Competing Risks Step-Stress Model with Lagged Effect under Gompertz Distribution

In many survival analysis studies, the failure of a product may be attributed to one of several competing risks. In addition, if survival time is long, researchers can adopt accelerated life tests, causing devices to fail more quickly. One popular type of accelerated life tests is the step-stress test, and in this test, the stress level changes at a predetermined point time. The manner that stress levels change abruptly and increase discontinuously has been studied extensively. This paper considers a more realistic situation where the effect of stress increases cannot be achieved all at once, but with a lag time, and we propose a step-stress model consisting of two independent competing risks with a lag period in which the failure time caused by different risks at different stress levels obey Gompertz distribution, and the range of lag period is predetermined. The unknown parameters are estimated by maximum likelihood estimation and least squares estimation. For comparison, asymptotic confidence intervals and percentile bootstrap confidence intervals are constructed. By using Monte-Carlo simulations, we obtain the means and mean square errors of the maximum likelihood estimates and the least squares estimates, as well as the mean lengths and coverage rates of the two confidence intervals, which show the performance of various methods. Finally, in order to illustrate the model and proposed methods, we analyze a dataset from a solar energy experiment.

Exponentiated Generalized Inverted Gompertz Distribution: Properties and Estimation Methods with Applications to Symmetric and Asymmetric Data

In this article, a new four-parameter lifetime model called the exponentiated generalized inverted Gompertz distribution is studied and proposed. The newly proposed distribution is able to model the lifetimes with upside-down bathtub-shaped hazard rates and is suitable for describing the negative and positive skewness. A detailed description of some various properties of this model, including the reliability function, hazard rate function, quantile function, and median, mode, moments, moment generating function, entropies, kurtosis, and skewness, mean waiting lifetime, and others are presented. The parameters of the studied model are appreciated using four various estimation methods, the maximum likelihood, least squares, weighted least squares, and Cramér-von Mises methods. A simulation study is carried out to examine the performance of the new model estimators based on the four estimation methods using the mean squared errors (MSEs) and the bias estimates. The flexibility of the proposed model is clarified by studying four different engineering applications to symmetric and asymmetric data, and it is found that this model is more flexible and works quite well for modeling these data.

A COMPARISON OF ESTIMATION METHODS FOR ONE PARAMETER INVERSE GOMPERTZ DISTRIBUTION

In this paper, we compare the methods of estimation for one parameter lifetime distribution, which is a special case of inverse Gompertz distribution. We discuss five different estimation methods such as maximum likelihood method, least-squares method, weighted least-squares method, the method of Anderson-Darling, and the method of Crámer–von Mises. It is evaluated the performances of these estimators via Monte Carlo simulations according to the bias and mean-squared error. Furthermore, two real data applications are performed.

Theory and Applications of the Unit Gamma/Gompertz Distribution

Unit distributions are commonly used in probability and statistics to describe useful quantities with values between 0 and 1, such as proportions, probabilities, and percentages. Some unit distributions are defined in a natural analytical manner, and the others are derived through the transformation of an existing distribution defined in a greater domain. In this article, we introduce the unit gamma/Gompertz distribution, founded on the inverse-exponential scheme and the gamma/Gompertz distribution. The gamma/Gompertz distribution is known to be a very flexible three-parameter lifetime distribution, and we aim to transpose this flexibility to the unit interval. First, we check this aspect with the analytical behavior of the primary functions. It is shown that the probability density function can be increasing, decreasing, “increasing-decreasing” and “decreasing-increasing”, with pliant asymmetric properties. On the other hand, the hazard rate function has monotonically increasing, decreasing, or constant shapes. We complete the theoretical part with some propositions on stochastic ordering, moments, quantiles, and the reliability coefficient. Practically, to estimate the model parameters from unit data, the maximum likelihood method is used. We present some simulation results to evaluate this method. Two applications using real data sets, one on trade shares and the other on flood levels, demonstrate the importance of the new model when compared to other unit models.