This paper investigates the statistical inference of inverse power Lomax distribution parameters under progressive first-failure censored samples. The maximum likelihood estimates (MLEs) and the asymptotic confidence intervals are derived based on the iterative procedure and asymptotic normality theory of MLEs, respectively. Bayesian estimates of the parameters under squared error loss and generalized entropy loss function are obtained using independent gamma priors. For Bayesian computation, Tierney–Kadane’s approximation method is used. In addition, the highest posterior credible intervals of the parameters are constructed based on the importance sampling procedure. A Monte Carlo simulation study is carried out to compare the behavior of various estimates developed in this paper. Finally, a real data set is analyzed for illustration purposes.

In studying the “11/8-Conjecture” on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of $\operatorname {Pin}(2)$ -equivariant stable maps between certain representation spheres. A precise answer of Furuta’s problem was later conjectured by Jones. In this paper, we completely resolve Jones conjecture by analyzing the $\operatorname {Pin}(2)$ -equivariant Mahowald invariants. As a geometric application of our result, we prove a “10/8+4”-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from $B\operatorname {Pin}(2)$ and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the $j$ -based Atiyah–Hirzebruch spectral sequence.

Entropy measures the uncertainty associated with a random variable. It has important applications in cybernetics, probability theory, astrophysics, life sciences and other fields. Recently, many authors focused on the estimation of entropy with different life distributions. However, the estimation of entropy for the generalized Bilal (GB) distribution has not yet been involved. In this paper, we consider the estimation of the entropy and the parameters with GB distribution based on adaptive Type-II progressive hybrid censored data. Maximum likelihood estimation of the entropy and the parameters are obtained using the Newton–Raphson iteration method. Bayesian estimations under different loss functions are provided with the help of Lindley’s approximation. The approximate confidence interval and the Bayesian credible interval of the parameters and entropy are obtained by using the delta and Markov chain Monte Carlo (MCMC) methods, respectively. Monte Carlo simulation studies are carried out to observe the performances of the different point and interval estimations. Finally, a real data set has been analyzed for illustrative purposes.

In this paper, we investigate parameter inference for four-unit hybrid systems based on Type-II progressively hybrid censored and masked data from constant-stress partially accelerated life test. It is assumed that the unit lifetime in systems has exponential hazard rate. The maximum likelihood estimations (MLEs) for unknown parameters and acceleration factor are obtained. The observed Fisher information matrix, as well as the asymptotic variance-covariance matrix of the MLEs are derived. Approximate confidence intervals (CIs) and studentized-t bootstrap CIs for the parameters are presented based on normal approximation to the asymptotic distribution of MLEs and bootstrap method, respectively. A Monte Carlo simulation study is carried out to investigate the performance of the MLEs and to compare the effects of two different CIs in terms of interval length and coverage percentage for the parameters.

In this paper, we investigate a step-stress partially accelerated lifetime test for the four-component hybrid systems with Type-II progressive hybrid censoring scheme while the life time of system component follows exponential failure rate. In many cases, the exact component causing the system failure cannot be identified and the cause of failure is masked. Based on Type-II progressively hybrid censored and masked data, the maximum likelihood estimations for unknown parameters and acceleration factor are obtained. In addition, approximate confidence interval and bootstrap confidence interval are presented by using the asymptotic distributions of the maximum likelihood estimations for unknown parameters and bootstrap method, respectively. Finally, the proposed method is illustrated through the simulation studies.

This paper deals with the Bayesian inference on step-stress partially accelerated life tests using Type II progressive censored data in the presence of competing failure causes. Suppose that the occurrence time of the failure cause follows Pareto distribution under use stress levels. Based on the tampered failure rate model, the objective Bayesian estimates, Bayesian estimates, and E-Bayesian estimates of the unknown parameters and acceleration factor are obtained under the squared loss function. To evaluate the performance of the obtained estimates, the average relative errors (AREs) and mean squared errors (MSEs) are calculated. In addition, the comparisons of the three estimates of unknown parameters and acceleration factor for different sample sizes and different progressive censoring schemes are conducted through Monte Carlo simulations.