Characterizations of continuous distributions through inequalities involving the expected values of selected functions

summary:Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser's function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via $w(\cdot )$-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable $X$ is provided

Theoretical aspects of total time on test transform of weighted variables and applications

summary:Although the total time on test (\textit{TTT}) transform is not a newly discovered concept, it has many applications in various fields. On the other hand, weighted distributions are extensively developed by the statisticians to tackle the insufficiency of the standard statistical distributions in modeling the arising data from real-world problems in the contexts like medicine, ecology, and reliability engineering. This paper develops the \textit{TTT} transform for the weighted random variables and investigates the behavior of the failure rate function of such variables based on the \textit{TTT} transform. In addition, the conditions for establishing the $TTT$ transform ordering for weight variables and its relationship with some stochastic orders have been investigated, and the conditions for establishing the \textit{TTT} transform order as well as the presentation of the new better than used in total time on test transform (\textit{NBUT}) class of the weighted variables have also been studied. Finally, by analyzing the real data sets, applications of the transform introduced in the fit of a model is presented, and it is shown that weighted models have a significant advantage over the base models

Preventive replacement for belligerent systems

A mortar is commonly used as an indirect firing weapon to support close fires with a variety of ammunition. There are mortar weapons with various shells. Each type of shells fired by mortars does damage to a weapon when the total damage on a mortar weapon reaches the tolerance limit, the mortar weapon either fails or explodes, leading to a compulsory replacement which is costly. In order to maintain the mortar weapons and archers in wars, a research was conducted to find the best number of mortar shells that will be fired until a preventive replacement for mortar weapons is implemented

A NEW GENERALIZED VARENTROPY AND ITS PROPERTIES

The variance of Shannon information related to the random variable $X$, which is called varentropy, is a measurement that indicates, how the information content of $X$ is scattered around its entropy and explains its various applications in information theory, computer sciences, and statistics. In this paper, we introduce a new generalized varentropy based on the Tsallis entropy and also obtain some results and bounds for it. We compare the varentropy with the Tsallis varentropy. Moreover, we explain the Tsallis varentropy of the order statistics and analyse this concept in residual (past) lifetime distributions and then introduce two new classes of distributions by them

A comparative study of the Gini coefficient estimators based on the linearization and U-statistics Methods

In this paper, we consider two well-known methods for analysis of the Gini index, which are U-statistics and linearization for some incomedistributions. In addition, we evaluate two different methods for some properties of their proposed estimators. Also, we compare two methods with resampling techniques in approximating some properties of the Gini index. A simulation study shows that the linearization method performs 'well' compared to the Gini estimator based on U-statistics. A brief study on real data supports our findings.En este artículo consideramos dos métodos ampliamente conocidos para en análisis del índice Gini, los cuales son U-statistics y linealización. Adicionalmente, evaluamos los dos métodos diferentes con base en las propiedades de los estimadores propuestos sobre distribuciones de la renta. También comparamos los métodos con técnicas de remuestreo aproximando algunas propiedades del índice Gini. Un estudio de simulación muestra que el método de linealización se comporta ``bien'' comparado con el método basado en U-statistics. Un corto estudio de datos reales confirma nuestro resultado

On lower bounds for the variance of functions of random variables

summary:In this paper, we obtain lower bounds for the variance of a function of random variables in terms of measures of reliability and entropy. Also based on the obtained characterization via the lower bounds for the variance of a function of random variable $X$, we find a characterization of the weighted function corresponding to density function $f(x)$, in terms of Chernoff-type inequalities. Subsequently, we obtain monotonic relationships between variance residual life and dynamic cumulative residual entropy and between variance past lifetime and dynamic cumulative past entropy. Moreover, we find lower bounds for the variance of functions of weighted random variables with specific weight functions applicable in reliability under suitable conditions

Reversed preservation of stochastic orders for random minima and maxima with applications

Random minima (maxima), TTT-transform order, right spread order, increasing convex (concave) order, DMRL, NBUC, NBU(2), NBUT,

Discrete likelihood ratio order for power series distribution

Es bien conocido en la literatura que algunas distribuciones discretaspertenecen a la familia de distribuciones de series de potencias (PSD, powerseries distributions por sus siglas en inglés). Por lo tanto, es útil estudiaralgunas condiciones para establecer el orden de la razón de verosimilitudpara esta familia. En este artículo, se estudian las condiciones para algunoscasos de la familia PSD bajo las cuales se mantiene el orden de la razónde verosimilitud. Otros autores han introducido y estudiado el orden de larazón de verosimilitud proporcional como una extensión del orden de razónde verosimilitud para variables aleatorias continuas. Aquí, se presenta el orden de razón de verosimilitud proporcional para variables aleatorias discretasy se estudian para la familia PSD