243 research outputs found
Weighted mean inactivity time function with applications
The concept of mean inactivity time plays a crucial role in reliability, risk
theory and life testing. In this regard, we introduce a weighted mean
inactivity time function by considering a non-negative weight function. Based
on this function, we provide expressions for the variance of transformed random
variable and the weighted generalized cumulative entropy. The latter concept is
an important measure of uncertainty which is shift-dependent and is of interest
in certain applied contexts, such as reliability or mathematical neurobiology.
Moreover, based on the comparison of mean inactivity times of a certain
function of two lifetime random variables, we introduce and study a new
stochastic order in terms of the weighted mean inactivity time function.
Several characterizations and preservation properties of the new order under
shock models, random maxima and renewal theory are discussed.Comment: 25 page
Weighted Cumulative Past Extropy and Its Inference
This paper introduces and studies a new generalization of cumulative past extropy called
weighted cumulative past extropy (WCPJ) for continuous random variables. We explore the following:
if the WCPJs of the last order statistic are equal for two distributions, then these two distributions will
be equal. We examine some properties of the WCPJ, and a number of inequalities involving bounds
for WCPJ are obtained. Studies related to reliability theory are discussed. Finally, the empirical
version of the WCPJ is considered, and a test statistic is proposed. The critical cutoff points of the
test statistic are computed numerically. Then, the power of this test is compared to a number of
alternative approaches. In some situations, its power is superior to the rest, and in some other settings,
it is somewhat weaker than the others. The simulation study shows that the use of this test statistic
can be satisfactory with due attention to its simple form and the rich information content behind it
Weak Dependence Notions and Their Mutual Relationships
New weak notions of positive dependence between the components X and Y of a random pair (X,Y) have been considered in recent papers that deal with the effects of dependence on conditional residual lifetimes and conditional inactivity times. The purpose of this paper is to provide a structured framework for the definition and description of these notions, and other new ones, and to describe their mutual relationships. An exhaustive review of some well-know notions of dependence, with a complete description of the equivalent definitions and reciprocal relationships, some of them expressed in terms of the properties of the copula or survival copula of (X,Y), is also 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 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
Weighted Fractional Generalized Cumulative Past Entropy
In this paper, we introduce weighted fractional generalized cumulative past
entropy of a nonnegative absolutely continuous random variable with bounded
support. Various properties of the proposed weighted fractional measure are
studied. Bounds and stochastic orderings are derived. A connection between the
proposed measure and the left-sided Riemann-Liouville fractional integral is
established. Further, the proposed measure is studied for the proportional
reversed hazard rate models. Next, a nonparametric estimator of the weighted
fractional generalized cumulative past entropy is suggested based on the
empirical distribution function. Various examples with a real life data set are
considered for the illustration purposes. Finally, large sample properties of
the proposed empirical estimator are studied.Comment: 23 pages, 8 figure
Past Lifetime and Inactivity Time: from Entropy to Coherent Systems
Information Theory was originally proposed by Claude Shannon in 1948 in the landmark paper entitled "A Mathematical Theory of Communication". In this paper the concept of entropy was adopted for the first time in a field other than thermodynamics and statistical mechanics. Since then, the interest in entropy has grown more and
more and the current literature now focuses mainly on the analysis of residual lifetime. However, in recent years the interest has 'changed direction'. New notions of entropy have been introduced and are used to describe the past lifetime and the inactivity time of a given system or of a component that is found not to be working at the current time. Moreover inferences about the history of a system may be of interest in real life situations. So, the past lifetime and the inactivity time can also be analysed in the context of the theory of coherent systems
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