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In the present paper, we study the information generating (IG) function and relative information generating (RIG) function measures associated with maximum and minimum ranked set sampling (RSS) schemes with unequal sizes. We also examine the IG measures for simple random sampling (SRS) and provide some comparison results between SRS and RSS procedures in terms of dispersive stochastic ordering. Finally, we discuss the RIG divergence measure between SRS and RSS frameworks.

Information generating functions (IGFs) have been of great interest to researchers due to their ability to generate various information measures. The IGF of an absolutely continuous random variable (see Golomb, S. (1966). The information generating function of a probability distribution. IEEE Transactions in Information Theory , 12(1), 75–77) depends on its density function. But, there are several models with intractable cumulative distribution functions, but do have explicit quantile functions. For this reason, in this work, we propose quantile version of the IGF, and then explore some of its properties. Effect of increasing transformations on it is then studied. Bounds are also obtained. The proposed generating function is studied especially for escort and generalized escort distributions. Some connections between the quantile-based IGF (Q-IGF) order and well-known stochastic orders are established. Finally, the proposed Q-IGF is extended for residual and past lifetimes as well. Several examples are presented through out to illustrate the theoretical results established here. An inferential application of the proposed methodology is also discussed

In this work, we first consider the discrete version of information generating function and develop some new results for it. We then propose Jensen-discrete information generating (JDIG) function as a generalized measure, which is connected to Shannon entropy, fractional Shannon entropy, Gini–Simpson index (Gini entropy), extropy, Jensen–Shannon entropy, Jensen–Gini–Simpson index and Jensen-extropy measures. Finally, for illustrative purpose, we use a real example from image processing and provide some numerical results in terms of the Jensen-discrete information generating function and demonstrate that the JDIG measure introduced here is an useful criteria for measuring similarity between two images.

Evaluating the negation of an uncertain event is an open issue. Yager (IEEE Trans Fuzzy Syst 23:1899–1902, 2004) suggested a transformation for evaluating the negation of a probability distribution. He used the idea that any event whose outcome is not certain can be negated by supporting the occurrence of other events with no bias or prejudice for any particular outcome. Various authors have tried to generalize the negation transformation proposed by Yager (IEEE Trans Fuzzy Syst 23:1899–1902, 2004). However, we need to focus on developing the basic structure of negation so that the behaviour of the process modelled by negation transformation can be understood in detail. Yager’s negation is based on distribution of maximum entropy. If a probability distribution is uncertain(a state other than maximum entropy), the more the iterations of negation, the more uncertain this probability event becomes, eventually converging to a homogeneous state, i.e. maximum entropy. In other words, it is the realization of the process. What is noted that during each negation, Yager’s method ensures that the negation is intuitive; the next negation weakens the probability of the event occurring in the previous step. Since negation involves reallocation of probabilities at each step in such a way that the reallocation at each step can be determined from the reallocation at the previous step, it is clear that Yager’s negation has various attributes similar to that of a Markov chain. In the present work, we have shown that Yager’s definition of negation can be modelled as a Markov chain which is irreducible, aperiodic with no absorbing states. Two examples have been discussed to strengthen and support the analytical results. Also, we have defined an information generating function (IGF) whose derivative evaluated at specific points gives the moments of the self-information of negation of a probability distribution. The properties of the generating function along with its relationship with the information generating function proposed by S. Golomb (IEEE Trans Inf Theory 12:75–77, 1966) have been explored. A closer look at the properties of IGF confirms the existence of Markovian structure of Yager’s negation.

Upon the motivation of unstable climatic conditions of the world like excess of rains, drought and huge floods, we introduce a versatile hydrologic probability model with two scale parameters. The proposed model contains Lindley and exponentiated exponential (Lindley in J R Stat Soc Ser B 20:102–107, 1958; Gupta and Kundu in Biom J 43(1):117–130, 2001) distributions as special cases. Various properties of the distribution are obtained, such as shapes of the density and hazard functions, moments, mean deviation, information-generating function, conditional moments, Shannon entropy, L-moments, order statistics, information matrix and characterization via hazard function. Parameters are estimated via maximum likelihood estimation method. A simulation scheme is provided for generating the random data from the proposed distribution. Four data sets are used for comparing the proposed model with a set of well-known hydrologic models, such as generalized Pareto, log normal (3), log Pearson type III, Kappa(3), Gumbel, generalized logistic and generalized Lindley distributions, using some goodness-offit tests. These comparisons render the proposed model suitable and representative for hydrologic data sets with least loss of information attitude and a realistic return period, which render it as an appropriate alternate of the existing hydrologic models. Supplementary materials for this paper are available online.

In this paper we discuss four information theoretic ideas and present their implications to statistical inference: (1) Fisher information and divergence generating functions, (2) information optimum unbiased estimators, (3) information content of various statistics, (4) characterizations based on Fisher information.

It is well known that the traditional Jensen inequality is proved by lower bounding the given convex function, f(x), by the tangential affine function that passes through the point (E{X},f(E{X})), where E{X} is the expectation of the random variable X. While this tangential affine function yields the tightest lower bound among all lower bounds induced by affine functions that are tangential to f, it turns out that when the function f is just part of a more complicated expression whose expectation is to be bounded, the tightest lower bound might belong to a tangential affine function that passes through a point different than (E{X},f(E{X})). In this paper, we take advantage of this observation by optimizing the point of tangency with regard to the specific given expression in a variety of cases and thereby derive several families of inequalities, henceforth referred to as “Jensen-like” inequalities, which are new to the best knowledge of the author. The degree of tightness and the potential usefulness of these inequalities is demonstrated in several application examples related to information theory.

The robustness of complex networks against node failure and malicious attack has been of interest for decades, while most of the research has focused on random attack or hub-targeted attack. In many real-world scenarios, however, attacks are neither random nor hub-targeted, but localized, where a group of neighboring nodes in a network are attacked and fail. In this paper we develop a percolation framework to analytically and numerically study the robustness of complex networks against such localized attack. In particular, we investigate this robustness in Erd s-Rényi networks, random-regular networks, and scale-free networks. Our results provide insight into how to better protect networks, enhance cybersecurity, and facilitate the design of more robust infrastructures.

Current network models assume one type of links to define the relations between the network entities. However, many real networks can only be correctly described using two different types of relations. Connectivity links that enable the nodes to function cooperatively as a network and dependency links that bind the failure of one network element to the failure of other network elements. Here we present an analytical framework for studying the robustness of networks that include both connectivity and dependency links. We show that a synergy exists between the failure of connectivity and dependency links that leads to an interative process of cascading failures that has a devastating effect on the network stability. We present exact analytical results for the dramatic change in the network behavior when introducing dependency links. For a high density of dependency links, the network disintegrates in a form of a first-order phase transition, whereas for a low density of dependency links, the network disintegrates in a second-order transition. Moreover, opposed to networks containing only connectivity links where a broader degree distribution results in a more robust network, when both types of links are present a broad degree distribution leads to higher vulnerability.

Waiting line problems with server vacation have envisaged with increasing complexities and their explicit transient solutions are rigorous in computations, at the same time such solutions are valued for studying the dynamical behaviour of queuing systems over a finite period predominantly utilizes within the state-of-art design process for a real time system. Keeping this fact in mind we adopt continued fractions and generating function to derive explicit expressions for transient state probabilities. In this paper, we consider the waiting line problem with a single server which adopts the multi vacations policy. We analyzed the transient part for a single server multi vacations queue with discouragement . It is also obtained the expected value of the state of the system using stationary queue size distribution, which gives a quick glance of a system performance.