arXiv subjects

P. Vellaisamy

Publications and source records attributed to P. Vellaisamy.

43 records · Page 3Linked to original sources

arXiv2011

Average Collapsibility of Some Association Measures

Collapsibility deals with the conditions under which a conditional (on a covariate W) measure of association between two random variables X and Y equals the marginal measure of association, under the assumption of homogeneity over the covariate. In this paper, we discuss the average collapsibility of certain well-known measures of association, and also with respect to a new measure of association. The concept of average collapsibility is more general than collapsibility, and requires that the conditional average of an association measure equals the corresponding marginal measure. Sufficient conditions for the average collapsibility of the measures under consideration are obtained. Some difficult, but interesting, counter-examples are constructed. Applications to linear, Poisson, logistic and negative binomial regression models are addressed. An extension to the case of multivariate covariate W is also discussed.

math.ST↗

arXiv2011

A-Collapsibility of Distribution Dependence and Quantile Regression Coefficients

The Yule-Simpson paradox notes that an association between random variables can be reversed when averaged over a background variable. Cox and Wermuth (2003) introduced the concept of distribution dependence between two random variables X and Y, and developed two dependence conditions, each of which guarantees that reversal cannot occur. Ma, Xie and Geng (2006) studied the collapsibility of distribution dependence over a background variable W, under a rather strong homogeneity condition. Collapsibility ensures the association remains the same for conditional and marginal models, so that Yule-Simpson reversal cannot occur. In this paper, we investigate a more general condition for avoiding effect reversal: A-collapsibility. The conditions of Cox and Wermuth imply A-collapsibility, without assuming homogeneity. In fact, we show that, when W is a binary variable, collapsibility is equivalent to A-collapsibility plus homogeneity, and A-collapsibility is equivalent to the conditions of Cox and Wermuth. Recently, Cox (2007) extended Cochran's result on regression coefficients of conditional and marginal models, to quantile regression coefficients. The conditions of Cox and Wermuth are sufficient for A-collapsibility of quantile regression coefficients. If the conditional distribution of W, given Y = y and X = x, belong to one-dimensional natural exponential family, they are also necessary. Some applications of A-collapsibility include the analysis of a contingency table, linear regression models and quantile regression models.

stat.ME↗

arXiv2011

Time-Changed Poisson Processes

We consider time-changed Poisson processes, and derive the governing difference-differential equations (DDE) these processes. In particular, we consider the time-changed Poisson processes where the the time-change is inverse Gaussian, or its hitting time process, and discuss the governing DDE's. The stable subordinator, inverse stable subordinator and their iterated versions are also considered as time-changes. DDE's corresponding to probability mass functions of these time-changed processes are obtained. Finally, we obtain a new governing partial differential equation for the tempered stable subordinator of index $0<β<1,$ when $β$ is a rational number. We then use this result to obtain the governing DDE for the mass function of Poisson process time-changed by tempered stable subordinator. Our results extend and complement the results in Baeumer et al. \cite{B-M-N} and Beghin et al. \cite{BO-1} in several directions.

math.PR↗

arXiv2011

The fractional Poisson process and the inverse stable subordinator

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time.

math.PR↗

arXiv2010

Compound Poisson and signed compound Poisson approximations to the Markov binomial law

Compound Poisson distributions and signed compound Poisson measures are used for approximation of the Markov binomial distribution. The upper and lower bound estimates are obtained for the total variation, local and Wasserstein norms. In a special case, asymptotically sharp constants are calculated. For the upper bounds, the smoothing properties of compound Poisson distributions are applied. For the lower bound estimates, the characteristic function method is used.

math.ST↗

arXiv2009

Fractional Cauchy problems on bounded domains

Fractional Cauchy problems replace the usual first-order time derivative by a fractional derivative. This paper develops classical solutions and stochastic analogues for fractional Cauchy problems in a bounded domain $D\subset\mathbb{R}^d$ with Dirichlet boundary conditions. Stochastic solutions are constructed via an inverse stable subordinator whose scaling index corresponds to the order of the fractional time derivative. Dirichlet problems corresponding to iterated Brownian motion in a bounded domain are then solved by establishing a correspondence with the case of a half-derivative in time.

math.PR↗

arXiv2009

Fractional Normal Inverse Gaussian Process

Normal inverse Gaussian (NIG) process was introduced by Barndorff-Nielsen (1997) by subordinating Brownian motion with drift to an inverse Gaussian process. Increments of NIG process are independent and stationary. In this paper, we introduce dependence between the increments of NIG process, by subordinating fractional Brownian motion to an inverse Gaussian process and call it fractional normal inverse Gaussian (FNIG) process. The basic properties of this process are discussed. Its marginal distributions are scale mixtures of normal laws, infinitely divisible for the Hurst parameter 1/2<=H< 1 and are heavy tailed. First order increments of the process are stationary and possess long-range dependence (LRD) property. It is shown that they have persistence of signs LRD property also. A generalization of the FNIG process called n-FNIG process is also discussed which allows Hurst parameter H in the interval (n-1, n). Possible applications to mathematical finance and hydraulics are also pointed out

math.PR↗