A Bayesian Inference Approach for Bivariate Weibull Distributions Derived from Roy and Morgenstern Methods

Bivariate lifetime distributions are of great importance in studies related to interdependent components, especially in engineering applications. In this paper, we introduce two bivariate lifetime assuming three- parameter Weibull marginal distributions. Some characteristics of the proposed distributions as the joint survival function, hazard rate function, cross factorial moment and stress-strength parameter are also derived. The inferences for the parameters or even functions of the parameters of the models are obtained under a Bayesian approach. An extensive numerical application using simulated data is carried out to evaluate the accuracy of the obtained estimators to illustrate the usefulness of the proposed methodology. To illustrate the usefulness of the proposed model, we also include an example with real data from which it is possible to see that the proposed model leads to good fits to the data.

A Poisson XLindley Distribution with Applications

In this paper, a Poisson XLindley distribution (PXLD) has been obtained by compounding Poisson (PD) distribution with a continuous distribution. A general expression for its rth factorial moment about origin has been derived and hence its raw moments and central moments are obtained. The expressions for its coefficient of variation, skewness, kurtosis and index of dispersion have also been given. In particular, the method of maximum likelihood and the method of moments for the estimation of its parameters have been discussed. Finally, two real-life data sets are analyzed to investigate the suitability of the proposed distribution in modeling a real data set on Nipah virus infection, number of Hemocytometer yeast cell count data and epileptic seizure counts data.

ON ASYMPTOTIC RELATIONS FOR THE CRITICAL GALTON – WATSON PROCESS

Determining the asymptotics of the continuation probability for a Galton–Watson branching process is one of the most important problems in the theory of branching processes. This problem was solved by A.N. Kolmogorov (1938) in the case when the process starts with a single particle, and the classical result is obtained. A similar result for continuous branching processes was proved by B.A. Sevastyanov (1951). The next term in the expansion for continuous branching processes was obtained by V.M. Zolotarev (1957). The next term in the expansion for continuous branching processes in the critical case was obtained by V.P. Chistyakov (1957); the asymptotic expansion in the subcritical case under the condition of finiteness of the k-factorial moment was obtained by R. Mukhamedkhanova (1966). Asymptotic expansions for discrete branching processes in the subcritical and supercritical cases, provided that any m-factorial moment is finite, were obtained by S.V. Nagaev and R. Mukhamedkhanova (1966). In the critical case, the weak convergence of the conditional distribution of the quantity P(Z(n) > 0)Z(n) under the condition Z(n) > 0 to the exponential distribution was proved by A.M. Yaglom (1947) for processes starting with a single particle in the case of finiteness of the third moment of the number of generations. Subsequently, Spitzer, Kesten, and Ney (1966) proved this result under the condition that the second moment is finite. A similar result for branching processes with continuous parameters was established by V.M. Zolotarev (1957). In this paper, we study the asymptotics of the probability of continuation of the critical Galton-Watson process, starting with η particles. In addition, we prove an analogue of Yaglom’s theorem for critical Galton – Watson processes starting with a random number of particles.

Modeling multi-tier heterogeneous small cell networks: rate and coverage performance

The rapid growth of small cells is driving cellular network toward randomness and heterogeneity. The multi-tier heterogeneous network (HetNet) addresses the massive connectivity demands of the emerging cellular networks. Cellular networks are usually modeled by placing each tier (e.g macro, pico and relay nodes) deterministically on a grid which ignores the spatial randomness of the nodes. Several works were idealized for not capturing the interference which is a major performance bottleneck. Overcoming such limitation by realistic models is much appreciated. Multi-tier relay cellular network is studied in this paper, In particular, we consider $${\mathscr {K}}$$ K -tier transmission modeled by factorial moment and stochastic geometry and compare it with a single-tier, traditional grid model and multi-antenna ultra-dense network (UDN) model to obtain tractable rate coverage and coverage probability. The locations of the relays, base stations, and users nodes are modeled as a Poisson Point Process. The results showed that the proposed model outperforms the traditional multi-antenna UDN model and its accuracy is confirmed to be similar to the traditional grid model. The obtained results from the proposed and comparable models demonstrate the effectiveness and analytical tractability to study the HetNet performance.

Asymptotic Analysis of the kth Subword Complexity

Patterns within strings enable us to extract vital information regarding a string’s randomness. Understanding whether a string is random (Showing no to little repetition in patterns) or periodic (showing repetitions in patterns) are described by a value that is called the kth Subword Complexity of the character string. By definition, the kth Subword Complexity is the number of distinct substrings of length k that appear in a given string. In this paper, we evaluate the expected value and the second factorial moment (followed by a corollary on the second moment) of the kth Subword Complexity for the binary strings over memory-less sources. We first take a combinatorial approach to derive a probability generating function for the number of occurrences of patterns in strings of finite length. This enables us to have an exact expression for the two moments in terms of patterns’ auto-correlation and correlation polynomials. We then investigate the asymptotic behavior for values of k = Θ ( log n ) . In the proof, we compare the distribution of the kth Subword Complexity of binary strings to the distribution of distinct prefixes of independent strings stored in a trie. The methodology that we use involves complex analysis, analytical poissonization and depoissonization, the Mellin transform, and saddle point analysis.

Cluster Expansions for GIBBS Point Processes

We provide a sufficient condition for the uniqueness in distribution of Gibbs point processes with non-negative pairwise interaction, together with convergent expansions of the log-Laplace functional, factorial moment densities and factorial cumulant densities (correlation functions and truncated correlation functions). The criterion is a continuum version of a convergence condition by Fernández and Procacci (2007), the proof is based on the Kirkwood–Salsburg integral equations and is close in spirit to the approach by Bissacot, Fernández, and Procacci (2010). In addition, we provide formulas for cumulants of double stochastic integrals with respect to Poisson random measures (not compensated) in terms of multigraphs and pairs of partitions, explaining how to go from cluster expansions to some diagrammatic expansions (Peccati and Taqqu, 2011). We also discuss relations with generating functions for trees, branching processes, Boolean percolation and the random connection model. The presentation is self-contained and requires no preliminary knowledge of cluster expansions.

Chemical Kinetics Roots and Methods to Obtain the Probability Distribution Function Evolution of Reactants and Products in Chemical Networks Governed by a Master Equation

In this paper first, we review the physical root bases of chemical reaction networks as a Markov process in multidimensional vector space. Then we study the chemical reactions from a microscopic point of view, to obtain the expression for the propensities for the different reactions that can happen in the network. These chemical propensities, at a given time, depend on the system state at that time, and do not depend on the state at an earlier time indicating that we are dealing with Markov processes. Then the Chemical Master Equation (CME) is deduced for an arbitrary chemical network from a probability balance and it is expressed in terms of the reaction propensities. This CME governs the dynamics of the chemical system. Due to the difficulty to solve this equation two methods are studied, the first one is the probability generating function method or z-transform, which permits to obtain the evolution of the factorial moment of the system with time in an easiest way or after some manipulation the evolution of the polynomial moments. The second method studied is the expansion of the CME in terms of an order parameter (system volume). In this case we study first the expansion of the CME using the propensities obtained previously and splitting the molecular concentration into a deterministic part and a random part. An expression in terms of multinomial coefficients is obtained for the evolution of the probability of the random part. Then we study how to reconstruct the probability distribution from the moments using the maximum entropy principle. Finally, the previous methods are applied to simple chemical networks and the consistency of these methods is studied.