%0 Journal Article
%T The BALM Copula
%A Boyan Dimitrov
%A Nikolai Kolev
%J International Journal of Stochastic Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/652364
%X The class of probability distributions possessing the almost-lack-of-memory property appeared about 20 years ago. It reasonably took place in research and modeling, due to its suitability to represent uncertainty in periodic random environment. Multivariate version of the almost-lack-of-memory property is less known, but it is not less interesting. In this paper we give the copula of the bivariate almost-lack-of-memory (BALM) distributions and discuss some of its properties and applications. An example shows how the Marshal-Olkin distribution can be turned into BALM and what is its copula. 1. Introduction The class of probability distributions called ˇ°almost-lack-of-memory (ALM) distributionsˇ± was introduced in Chukova and Dimitrov [1] as a counterexample of a characterization problem. Dimitrov and Khalil [2] found a constructive approach considering the waiting time up to the first success for extended in time Bernoulli trials. Similar approach was used in Dimitrov and Kolev [3] in sequences of extended in time and correlated Bernoulli trials. The fact that nonhomogeneous in time Poisson processes with periodic failure rates are uniquely related to the ALM distributions was established in Chukova et al. [4]. It gave impetus to several additional statistical studies on estimations of process parameters (see, e.g., [5, 6] to name a few) of these properties. Best collection of properties of the ALM distributions and related processes can be found in Dimitrov et al. [7]. Meanwhile, Dimitrov et al. [8] extended the ALM property to bivariate case and called the obtained class BALM distributions. For the BALM distributions, a characterization via a specific hyperbolic partial differential equation of order 2 was obtained in Dimitrov et al. [9]. Roy [10] found another interpretation of bivariate lack-of-memory (LM) property and gave a characterization of class of bivariate distributions via survival functions possessing that LM property for all choices of the participating in it four nonnegative arguments. One curious part of the BALM distributions is that the two components of the 2-dimensional vector satisfy the properties characterizing the bivariate exponential distributions with independent components only in the nodes of a rectangular grid in the first quadrant. However, inside the rectangles of that grid any kind of dependence between the two components may hold. In addition, the marginal distributions have periodic failure rates. This picture makes the BALM class attractive for modeling dependences in investment portfolios, financial mathematics, risk
%U http://www.hindawi.com/journals/ijsa/2013/652364/