8,190 research outputs found
Extension of the past lifetime and its connection to the cumulative entropy
Given two absolutely continuous nonnegative independent random variables, we
define the reversed relevation transform as dual to the relevation transform.
We first apply such transforms to the lifetimes of the components of parallel
and series systems under suitably proportionality assumptions on the hazards
rates. Furthermore, we prove that the (reversed) relevation transform is
commutative if and only if the proportional (reversed) hazard rate model holds.
By repeated application of the reversed relevation transform we construct a
decreasing sequence of random variables which leads to new weighted probability
densities. We obtain various relations involving ageing notions and stochastic
orders. We also exploit the connection of such a sequence to the cumulative
entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative
formulae for computing the mean and the cumulative entropy of the random
variables of the sequence are finally investigated
A Shift-Dependent Measure of Extended Cumulative Entropy and Its Applications in Blind Image Quality Assessment
Recently, Tahmasebi and Eskandarzadeh introduced a new extended cumulative entropy (ECE). In this paper, we present results on shift-dependent measure of ECE and its dynamic past version. These results contain stochastic order, upper and lower bounds, the symmetry property and some relationships with other reliability functions. We also discuss some properties of conditional weighted ECE under some assumptions. Finally, we propose a nonparametric estimator of this new measure and study its practical results in blind image quality assessment
Extended fractional cumulative past and paired phi-entropy measures
Very recently, extended fractional cumulative residual entropy (EFCRE) has
been proposed by Foroghi et al. (2022). In this paper, we introduce extended
fractional cumulative past entropy (EFCPE), which is a dual of the EFCRE. The
newly proposed measure depends on the logarithm of fractional order and the
cumulative distribution function (CDF). Various properties of the EFCPE have
been explored. This measure has been extended to the bivariate setup.
Furthermore, the conditional EFCPE is studied and some of its properties are
provided. The EFCPE for inactivity time has been proposed. In addition, the
extended fractional cumulative paired phi-entropy has been introduced and
studied. The proposed EFCPE has been estimated using empirical CDF.
Furthermore, the EFCPE is studied for coherent systems. A validation of the
proposed measure is provided using logistic map. Finally, an application is
reported
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
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
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
General weighted information and relative information generating functions with properties
In this work, we propose two information generating functions: general
weighted information and relative information generating functions, and study
their properties. The effects under monotone transformations of the proposed
generating functions are explored. Bounds are obtained. Further, these
information generating functions are studied for escort, generalized escort and
mixture distributions. The residual versions of the newly proposed generating
functions are considered and several properties are proposed
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