Hazard Rate Order Research Articles

Three Parameter Transmuted Exponential Distribution

Many distribution functions, can be explored in numerous dimensions with their extended form. In this paper an extended form of exponential distribution is studied by the quadratic rank transmutation map, called three parameter transmuted exponential distribution, where the base distribution is exponential distribution with two parameters \(\theta\) > 0 and \(\beta\) > 0 , shape and location parameters respectively. Some statistical properties have been studied for the said distribution including moments, quantile function, moment generating function, reliability analysis, mills ratio, reverse hazard rate function, order statistics, Bonferroni and Lorenz curves and indices, mean deviation about mean and median, estimation of parameter. The applicability have been explored by comparing value of -2ℓ, AIC and BIC using real data set.

Stochastic comparisons of lifetimes of fail-safe systems with dependent and heterogeneous components under random shocks

A fail-safe system is a -out-of-n system whose lifetime is represented by the second-order statistic. This work studies stochastic comparisons of lifetimes of fail-safe systems with dependent and heterogeneous components, where the components are subjected to random shocks instantaneously. The results are derived for a general semiparametric family of distributions of the component lifetimes. We derive sufficient conditions for the comparison results under the usual stochastic order, and we provide counterexamples to show that under those conditions, similar comparisons cannot be made under the hazard rate order.

Stochastic comparisons of largest claim amount from heterogeneous and dependent insurance portfolios

This paper investigates the usual stochastic and hazard rate orders between the largest claim amounts from two sets of heterogeneous and dependent insurance portfolios. Sufficient conditions are established in terms of the dependence structure and the heterogeneity among claim severities when the occurrence probabilities are stochastically arrangement increasing. These newly established results generalize or complement some earlier results studied in Balakrishnan et al. (2018b) and Zhang et al. (2019). Numerical examples are provided to illustrate the theoretical findings.

The Cubic Rank Transmuted Gumbel Distribution

A Cubic Rank Transmuted Gumbel distribution (CTGD) in this research is extend the work of cubic transmuted distribution families. CTGD improves the flexibility of transmuted distributions and allows for the modeling of more complex data. Its hazard rate function, moment-generating function, moments, quantile function, entropy and order statistics, are only a few of the key statistical characteristics that we examine. Finally, the Cubic Transmuted Gumbel Distribution is applied to three real datasets to test its applicability and evaluate how well estimate approaches function for the CTGD, Gumbel(G), and transmuted Gumbel (TG) distributions. The observed results demonstrated that, for the used data sets, CTGD provides a superior fit than G and TG distributions.

HR and RHR orderings of extremes of dependent variables under Archimedean copula

In this article, we discuss sufficient conditions for the hazard rate order on smallest order statistics for variables with Archimedean survival copula and having modified proportional hazard rates (modified scale) model. We also establish the reversed hazard rate order on largest order statistics from variables with Archimedean copula and having modified proportional reversed hazard rates model. The results established here extend the recent results of Li and Li (2019).

Stochastic comparisons of second-order statistics from dependent and heterogenous modified proportional hazard rate observations

This manuscript studies the stochastic comparisons of the second-order statistics from dependent or independent and heterogeneous modified proportional hazard rate observations. Some sufficient conditions on the usual stochastic order of the second-order statistics from dependent and heterogeneous observations are established under Archimedean copula. Some sufficient conditions are also provided in the hazard rate order of the second-order statistics arising from two sets of independent and heterogeneous or multiple-outlier modified proportional hazard rate observations. Some numerical examples are given to illustrate the theoretical findings.

A New Version of Cubic Rank Transmuted Gumbel Distribution

A Cubic Rank Transmuted Gumbel distribution (CTGD) in this article is a new generalization of the Gumbel distribution based on a cubic ranking transmutation map. Examined are the cubic transmuted Gumbel model's fundamental statistical properties, such as its hazard rate function, moment-generating function, moments, characteristic function, quantile function, entropy, and order statistics. Finally, the usefulness and applicability of the CTGD using two real data sets about waiting time at a bank is described and Wheaton River flood, and the fit has been compared with Gumbel distribution (GD) and transmuted Gumbel distribution (TGD). The results show that the proposed model provides a superior fit than transmuted Gumbel distributions and Gumbel distributions.

Answer to an open problem about stochastic comparison of parallel systems with geometric components

Mao and Hu [5] left an open problem about the hazard rate order between two parallel systems of n geometric components. Du et al. [3] solved the problem for n=2. Wang [6] gave the answer to the problem for n up to 9. In this short note, we solve the open problem for any n.

Modelling dispersed count data under various shapes of failure rates: A discrete probability analogue of odd Lomax generator

In this article, we introduce a discrete analogue of odd Lomax generator of distributions. The new discrete class can be utilized as a probabilistic tool to generalize any discrete baseline model. After proposing the new class, two special discrete models are investigated and discussed in detail. Some mathematical and statistical properties including, probability mass function, hazard rate function, quantile, crude moments, index of dispersion, entropies, order statistics, and L-moment statistics, are derived. It is found that the presented discrete class can be used to model symmetric and asymmetric data under different types of kurtosis shapes. It can be utilized to explain and analyze overdispersion data with extreme, zero-inflated or outliers? observations. Furthermore, it can be applied to discuss various shapes of hazard rates including monotone increasing, monotone decreasing, unimodal, bathtub, unimodal-bathtub, among others. We discuss the estimation of the class parameters by the maximum likelihood approach. The performance of the estimation method is tested via Markov chain Monte Carlo (MCMC) simulation technique. Finally, to demonstrate the proposed methodology in a real-life scenario, three real data sets are considered to show the applicability of the proposed generator.

Stochastic comparison of the second-largest order statistics from heterogeneous models

The second-largest order statistic is of special importance in reliability theory since it represents the time to failure of a $ 2 $-out-of-$ n $ system. Consider two $ 2 $-out-of-$ n $ systems with heterogeneous random component lifetimes, following a general family of exponentiated location-scale distributions. In this article, usual stochastic order between the lifetimes of systems has been established when the components are interdependent. For the independent heterogeneous distributions, sufficient conditions under which reversed hazard rate order between the second-largest order statistics holds are investigated. To illustrate theoretical findings, some examples are considered.

A New Stochastic Order of Multivariate Distributions: Application in the Study of Reliability of Bridges Affected by Earthquakes

In this article, we introduce and study a new stochastic order of multivariate distributions, namely, the conditional likelihood ratio order. The proposed order and other stochastic orders are analyzed in the case of a bivariate exponential distributions family. The theoretical results obtained are applied for studying the reliability of bridges affected by earthquakes. The conditional likelihood ratio order involves the multivariate stochastic ordering; it resembles the likelihood ratio order in the univariate case but is much easier to verify than the likelihood ratio order in the multivariate case. Additionally, the likelihood ratio order in the multivariate case implies this ordering. However, the conditional likelihood ratio order does not imply the weak hard rate order, and it is not an order relation on the multivariate distributions set. The new conditional likelihood ratio order, together with the likelihood ratio order and the weak hazard rate order, were studied in the case of the bivariate Marshall–Olkin exponential distributions family, which has a lack of memory type property. At the end of the paper, we also presented an application of the analyzed orderings for this bivariate distributions family to the study of the effects of earthquakes on bridges.

Odd Lomax Generalized Exponential Distribution: Application to Engineering and COVID-19 data

This paper proposes the 4-parameter odd Lomax generalized exponential distribution for the study of engineering and COVID-19 data. The statistical and mathematical properties of this distribution such as a linear representation of the probability density function, survival function, hazard rate function, moments, quantile function, order statistics, entropy, mean deviation, characteristic function, and average residual life function are established. The estimates of parameters of the proposed distribution are obtained using maximum likelihood estimation (MLE), Maximum product spacings (MPS), least-square estimation (LSE), and Cramer-Von-Mises estimation (CVME) methods. A Monte-Carlo simulation experiment is carried out to study the MLEs. The applicability of the proposed distribution is evaluated using two real datasets related to engineering and COVID-19. All the computational work was performed in R programming software.

On stochastic ordering among extreme shock models

AbstractIn the usual shock models, the shocks arrive from a single source. Bozbulut and Eryilmaz [(2020). Generalized extreme shock models and their applications. Communications in Statistics – Simulation and Computation49(1): 110–120] introduced two types of extreme shock models when the shocks arrive from one of $m\geq 1$ possible sources. In Model 1, the shocks arrive from different sources over time. In Model 2, initially, the shocks randomly come from one of $m$ sources, and shocks continue to arrive from the same source. In this paper, we prove that the lifetime of Model 1 is less than Model 2 in the usual stochastic ordering. We further show that if the inter-arrival times of shocks have increasing failure rate distributions, then the usual stochastic ordering can be generalized to the hazard rate ordering. We study the stochastic behavior of the lifetime of Model 2 with respect to the severity of shocks using the notion of majorization. We apply the new stochastic ordering results to show that the age replacement policy under Model 1 is more costly than Model 2.

Stochastic comparison of the second-order statistics arising from exponentiated location-scale model

In this article, we consider stochastic comparisons between second-order statistics arising from general exponentiated location-scale models. When the random variables are independent, we establish usual stochastic and hazard rate orders between second-order statistics. Further, similar ordering results are obtained when the random observations are dependent. Some applications of the established results are presented.

Discussion of signature‐based models of preventive maintenance

Discussion of signature‐based models of preventive maintenance

Reliability analysis of fail-safe systems with heterogeneous and dependent components subject to random shocks

In many significant safety systems, the fault tolerance of a fail-safe system has been a fundamental design attributed to obtaining more higher reliability. However, in most practical scenarios, the components in a fail-safe system suffer from additional random shocks, and one component’s failure may increase the failure probability of other surviving components. This article analyzes the effects of survived probabilities and components lifetimes on the reliability of fail-safe systems with statistically dependent components subject to random shocks in terms of the usual stochastic and hazard rate orderings. Sufficient conditions are established by means of transformation functions, underlying component lifetime distributions, and majorization orders. Some numerical examples are also presented for illustrating our theoretical findings.

Stochastic Comparisons of Convolution of Independent Random Variables in the Scale Model

This paper deals with some stochastic comparisons of convolution of random variables comprising scale variables. Sufficient conditions are established for these convolutions' likelihood ratio ordering and hazard rate order. The results established in this paper generalize some known results in the literature. Several examples are also presented for more illustrations.

Stochastic Comparison of (n-1)-out-of-n Systems from Multiple-Outlier Modified Proportional Hazard Rates Components in terms of Hazard Rate Order

Stochastic Comparison of (n-1)-out-of-n Systems from Multiple-Outlier Modified Proportional Hazard Rates Components in terms of Hazard Rate Order

A New Family of Odd Nakagami Exponential (NE-G) Distributions

In this study, a new family of odd nakagami exponential (NE-G) distributions is introduced and investigated as a new generator of continuous distributions. Quantile, hazard rate function, moments, incomplete moments, order statistics, and entropies are only a few of the statistical features that are investigated. A unique model is presented and thoroughly examined. To estimate model parameters based on describing real-life data sets, the maximum likelihood method is applied. The bias and mean square error of maximum likelihood estimators are investigated using a comprehensive simulation exercise. Finally, the new family adaptability is demonstrated via application to real-world data sets.

Stochastic comparisons on extremes of burr type XII samples associated with Archimedean copula and heterogeneous shape parameters

This study deals with random variables from Burr Type XII samples having heterogeneous shape parameters and homogeneous frailty parameters. For samples whose survival copula (whose copula) is Archimedean copula, sufficient conditions based on the heterogeneity of shape parameter vector for the stochastic tail ordering and usual stochastic ordering between sample minimums (sample maximums) are provided. As for independent samples, the convex transform order and tail hazard rate order (tail reversed hazard rate order) between sample minimums (maximums) are discussed. Several numerical examples are also presented to illustrate the theoretical findings.