Neelesh Shankar Upadhye

In this article, we first propose the modified Hannan–Rissanen Method for estimating the parameters of autoregressive moving average (ARMA) process with symmetric stable noise and symmetric stable generalized autoregressive conditional heteroskedastic (GARCH) noise. Next, we propose the modified empirical characteristic function method for the estimation of GARCH parameters with symmetric stable noise. Further, we show the efficiency, accuracy and simplicity of our methods with Monte-Carlo simulation. Finally, we apply our proposed methods to model the financial data.

In this study, we construct a linear recurrent fractal interpolation function (RFIF) with variable scaling parameters for data set with α -stable noise (a generalization of Gaussian noise) on its ordinate, which captures the uncertainty at any missing or unknown intermediate point. The propagation of uncertainty in this linear RFIF is investigated, and a method for estimating parameters of the uncertainty at any interpolated value is provided. Moreover, a simulation study to visualize uncertainty for interpolated values is presented.

In this article, we first review the connection between Lévy processes and infinitely divisible random variables, and discuss the classification of infinitely divisible distributions. Next, we establish a Stein identity for an infinitely divisible random variable via the Lévy-Khintchine representation of the characteristic function. In particular, we establish and unify the Stein identities for an α-stable random variable available in the existing literature. Next, we derive the solutions of the Stein equations and its regularity estimates. Further, we derive error bounds for α-stable approximations, and also obtain rates of convergence results in Wasserstein-δ, δ < α for α ∈ (0, 1) and Wasserstein-type distances for α ∈ (1, 2). Finally, we compare these results with existing literature.

In this work, we analyze the maximum number of wireless transmitters (nodes) that can be scheduled subject to interference constraints across the nodes. Given a set of nodes, the problem reduces to finding the maximum cardinality of a subset that can concurrently transmit without violating interference constraints. The resulting packing problem is a binary optimization problem, which is NP hard. We propose a semi-definite (SDP) relaxation for the NP hard problem and discuss the algorithm and the quality of the relaxation by providing approximation ratios for the relaxation.

In this paper, we mainly focus on the sums of non-negative integer-valued 1-dependent random variables and its approximation to the power series distribution. We first discuss some relevant results for power series distribution such as the Stein operator, uniform and non-uniform bounds on the solution of the Stein equation. Using Stein’s method, we obtain error bounds for the approximation problem considered. The obtained results can also be applied to the sums of m-dependent random variables via appropriate rearrangements of random variables. As special cases, we discuss two applications, namely, 2-runs and (k1,k2)-runs, and compare our bounds with existing bounds.

It is shown that the tail behavior of the function of nonnegative random variables can be characterized using deterministic functions satisfying certain properties. Also, the upper and lower bounds for the tail of product of random variables are given. Applications of these results are given to some of the well-known models in economics and risk theory.

In this article, we obtain a Stein operator for the sum of n independent random variables (rvs) which is shown as the perturbation of the negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three-parameter approximation for such a sum is considered and is shown to improve the existing bounds in the literature. Finally, an application of our results to a function of waiting time for (k1, k2)-events is given.

In this work, we consider compound negative binomial and compound Poisson approximations to the generalized Poisson-binomial distribution. We derive some total variation upper bounds which improve on the existing results in terms of the order of approximation. An application is also discussed. © 2012 Elsevier B.V.

In this article, we study the Poisson process of order (Formula presented.) (PPoK) time-changed with an independent Lévy subordinator and its inverse, which we call, respectively, as TCPPoK-I and TCPPoK-II, through various distributional properties, long-range dependence and limit theorems for the PPoK and the TCPPoK-I. Further, we study the governing difference-differential equations of the TCPPoK-I for the case inverse Gaussian subordinator. Similarly, we study the distributional properties, asymptotic moments and the governing difference-differential equation of TCPPoK-II. As an application to ruin theory, we give a governing differential equation of ruin probability in insurance ruin using these processes. Finally, we present some simulated sample paths of both the processes.

The distribution of (k1,k2)-runs is well-known (Dafnis et al., 2010), under independent and identically distributed (i.i.d.) setup of Bernoulli trials but is intractable under non i.i.d. setup. Hence, it is of interest to find a suitable approximate distribution for (k1,k2)-runs, under non i.i.d. setup, with reasonable accuracy. In this paper, pseudo-binomial approximation to (k1,k2)-runs is considered using total variation distance. The approximation results derived are of optimal order and improve the existing results.