Residual and Past Entropy for Concomitants of Ordered Random Variables of Morgenstern Family
For a system, which is observed at time t, the residual and past entropies measure the uncertainty about the remaining and the past life of the distribution, respectively. In this paper, we have presented the residual and past entropy of Morgenstern family based on the concomitants of the different types of generalized order statistics (gos) and give the linear transformation of such model. Characterization results for these dynamic entropies for concomitants of ordered random variables have been considered.
The Morgenstern family discussed in  provides a flexible family that can be used in such contexts, which is specified by the distribution function (df) and the probability density function (pdf), respectively, as follows:where , and , and , are the marginal pdf and df of and , respectively. The parameter is known as the dependence parameter of the random variables and . If is zero, then and are independent.
The general theory of concomitants of order statistics has originally been studied by David et al. . Let , , be pairs of independent random variables from some bivariate population with df . Let be the th order statistics; then the value associated with is called the concomitant of the th order statistics and is denoted by . The pdf of is given by where is the pdf of .
The concept of gos was introduced by Kamps  and we refer to it as case-I of gos; Kamps and Cramer  have introduced a second model of gos which we refer to as case-II of gos. The concept of lower gos was given by Pawlas and Szynal , and later Burkschat et al.  introduced it as dual generalized order statistics (dgos) to enable a common approach to descendingly ordered random variables like reversed order statistics and lower records models.
For the Morgenstern family with pdf given by (2), the pdf of the concomitant of case-I of gos , , is given by Beg and Ahsanullah  as follows:and the pdf of the concomitant of case-I of dgos , , is given by Nayabuddin  as follows:where , , , , and . The pdf of the concomitant of case-II of gos , , is given by Mohie El-Din et al.  as follows:and the pdf of the concomitant of case-II of dgos , , is given bywhere , , , , , .
Let be an absolutely continuous nonnegative random variable having df and the survival function . Suppose denotes the lifetime of a component/system or of a living organism and denotes the lifetime density function. In information theory, Shannon entropy  plays a vital role in measuring the index of dispersion, volatility, or uncertainty associated with a random variable . Shannon entropy for a nonnegative continuous random variable is given by Ebrahimi  has defined the uncertainty of residual lifetime distributions by truncating the distributions below some point of a component as follows:where is the failure rate or hazard rate defined by . Clearly for , represents the Shannon uncertainty contained in .
Di Crescenzo and Longobardi  have introduced past entropy over , since it is reasonable to presume that in many realistic situations uncertainty is not necessarily related to the future but can also refer to the past. They have also shown the necessity of past entropy and its relation with the residual entropy. If denotes the lifetime of an item or of a living organism, then past entropy (or uncertainty of lifetime distribution) of an item is defined aswhere is the reversed hazard rate of given by . In this paper, we will obtain and study the residual and past entropy of the Morgenstern family for concomitants of ordered random variables. We will also consider the characterization results based on the entropy function for concomitants of ordered random variables based on residual lifetime distribution and the past life distribution. The organization of the paper is as follows. In Section 2, we obtain the distribution function for concomitants of ordered random variables of the Morgenstern family. In Section 3, we obtain the residual and past entropy of our model and study the linear transformation and the upper bound for concomitants of ordered random variables. Some characterization results are presented in Section 4. Finally, some conclusions and comments are given in Section 5.
2. Distribution Function for Concomitants of Ordered Random Variables
Let , , be pairs of independent continuous nonnegative random variables from some bivariate population with df . From (1), the conditional distribution function of given is given byFor the Morgenstern family with conditional distribution function given by (11), the df of the concomitant of case-I of gos , , is given bywhere is the pdf of case-I of gos (see Kamps ), the df of the concomitant of case-I of dgos , , is given bywhere is the pdf of case-I of dgos (see Burkschat et al. ), the df of the concomitant of case-II of gos , , is given bywhere is the pdf of case-II of gos (see Kamps and Cramer ), and the df of the concomitant of case-II of dgos , , is given by where is the pdf of case-II of dgos (see Kamps and Cramer  and Burkschat et al. ).
Equations (4) to (7) and (12) to (15) can be combined as follows:whereIn the following section we will use the last equations to obtain and study the residual and past entropy of concomitants for Morgenstern family based on the types of gos.
3. Residual and Past Entropy
The following proposition gives the values of the functions and under linear transformation.
Proposition 3. For any absolutely continuous random variable , which is the concomitant of th ordered random variable of Morgenstern family defined in (18), define , where and are constants. Then, for , (1) (2)
3.1. Upper Bound for Residual and Past Entropy
We derive the upper bound for the residual and past entropy under the condition that the pdf for , defined in (18), is less than 1. Note that We know that, for , . Using this we get Hence,with equality in which when , is the Shannon entropy based on the concomitant of ordered random variables.
Next, we calculate an upper bound for the past entropy. We have We know that, for , . Using this we getHence,with equality in which when , is the Shannon entropy based on the concomitant of ordered random variables.
4. Some Characterization Results
Gupta et al.  have studied characterizations of residual and past entropy of order statistics. Here, in this section, we present some characterization results based on residual and past entropy of the concomitant of ordered random variables.
Consider a problem of finding sufficient condition for the uniqueness of the solution of the initial value problem (IVP): where is a given function of two variables whose domain is a region , is a specified point in , and is the unknown function. By the solution of the IVP on an interval , we mean a function such that (i) is differentiable on , (ii) the growth of lies in , (iii) , and (iv) , for all . The following theorem together with other results will help in proving our characterization result.
Theorem 4. Let the function be defined and continuous in a domain , and let satisfy a Lipschitz condition (with respect to ) in ; namely, for every point and in . Then the function satisfies the initial value problem and , , is unique.
Proof (see Gupta and Kirmani ). For any function of two variables defined in , we now present a sufficient condition which guarantees that the Lipschitz condition is satisfied in .
Lemma 5. Suppose that the function is continuous in a convex region . Suppose further that exists and is continuous in . The function satisfies Lipschitz condition in .
Proof (see Gupta and Kirmani ). We now present our two characterization results.
Theorem 6. Let be a nonnegative continuous random variable, defined in (18), with distribution function defined in (17). Let the residual entropy of the concomitant of th ordered random variable be denoted by , . Then characterizes the distribution.
Proof. Suppose there exist two functions and such that for all . Thenwhere is the hazard rate of the concomitant of th ordered random variable, for . Differentiating the above equation with respect to and simplifying, we get Suppose now that Then, for all , for , where It follows from Theorem 4 and Lemma 5 that , which prove the characterization result.
Theorem 7. Let be a nonnegative continuous random variable, defined in (18), with distribution function defined in (17). Let the past entropy of the concomitant of th ordered random variable be denoted by , . Then characterizes the distribution.
Proof. By the same way of the last theorem, where is the reversed hazard rate of the concomitant of th ordered random variable, for .
Next we present a characterization result of the linear mean residual family of the concomitant of th ordered random variable based on Theorem 6. The linear mean residual life of the concomitant of th ordered random variable is given by , , . It can be verified that the corresponding failure rate is given by . The linear mean residual family of a distribution has been studied among others by Hall and Wellner , Oakes and Dasu , and Gupta and Kirmani . It includes the exponential distribution for and the power distribution for . Gupta et al.  have studied the linear mean residual life of a distribution based on order statistics.
Theorem 8. Let be a nonnegative continuous random variable, defined in (18). Then if and only if
Proof. From (9), we haveTo find , first, we want to find Taking the derivative of the above equation with respect to and evaluating at , we get . Substituting the pervious results into (40) we get To prove the converse, (35) gives Since , then we getwhose solution is
5. Conclusion and Comments
The different types of gos are considered in one model based on the concomitants for Morgenstern family. From this model, we derived an analytical expression of the residual and past lifetime distribution and find some formulas of it. The dynamic entropy measures based on the concomitant of ordered random variables characterize uniquely the underlying distribution and are also bounded above in terms of Shannon entropy of the concomitant of the different types of gos. Stochastic comparisons can also be studied easily for such model.
Conflict of Interests
The authors declare that they have no conflict of interests.
Nayabuddin, “Concomitants of dual generalized order statistics from Farlie Gumbel Morgenstern type bivariate power function distribution,” Journal of Global Research in Mathematical Archives, vol. 1, no. 8, pp. 79–90, 2013.View at: Google Scholar
M. M. Mohie El-Din, M. M. Ameina, and M. S. Mohamed, “Concomitants of case-II of generalized order statistics from Farlie-Gumbel-Morgenstern distributions,” Journal of Statistics Applications and Probability, vol. 3, no. 3, pp. 345–353, 2015.View at: Google Scholar