arXiv subjects

P. Vellaisamy

Publications and source records attributed to P. Vellaisamy.

At least 37 records · Page 2Linked to original sources

arXiv2017

Saigo Space-Time Fractional Poisson Process via Adomian Decomposition Method

We obtain the state probabilities of various fractional versions of the classical homogeneous Poisson process using an alternate and simpler method known as the Adomian decomposition method (ADM). Generally these state probabilities are obtained by evaluating probability generating function using Laplace transform. A generalization of the space and time fractional Poisson process involving the Caputo type Saigo differential operator is introduced and its state probabilities are obtained using ADM.

math.PR↗

arXiv2016

First-Exit Times of an Inverse Gaussian Process

The first-exit time process of an inverse Gaussian Lévy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail probabilities decay exponentially. These distribution functions can also be viewed as distribution functions of supremum of the Brownian motion with drift. The density function is shown to solve a fractional PDE and the result is also generalized to tempered stable subordinators. The subordination of this process to the Brownian motion is considered and the underlying PDE of the subordinated process is obtained. The infinite divisibility of the first-exit time of a $β$-stable subordinator is also discussed.

math.PR↗

arXiv2016

On Densities of the Product, Quotient and Power of Independent Subordinators

We obtain the closed form expressions for the densities of the product, quotient, power and scalar multiple of independent stable subordinators. Similar results for the independent inverse stable subordinators are discussed. This is achieved by expressing the densities of stable and inverse stable subordinators in terms of the Fox's $H$-function.

math.PR↗

arXiv2016

On Stein operators for discrete approximations

In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, Stein operators for certain compound distributions, where the random summand satisfies Panjer's recurrence relation, are derived. A well-known perturbation approach for Stein's method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

math.PR↗

arXiv2016

Space-fractional versions of the negative binomial and Polya-type processes

In this paper, we introduce a space fractional negative binomial (SFNB) process by subordinating the space fractional Poisson process to a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB process, where the fractional index follows a two-point distribution, is analyzed in detail. The connections of the SFNB process to a space fractional Polya-type process is also pointed out. Moreover, we define and study a multivariate version of the SFNB obtained by subordinating a $d$-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications of the SFNB process to the studies of population's growth and epidemiology are pointed out. Finally, we discuss an algorithm for the simulation of the SFNB process.

math.PR↗

arXiv2016

On the Long-range Dependence of Fractional Poisson and Negative Binomial Processes

We study the long-range dependence (LRD) of the increments of the fractional Poisson process (FPP), the fractional negative binomial process (FNBP) and the increments of the FNBP. We first point out an error in the proof of Theorem 1 of Biard and Saussereau (2014) and prove that the increments of the FPP has indeed the short-range dependence (SRD) property, when the fractional index $β$ satisfies $0<β<\frac{1}{3}$. We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.

math.PR↗

arXiv2015

Discrete approximations for sums of m-dependent random variables

Sums of of 1-dependent integer-valued random variables are approximated by compound Poisson, negative binomial and Binomial distributions and signed compound Poisson measures. Estimates are obtained for total variation and local metrics. The results are then applied to statistics of $m$-dependent $(k_1,k_2)$ events and 2-runs. Heinrich's method and smoothing properties of convolutions are used for the proofs.

math.ST↗

arXiv2015

The Unbiasedness Approach to Linear Regression Models

The linear regression models are widely used statistical techniques in numerous practical applications. The standard regression model requires several assumptions about the regres- sors and the error term. The regression parameters are estimated using the least-squares method. In this paper, we consider the regression model with arbitrary regressors and with- out the error term. An explicit expression for the regression parameters vector is obtained. The unbiasedness approach is used to estimate the regression parameters and its various properties are investigated. It is shown that the resulting unbiased estimator equals the least-squares estimator for the fixed design model. The analysis of residuals and the regres- sion sum of squares can be carried out in a natural way. The unbiased estimator of the dispersion matrix of the unbiased estimator is also obtained. Applications to AR(p) model and numerical examples are also discussed.

stat.ME↗

arXiv2015

The I-Function Distribution and its Extensions

In this paper we introduce a new probability distribution on (0,1), associated with the I-function, namely, the I-function distribution. This distribution generalizes several known distributions with positive support. It is also shown that the distribution of products, quotients and powers of independent I-function variates are I-function variates. Another distribution called the I-function inverse Gaussian distribution is also introduced and studied.

math.PR↗

arXiv2014

Fractional Negative Binomial and Polya Processes

In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. First, it is shown that the one-dimensional distributions of the FPP are not infinitely divisible. The long-range dependence of the FNBP, the short-range dependence of its increments and the infinite divisibility of the FPP and the FNBP are investigated. Also, the space fractional Polya process (SFPP) is defined by replacing the rate parameter $λ$ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to $pde$'$s$ governing the density of the FNBP and the SFPP are also investigated.

math.PR↗

arXiv2014

Inverse Tempered Stable Subordinators

We consider the first-hitting time of a tempered $β$-stable subordinator, also called inverse tempered stable (ITS) subordinator. The density function of the ITS subordinator is obtained, for the index of stability $β\in (0,1)$. The series representation of the ITS density is also obtained, which could be helpful for computational purposes. The asymptotic behaviors of the $q$-th order moments of the ITS subordinator are investigated. In particular, the limiting behaviors of the mean of the ITS subordinator is given. The limiting form of the ITS density, as the space variable $x\rightarrow 0$, and its $k$-th order derivatives are obtained. The governing PDE for the ITS density is also obtained. The corresponding known results for inverse stable subordinator follow as special cases.

math.PR↗

arXiv2014

Some Fractional Calculus results associated with the $I$-Function

The effect of Marichev-Saigo-Maeda (MSM) fractional operators involving third Appell function on the $I$ function is studied. It is shown that the order of the $I$-function increases on application of these operators to the power multiple of the $I$-function. The Caputo-type MSM fractional derivatives are introduced and studied for the $I$-function. As special cases, the corresponding assertions for Saigo and Erdélyi-Kober fractional operators are also presented. The results obtained in this paper generalize several known results obtained recently in the literature.

math.CA↗

arXiv2014

The Generalized $K$-Wright Function and Marichev-Saigo-Maeda Fractional Operators

In this paper, the generalized fractional operators involving Appell's function $F_3$ in the kernel due to Marichev-Saigo-Maeda are applied to the generalized $K$-Wright function. These fractional operators when applied to power multipliers of the generalized $K$-Wright function ${}_{p}Ψ^k_q$ yields a higher ordered generalized $K$-Wright function, namely, ${}_{p+3}Ψ^k_{q+3}$. The Caputo-type modification of Marichev-Saigo-Maeda fractional differentiation is introduced and the corresponding assertions for Saigo and Erdélyi-Kober fractional operators are also presented. The results derived in this paper generalize several recent results in the theory of special functions.

math.CA↗

arXiv2014

On Probabilistic Proofs of Certain Binomial Identities

We give a simple statistical proof of a binomial identity, by evaluating the Laplace transform of the maximum of n independent exponential random variables in two different ways. As a by product, we obtain a simple proof of an interesting result concerning the exponential distribution. The connections between a probabilistic approach and the statistical approach are discussed, which explains why certain binomial identities admit probabilistic interpretations. In the process, several new binomial identities are also obtained and discussed.

math.ST↗

arXiv2014

On the Construction and the Cardinality of Finite $σ$-Fields

In this note, we first discuss some properties of generated $σ$-fields and a simple approach to the construction of finite $σ$-fields. It is shown that the $σ$-field generated by a finite class of $σ$-distinct sets which are also atoms, is the same as the one generated by the partition induced by them. The range of the cardinality of such a generated $σ$-field is explicitly obtained. Some typical examples and their complete forms are discussed. We discuss also a simple algorithm to find the exact cardinality of some particular finite $σ$-fields. Finally, an application of our results to statistics, with regard to independence of events, is pointed out.

math.PR↗

arXiv2014

Simpson's Paradox and Collapsibility

Simpson's paradox and collapsibility are two closely related concepts in the context of data analysis. While the knowledge about the occurrence of Simpson's paradox helps a statistician to draw correct and meaningful conclusions, the concept of collapsibility deals with dimension-reduction aspects, when Simpson's paradox does not occur. We discuss in this paper in some detail the nature and the genesis of Simpson's paradox with respect to well-known examples and also various concepts of collapsiblity. The main aim is to bring out the close connections between these two phenomena, especially with regard to the analysis of contingency tables, regression models and a certain measure of association or a dependence function. There is a vast literature on these topics and so we focus only on certain aspects, recent developments and some important results in the above-mentioned areas.

math.ST↗

arXiv2014

A Compound Poisson Convergence Theorem for Sums of $m$-Dependent Variables

We prove the Simons-Johnson theorem for the sums $S_n$ of $m$-dependent random variables, with exponential weights and limiting compound Poisson distribution $\CP(s,λ)$. More precisely, we give sufficient conditions for $\sum_{k=0}^\infty\ee^{hk}\ab{P(S_n=k)-\CP(s,λ)\{k\}}\to 0$ and provide an estimate on the rate of convergence. It is shown that the Simons-Johnson theorem holds for weighted Wasserstein norm as well. %limiting sum of two Poisson variables defined on %different lattices. The results are then illustrated for $N(n;k_1,k_2)$ and $k$-runs statistics.

math.ST↗

arXiv2013

Non-uniform approximations for sums of discrete m-dependent random variables

Non-uniform estimates are obtained for Poisson, compound Poisson, translated Poisson, negative binomial and binomial approximations to sums of of m-dependent integer-valued random variables. Estimates for Wasserstein metric also follow easily from our results. The results are then exemplified by the approximation of Poisson binomial distribution, 2-runs and $m$-dependent $(k_1,k_2)$-events.

math.PR↗