoring (PFFC) for generalized Pareto distribution (GPD). Characterization for GPD using RR of single and product MGF of PFFC are also obtained. Further, the results are specialized to the progressively type-II right censored (PTIIRCOS).

Characterization of a probability distribution is the investigation of those unique properties enjoyed by that distribution. In this article the general condition moments technique is used to obtain some new characterization results for the skew-normal distribution based on the order statistics as well as the record values of a random sample drawn from this distribution. The achieved results can be used to improve fitting and modeling skew data using the skew-normal distribution. Further, the new results are specialized to the standard normal distribution.

This study explores the development of recurrence relations (RRs) for single moments (SMs) and product moments (PMs) based on progressive first-failure censored data (PFFCD) using the extended power Lindley distribution (EPLD). These RRs enable efficient computation of key statistical measures, such as means and variances, across varying sample sizes, reducing computational effort. The proposed approach extends existing methodologies and adapts to progressive type-II censoring schemes (PTIICS), broadening its applicability. This advancement enhances analytical frameworks and computational tools for statistical inference in models involving censored data.