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International Journal of Stochastic Analysis
Volume 2013 (2013), Article ID 652364, 6 pages
The BALM Copula
1Department of Applied Mathematics, Kettering University, Flint, MI 48504, USA
2Department of Statistics, University of Sao Paulo, CP 66281, 05311-970 Sao Paulo, SP, Brazil
Received 25 April 2013; Revised 26 August 2013; Accepted 28 August 2013
Academic Editor: Nikolai Leonenko
Copyright © 2013 Boyan Dimitrov and Nikolai Kolev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The class of probability distributions called ####^~^~^~^~^~^####x201c;almost-lack-of-memory (ALM) distributions####^~^~^~^~^~^####x201d; was introduced in Chukova and Dimitrov  as a counterexample of a characterization problem. Dimitrov and Khalil  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  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. . 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. . Meanwhile, Dimitrov et al.  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. . Roy  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 studies, and more (see ) where bivariate models are used. Later, the copula approach in modeling dependences , its potential use in financial mathematics (as proposed in ), and some new ideas expressed in Kolev et al.  encouraged us to focus our attention on the construction of copula for the BALM distributions. The results on an extension of the multiplicative lack of memory by Dimitrov and von Collani  played a key role in the construction use in this paper.
Here, in the introduction part we give the basic definition and some of the main properties of the univariate ALM distributions needed to build a quick vision on this subject in the multivariate extensions too. Then we present our main results.
Definition 1. A nonnegative random variable possesses the almost-lack-of-memory (ALM) property if there exists an infinite sequence of distinct numbers , such that the lack-of-memory property holds for every member , and all .
In general, ALM distributions may be discrete, continuous, or of mixed type. But in all the cases the sequence is a lattice of step . Certain explicit forms of ALM distributions related to random processes are based on an assumption of independence of the uncertainty over nonoverlapping time intervals. It is known (see ) that random variable (r.v.) with the ALM property admits a representation as a sum of two independent componentsas follows: Here is a geometrically distributed r.v. with some parameter , and has an arbitrary distribution over the interval . If we denote by the survival function (briefly, SF) of in the presentation (2), then the notation of the class of distribution functions for the r.v. above shows the main parameters of the family. These parameters are , and an SF with support on . An important relationship here is expressed by the equation . An explanation of the parameter can be found in the fact that the failure rate function is a periodic function of period . The distribution function can be arbitrary; it just needs to have support on the interval . Dimitrov and Khalil  in their constructive approach explained the parameter by the duration of the extended in time Bernoulli trials (the time until a trial is completed), and is the time within a trial when the success is noticed under the condition that success occurs.
The exponential and geometric distributions are specific particular cases when special relationships between parameters , and are met.
In the multivariate, case there are several approaches to define bivariate lack-of-memory property. The options generate more ways to introduce the concept of multivariate lack-of-memory classes of probability distributions; see, for example, Roy  and the references therein. Dimitrov et al.  proposed a two-dimensional notion of the ALM property by using the simplest bivariate LM property that characterizes the bivariate exponential distributions with independent components. The following definition explains it.
Definition 2. A pair of nonnegative r.v.####^~^~^~^~^~^####x2019;s possesses the bivariate almost-lack-of-memory (BALM) property if there exist two infinite sequences of distinct numbers , and , such that the lack-of-memory property holds for all members of the two sequences .
The properties of BALM distributions are similar to those of the univariate ALM class. It appears that in order to avoid the known exponential characterization, the sequences and have the form with some and with some . The points with coordinates form nodes of a lattice in the first quadrant in the plane , and these nodes partition the first quadrant into rectangles equal to the rectangle . In particular, an analogous representation to () is obtained. Any two-dimensional random vectors with a BALM distribution can be presented as a sum of two independent bivariate vectors as follows: The first component in (5) is defined by an arbitrary survival function with a support on the rectangle . The coordinate variables in the second component are independent geometrically distributed r.v.'s over the sets and , with parameters and , respectively. The BALM property required by Definition 2 holds for all , and for with nonnegative integers and .
Important fact here is that the marginal distributions of the BALM random vectors are univariate ALM distributions (see ). The two components and may be dependent inside rectangles of the form , but are independent at the nodes for any integers . Dependence on the noted rectangles emulates the dependence between the pair on the rectangle with the vertex at the origin. This statement can be noticed in the form of the BALM survival function which is given by Here is notation for the integer part of any nonnegative number inside the brackets. The in (6) are parameters with values on (0,1). Actually, it occurs that these parameters are related to the components of the random vector by the equalities is the SF of a random vector with probability 1 located on the rectangle , and and are marginal survival functions of the components and obtained from the concordance conditions.
The components and have and distributions correspondingly. As possible area of applications of the class of BALM variables, we see the financial portfolio with two components and , where the actualization (renewals, updates) is performed when the value of investment becomes multiple to a quantity and the value of investment becomes multiple to a quantity
In parallel to the works on ALM distributions, researchers successfully were developing unifying approaches to model dependencies between components of multivariate distributions, namely, the use of copula. The copula approach enjoyed an incredible evolution during the last decades, motivated by its application in probability modeling of dependences in finances, insurance, economics (see [11####^~^~^~^~^~^####x2013;13] and references therein). The new investigations are related mainly to finding the copula for random vectors with components possessing given marginal distributions, whose mutual dependence follows certain correlation structure. A systematic exposition of the copula approach and its applications, as well as several modern copula concepts and graphical tools for studying the dependence phenomena, are presented in Nelsen . In Section 2 we obtain the copula representation of the BALM distributions. In Section 3 we present some properties of the copula itself. To assess better the features of this copula and the possible use of the BALM class of probability distributions, in the next section we list some of the properties of this class.
2. The Copula of the BALM Distributions
The structure of ALM distributions with dependent components is given by properties in Definition 2 and (5)-(6). The possible dependence between and comes through the potential dependence between the components of , acting on the rectangle . The next statement gives the corresponding copula representation for the class of BALM probability distributions.
Theorem 3. Let the random vector have the BALM distribution with SF as in (6), where and are numbers strictly between 0 and 1. The survival copula corresponding to SF in (6) is given by valid for all , when they satisfy
is the survival copula on associated to the survival function whose support is on the rectangle .
Proof. The explicit survival function of is written in (6). Supposedly that it is fulfilled
this SF can be written shorter as
The corresponding marginal survival functions found by the concordance equations are Both functions belong to the class of ALM marginal distributions. Moreover, it is fulfilled
Let be the copula of the pair , and let be the copula of the pair , with . Taking into account Sklar####^~^~^~^~^~^####x2019;s theorem , we get and also
Relations (9)####^~^~^~^~^~^####x2013;(11) show that using we obtain the corresponding copula
Here is the copula associated with the joint distribution of variables whose support is on the rectangle . The expression corresponds to . Under the conditions (9) the arguments in this copula on the right hand side of (8) are always between . After some algebra, the last relation gives
Finally, the survival copula producing the survival function by use of Sklar####^~^~^~^~^~^####x2019;s theorem like in the statement is obtained by applying . Also the result of Theorem 4 (ii) below is taken into account.
The usefulness of the copula of BALM distributions follows from numerous reasons. We list the following two possible situations.(1)Components in financial investments can be dependent and subject to the influence of periodic random environment, possibly with different periodicity . Then bivariate ALM models are appropriate in modeling such event. Copula may be used to model dependence between the two components.(2)In environmental processes as pollution, spread of diseases is two-dimensional process. The growth of dimensions of the infected area may be modeled by appropriate BALM random variable. Dependences on rectangles can be modeled by appropriate copula, and the marginal distributions then will make the transfer of the picture from copula to reality work.
The use of copula is as usually explained in the books (see references in Section 1).
Here especially, when evaluating dependent components inside one rectangle of interaction between and , this dependence is transferred from the properties of the survival copula on the rectangle . It will allow to get models of dependencies of periodic nature. This seems important in actuarial and financial practice.
3. Some Properties of the BALM Copula
The properties of the BALM distributions are discussed mainly in Dimitrov et al. . Most important facts are presented in Section 1. It is useful and recommended to the interested reader to review these and to put them in correspondence to the properties of the BALM copula. Main feature here is the fact that the additive LM property is transferred into multiplicative LM property (from the points to the points ). This idea of Galambos and Kotz  is used in Dimitrov and von Collani, , to transfer the ALM class of univariate distributions into the class of distributions with multiplicative almost-lack-of-memory distributions. The BALM property transfer into bivariate multiplicative almost-lack-of-memory property has not been discussed. It is partially described next.
Here we combine some results about the univariate distributions which possesses the multiplicative-almost-lack-of-memory (MALM) property as discussed by Dimitrov and von Collani  and the BALM distributions. The goal is to present some useful properties of the copula of BALM distributions. These can be easily verified by the use of the specific form (8) of the BALM copula.
We use notations for and for just to simplify the text. Denote by a bivariate random vector with survival function as the BALM copula in (8), with some , , and arbitrary joint survival copula on the rectangle .
Theorem 4. The random vector satisfies the equations as follows.(i)The marginal distribution of each component (,) is uniform on the interval , when and .(ii)The pair possesses the bivariate multiplicative lack-of-memory property at the points as shown by for every integers and every and ; that is, the pair possesses the bivariate multiplicative lack-of-memory property at the points . This is the meaning of the bivariate MALM property when are free. (But in the BALM copula they are engaged.)(iii)The random vector has the same distribution as the pair , where is the pair of r.v. with survival copula , and are independent geometrically distributed random variables of parameters and correspondingly that is, it is true that (iv)On the line segments , and the two components are mutually independent; that is, it is fulfilled for arbitrary .(v)Each of the components and of the copula possesses the univariate multiplicative almost-lack-of-memory property
Proof. The properties listed here follow from the specific choice of the parameters in the copula components and from the form (8) of the copula itself. Statement (ii) is shown as Example 3.1.(a) in Dimitrov and von Collani . (iv) Corresponds to Corollary 2 in Dimitrov et al.  and also from (refnodes). The others need to write the conditional probability explicitly according to the rules, and then use the analytic presentations of respective probabilities. The presentation in property (iii) follows from appropriate explicit expansion of the generating functions of the random variables on both sides in (20) to confirm their coincidence. We prove it here.
Consider the generating function on the left hand side in (iii) as follows: At the end we got the generating function of the right hand side of (iii).
We omit further details.
Notice that property (iii) is suitable for simulation purposes.
4. The Contorted Marshal-Olkin BALM Distribution and Its Survival Copula
As an example in this section we construct the contorted Marshal-Olkin (MO) BALM distribution, following the ideas in Section 1, and show what its respective survival copula is.
It is known that the bivariate Marshal-Olkin distribution has survival functions Here , and are parameters of the MO distribution. Its survival copula is presented by the expression
Define the random vector which has the survival function This random vector with probability 1 is located on the rectangle . Let in (6), where the SF is the one from (26). The obtained then in (6) survival copula defines a pair of random variables which has the contorted Marshal-Olkin distribution on the first quadrant of the plane.
The construction of the MO contorted BALM distribution at the beginning of this section shows one of the ways a class of known probability distributions with dependences over the entire positive quadrant can be converted into BALM distributions.
The copula of BALM distributions can be used in risk modeling when the risk components are dependent, and each component could be a subject to the influence of some specific periodic random environment. Portfolio with multivariate ALM properties is real. Modeling dependent risks with two components is a right step towards multi-component periodic risks studies. For instance, the construction of the MO contorted BALM distribution shows the ways a class of known probability distributions with dependences over the entire positive quadrant can be converted into BALM distributions. ALM distributions in dimensions higher than two are not studied yet. Results presented here may serve well for possible extensions in such direction.
The authors are thankful to FAPESP for Grant no. was 06/60952-1, which provided to Boyan Dimitrov, a fellowship that made the visiting collaboration and direct work real, as well as the results of the present paper.
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