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Abstract:
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In this thesis, the concept of reversed lack of memory property and its generalizations
is studied.We we generalize this property which involves operations different
than the ”addition”. In particular an associative, binary operator ” * ” is considered. The
univariate reversed lack of memory property is generalized using the binary operator
and a class of probability distributions which include Type 3 extreme value, power
function, reflected Weibull and negative Pareto distributions are characterized (Asha
and Rejeesh (2009)). We also define the almost reversed lack of memory property
and considered the distributions with reversed periodic hazard rate under the binary
operation. Further, we give a bivariate extension of the generalized reversed lack of
memory property and characterize a class of bivariate distributions which include the
characterized extension (CE) model of Roy (2002a) apart from the bivariate reflected
Weibull and power function distributions. We proved the equality of local proportionality
of the reversed hazard rate and generalized reversed lack of memory property. Study of uncertainty is a subject of interest common to reliability, survival analysis,
actuary, economics, business and many other fields. However, in many realistic
situations, uncertainty is not necessarily related to the future but can also refer to the
past. Recently, Di Crescenzo and Longobardi (2009) introduced a new measure of information
called dynamic cumulative entropy. Dynamic cumulative entropy is suitable
to measure information when uncertainty is related to the past, a dual concept of the
cumulative residual entropy which relates to uncertainty of the future lifetime of a system.
We redefine this measure in the whole real line and study its properties. We also
discuss the implications of generalized reversed lack of memory property on dynamic
cumulative entropy and past entropy.In this study, we extend the idea of reversed lack of memory property to the
discrete set up. Here we investigate the discrete class of distributions characterized by
the discrete reversed lack of memory property. The concept is extended to the bivariate
case and bivariate distributions characterized by this property are also presented. The
implication of this property on discrete reversed hazard rate, mean past life, and discrete past entropy are also investigated. |