**1. Introduction**

The notion of concomitants or induced order statistics arose in the early 1970s in the works of David [1] and Bhattacharya [2]. Briefly, when there is a sample from a bivariate distribution ordered by the first variate, the second variate paired with the r-th first variate is called the concomitant of the r-th-order statistic. Concomitants are important in situations in which are implied two characteristics and measuring one of them can influence the other. Therefore, they have applications in many fields such as selection procedures, inference problems, double sampling plans and systems reliability. For example, in [3,4] are studied from a reliability point of view complex systems with components which have two subcomponents that performs different tasks, and in [5], the distribution theory of lifetimes of two component systems is discussed. In studies regarding the concomitants there are two elements that have to be mentioned: the kind of dependence between first and second variate, and the kind of order for the first variate. The majority of studies are based on the hypothesis of simple order statistics, but there are also studies that assume different kinds of orders such as as record values order or generalized order statistics.

Generalized order statistics (GOS) was introduced by Kamps [6] and it is a unifying concept for various types of order statistics such as simple order statistics, record values, sequential order statistics.

In this paper, we focus on the concomitants of GOS and with the dependence structure between the first variate and the second variate given by the Fairlie–Gumbel–Morgenstern (FGM) family. This family is a flexible family of bivariate distributions used as a modeling tool for bivariate data in many fields [7], one such field being Reliability, see [3–5]. The FGM family has a simple analytical form, but it can describe only relatively weak dependence because the correlation coefficient between the two components cannot exceed 1/3. To prevent this limitation, extensions of FGM family have been proposed, for example, iterative FGM distributions or Huang–Kotz FGM distributions [8–12]. The results obtained in our paper will be generalized for these extensions of FGM family in a future work.

For the concomitants mentioned above, we recall and determine properties that some information measures have. The information measures that we deal with are in two

**Citation:** Suter, F.; Cernat, I.; Dr ˘agan, M. Some Information Measures Properties of the GOS-Concomitants from the FGM Family. *Entropy* **2022**, *24*, 1361. https://doi.org/ 10.3390/e24101361

Academic Editors: Karagrigoriou Alexandros, Makrides Andreas, Yong Deng and Takuya Yamano

Received: 19 July 2022 Accepted: 21 September 2022 Published: 26 September 2022

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categories, information measures related to Shannon entropy and information measures related to Tsallis entropy.

Since it was introduced in physics and adapted to information theory by Shannon in 1948, the concept of entropy has become more and more important in fields such as information theory, code theory, probability and statistics, reliability.

In probability and statistics, entropy measures the uncertainty associated to a random variable. Taking as a starting point Shannon entropy, a series of entropies have been defined as a generalization of it. For the concomitants of GOS from the FGM family, we will look at Shannon-derived and Tsallis-derived entropies, and our main aim is to determine Awad-type extensions for all the considered entropies, because Awad entropies do not have several drawbacks that Shannon entropy, for example, has: different systems with the same entropy, possible negative values for continuous distributions, different results in discrete and continuous case of linear random variable transformation, etc.

Furthermore, for the concomitants of GOS from FGM, we will determine not only entropies, but also other information measures such as Tsallis divergence and shift-invariant Fisher–Tsallis information number.

In the following sections, we recall some definitions and properties of GOS and their concomitants, in particular, when the bivariate distribution is in the FGM family. Then, we will discuss Shannon-type entropies, Tsallis-type entropies, Fisher information and divergences for concomitants of GOS from the FGM family. For these concomitants, in the last section, we will introduce new extensions and results on information measures.

#### **2. Generalized Order Statistics and Their Concomitants for the FGM Family**
