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In many physical situations, it is reasonable to assume that the α-quantile of past life functions is increasing. In this study, we propose a new estimator for an increasing α-quantile of the past life function for right-censored data. We investigate the asymptotic properties of the new estimator and explore its weak and strong convergence. The simulation results show that the new estimator is consistent and outperforms the usual estimator from the literature. As an illustrative example, a time event from a data set of the Mayo Clinic study on primary biliary cirrhosis was analyzed.

Models of various physical, biological, and artificial phenomena follow a power law over multiple magnitudes. This article presents a modified Pareto model with an upside-down shape and an adjustable right tail. The moments, quantiles, failure rate, mean residual life, and quantile residual life functions are examined. In addition, some stochastic ordering characteristics of the proposed model are investigated. The estimation of the parameters using the maximum likelihood estimator, the mean square error, and the Anderson–Darling estimator is explored, and a simulation study is conducted to analyze their behavior. Finally, we compare the proposed model with alternative methods for describing a dataset on the strength of carbon fiber and a dataset on customer waiting times in a bank.

The weighted generalized cumulative residual entropy is a recently defined dispersion measure. This article introduces a new uncertainty measure as a generalization of the weighted generalized cumulative residual entropy, called it the weighted fractional generalized cumulative residual entropy of a nonnegative absolutely continuous random variable, which equates to the weighted fractional Shannon entropy. Several stochastic analyses and connections of this new measure to some famous stochastic orders are presented. As an application, we demonstrate this measure in random minima. The new measure can be used to study the coherent and mixed systems, risk measure, and image processing.

The Pareto model corresponds to the power law widely used in physics, biology, and many other fields. In this article, a new generalized Pareto model with a heavy right tail is introduced and studied. It exhibits an upside-down bathtub-shaped failure rate (FR) function. The moments, quantiles, FR function, and mean remaining life function are examined. Then, its parameters are estimated by maximum likelihood, least squared error, and Anderson–Darling (a weighted least squared error) approaches. A simulation study is conducted to verify the efficiency and consistency of the discussed estimators. Analysis of Floyd River flood discharges in James, Iowa, USA, from 1935 to 1973 shows that the proposed model can be quite useful in real applications, especially for extreme value data.

In this study, we focused on investigating the properties of residual Tsallis entropy for order statistics. The reliability of engineering systems is highly influenced by order statistics, for example, when modeling the lifetime of a series system and the lifetime of a parallel system. The residual Tsallis entropy of the ith order statistic from a continuous distribution function and its deviation from the residual Tsallis entropy of the ith order statistics from a uniform distribution were investigated. In the mathematical framework, a method was provided to represent the residual Tsallis entropy of the ith order statistic in the continuous case with respect to the case where the distribution was uniform. This approach can provide insight into the behavior and properties of the residual Tsallis entropy for order statistics. We also investigated the monotonicity of the new uncertainty measure under different conditions. An investigation of these properties leads to a deeper understanding of the relationship between the position of the order statistics and the resulting Tsallis entropy. Finally, we presented the computational results and proposed estimators for estimating the residual Tsallis entropy of an exponential distribution. For this purpose, we derived a maximum likelihood estimator. [ABSTRACT FROM AUTHOR]

The Pareto distribution satisfies the power law, which is widely used in physics, biology, earth and planetary sciences, economics, finance, computer science, and many other fields. In this article, the logarithmic Pareto distribution, a logarithmic transformation of the Pareto distribution, is presented and studied. The moments, percentiles, skewness, kurtosis, and some dynamic measures such as hazard rate, mean residual life, and quantile residual life are discussed. The parameters were estimated by quantile and maximum likelihood methods. A simulation study was conducted to investigate the efficiency, consistency, and behavior of the maximum likelihood estimator. Finally, the proposed distribution was fitted to some datasets to show its usefulness.

In reliability engineering and survival analysis, quantile functions are fundamental and often the most natural way to represent probability distributions and data samples. In this paper, the α-quantile function of past lifetime was estimated for right-censored data by applying the Kaplan-Meier survival estimator. The weak convergence of the proposed estimator to a Gaussian process was investigated. A confidence interval for the α-quantile of the past life function that does not depend on the density function was proposed. The strong convergence of the estimator to a Gaussian process was also discussed. The properties of the estimator and the confidence interval were investigated in a simulation study. Finally, two real datasets were analyzed. [ABSTRACT FROM AUTHOR]

A fundamental factor in relevant applications is the predictability of the life cycle of a coherent system consisting of more than one component. In this context, we examine how entropy can be applied to evaluate the degree of predictability. In particular, in order to calculate the Tsallis entropy of the past life, we consider a scenario in which all components of the system fail at a given time t and use the system signature to calculate the Tsallis entropy of the past life. We examine a number of analytical results, e.g., expressions, thresholds and orders for the measure at issue in our study. The results may provide insights into the predictability of a coherent system's life cycle. [ABSTRACT FROM AUTHOR]

In this paper, we first establish upper stochastic bounds on the lifetime of a used cold standby system with arbitrary age, using the likelihood ratio order and the usual stochastic order. Then, stochastic comparisons are made between the lifetime of a used cold standby system with age t and the lifetime of a cold standby system consisting of used components with age t using the likelihood ratio order and the usual stochastic order. We use illustrative examples to explore the results presented. [ABSTRACT FROM AUTHOR]

For a given system observed at time t, the past entropy serves as an uncertainty measure about the past life-time of the distribution. We consider a coherent system in which there are n components that have all failed at time t. To assess the predictability of the life-time of such a system, we use the signature vector to determine the entropy of its past life-time. We explore various analytical results, including expressions, bounds, and order properties, for this measure. Our results provide valuable insight into the predictability of the coherent system's life-time, which may be useful in a number of practical applications. [ABSTRACT FROM AUTHOR]

The measurement of uncertainty across the lifetimes of engineering systems has drawn more attention in recent years. It is a helpful metric for assessing how predictable a system's lifetime is. In these circumstances, Renyi entropy, a Shannon entropy extension, is particularly appealing. In this paper, we develop the system signature to give an explicit formula for the Renyi entropy of the residual lifetime of a coherent system when all system components have lived to a time t. In addition, several findings are studied for the aforementioned entropy, including the bounds and order characteristics. It is possible to compare the residual lifespan predictability of two coherent systems with known signatures using the findings of this study. [ABSTRACT FROM AUTHOR]