On Reliability Estimation of Lomax Distribution under Adaptive Type-I Progressive Hybrid Censoring Scheme

Bayesian estimates involve the selection of hyper-parameters in the prior distribution. To deal with this issue, the empirical Bayesian and E-Bayesian estimates may be used to overcome this problem. The first one uses the maximum likelihood estimate (MLE) procedure to decide the hyper-parameters; while the second one uses the expectation of the Bayesian estimate taken over the joint prior distribution of the hyper-parameters. This study focuses on establishing the E-Bayesian estimates for the Lomax distribution shape parameter functions by utilizing the Gamma prior of the unknown shape parameter along with three distinctive joint priors of Gamma hyper-parameters based on the square error as well as two asymmetric loss functions. These two asymmetric loss functions include a general entropy and LINEX loss functions. To investigate the effect of the hyper-parameters’ selections, mathematical propositions have been derived for the E-Bayesian estimates of the three shape functions that comprise the identity, reliability and hazard rate functions. Monte Carlo simulation has been performed to compare nine E-Bayesian, three empirical Bayesian and Bayesian estimates and MLEs for any aforementioned functions. Additionally, one simulated and two real data sets from industry life test and medical study are applied for the illustrative purpose. Concluding notes are provided at the end.

Knowledge and Asymmetric Loss

This paper offers a novel account of practical factor effects on knowledge attributions that is consistent with the denial of contextualism, relativism and pragmatic encroachemt. The account goes as follows. Knowledge depends on factors like safety, reliability or probability. In many cases, it is uncertain just how safe, how reliably formed or how probable the target proposition is. This means that we have to estimate these quantities in order to form knowledge judgements. Such estimates of uncertain quantities are independently known to be affected by pragmatic factors. When overestimation is costlier than underestimation, for instance, we tend to underestimate the relevant quantity to avoid greater losses. On the suggested account, high stakes and other pragmatic factors induce such “asymmetric loss functions” on quantities like safety, reliability and probability. This skews our estimates of these quantities and thereby our judgements about knowledge. The resulting theory is an error-theory, but one that rationlizes the error in question.

Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions

The fatigue-life or Birnbaum–Saunders distribution is an asymmetrical model that has been widely applied in several areas of science and mainly in reliability. Although diverse methodologies related to this distribution have been proposed, the problem of determining the optimal sample size when estimating its mean has not yet been studied. In this paper, we derive a methodology to determine the optimal sample size under a decision-theoretic approach. In this approach, we consider symmetric and asymmetric loss functions for point and interval inference. Computational tools in the R language were implemented to use this methodology in practice. An illustrative example with real data is also provided to show potential applications.