Now showing 1 - 10 of 15
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    Estimation of the parameters of symmetric stable ARMA and ARMA–GARCH models
    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.
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    Distribution of Noise in Linear Recurrent Fractal Interpolation Functions for Data Sets with α -Stable Noise
    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.
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    A unified approach to Stein’s method for stable distributions
    (01-01-2022) ;
    Barman, Kalyan
    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.
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    Approximations related to the sums of m-dependent random variables
    (01-06-2022)
    Kumar, Amit N.
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    Vellaisamy, P.
    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.
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    Time-changed Poisson processes of order k
    (02-01-2020)
    Sengar, Ayushi S.
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    Maheshwari, A.
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    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.
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    Convoluted fractional Poisson process of order k
    (01-01-2023)
    Sengar, Ayushi S.
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    In this article, we define a convoluted fractional Poisson process of order k (CFPPoK), which is governed by the discrete convolution operator in the system of fractional differential equations. Next, we obtain its one-dimensional distribution by using the Laplace transform of its state probabilities. Various distributional properties, such as probability generating function, moment generating function and moments, are derived. A special case of CFPPoK, (say) convoluted Poisson process of order k (CPPoK) is studied and also established Martingale characterization for CPPoK. We further derive the covariance structure of CFPPoK and investigate the long-range dependence property.
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    Linear fractal interpolation function for data set with random noise
    Fractal interpolation is a contemporary technique to approximate numerous scientific experiments and natural phenomena. For data sets in 2, the simplest and easy-to-handle fractal interpolation functions (FIFs) are linear. In this study, we estimate probability distributions of linear FIFs for data sets with various types of random noise. In order to evaluate the distribution of any linear FIF associated with a prescribed data set having Student's t-distributed noise, we develop a technique to approximate the distribution of a linear combination of independent generalized Student's t-distributed random variables. In addition, we provide some statistical properties and numerical approximations of these linear fractal functions.
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    DISTRIBUTION of LINEAR FRACTAL INTERPOLATION FUNCTION for RANDOM DATASET with STABLE NOISE
    In this paper, we derive the probability distribution of linear fractal interpolation function (FIF) for a random dataset on R2, where randomness in the data is induced, using α-stable noise, in the second coordinate. In particular, we show that the distribution of linear FIF, at any point (on the first coordinate), is also stable, but with different parameters which can be estimated. Finally, we also provide statistical validity of our results.
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    Linear Recurrent Fractal Interpolation Function for Data Set with Gaussian Noise
    In this article, we use the linear recurrent fractal interpolation function approach to interpolate a data set with Gaussian noise on its ordinate. To investigate the variability at any intermediate point in the given noisy data set, we estimate the parameters of the probability distribution of the fractal function. In addition, we present a simulation study that experimentally confirms our theoretical findings.
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    Forecasting of symmetric α−stable autoregressive models by time series approach supported by artificial neural networks
    (01-06-2023) ; ;
    Wyłomańska, Agnieszka
    Recent research activities in forecasting suggest that artificial neural networks can be a promising alternative to the traditional linear models. However, no single model, either linear or nonlinear is capable of obtaining the forecasts accurately. In this paper, a hybrid methodology that combines symmetric α-stable autoregressive time series and artificial neural networks is proposed. The methodology is validated through Monte-Carlo simulations. Moreover, the new method is used to model real empirical data thus showing the usefulness of heavy-tailed models supported by artificial neural networks in statistical modeling.