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Palaniappan Vellaisamy

Bio: Palaniappan Vellaisamy is an academic researcher from Indian Institute of Technology Bombay. The author has contributed to research in topics: Poisson distribution & Negative binomial distribution. The author has an hindex of 19, co-authored 126 publications receiving 1683 citations. Previous affiliations of Palaniappan Vellaisamy include Indian Institutes of Technology & Michigan State University.


Papers
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TL;DR: It is established that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property.
Abstract: 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 $\beta$ satisfies $0<\beta<\frac{1}{3}$. We also establish that the FNBP has the LRD property, while the increments of the FNBP possesses the SRD property.

25 citations

Journal ArticleDOI
02 Jan 2018
TL;DR: In this article, the first-exit time process of an inverse Gaussian Levy process is considered and the one-dimensional distribution functions of the process are obtained, and the tail is not infinitely divisible.
Abstract: The first-exit time process of an inverse Gaussian Levy process is considered. The one-dimensional distribution functions of the process are obtained. They are not infinitely divisible and the tail...

24 citations

Journal ArticleDOI
TL;DR: In this article, the authors show that the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times, and that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit for random walks with uncorrelated jumps.

24 citations

Journal ArticleDOI
TL;DR: This paper introduces the simulation procedures for both processes and discusses the estimation schemes for their parameters, and studies the distributional properties of both subordinators, namely, IG and IIG processes, and also that of the FBM time changed by these subordinators.
Abstract: In this paper we study the fractional Brownian motion (FBM) time changed by the inverse Gaussian (IG) process and its inverse, called the inverse to the inverse Gaussian (IIG) process. Some properties of the time-changed processes are similar to those of the classical FBM, such as long-range dependence. However, one can also observe different characteristics that are not satisfied by the FBM. We study the distributional properties of both subordinators, namely, IG and IIG processes, and also that of the FBM time changed by these subordinators. We establish also the connections between the subordinated processes considered and the continuous-time random-walk model. For the application part, we introduce the simulation procedures for both processes and discuss the estimation schemes for their parameters. The effectiveness of these methods is checked for simulated trajectories.

24 citations

Journal ArticleDOI
TL;DR: In this paper, it is shown that the natural estimator is positively biased, and the uniformly minimum variance unbiased estimator (UMVE) of M is not minimax, while the minimazity of certain estimators of interest is investigated.
Abstract: Let л1 and л2 denote two independent gamma populations G(α1, p) and G(α2, p) respectively. Assume α(i=1,2)are unknown and the common shape parameter p is a known positive integer. Let Yi denote the sample mean based on a random sample of size n from the i-th population. For selecting the population with the larger mean, we consider, the natural rule according to which the population corresponding to the larger Yi is selected. We consider? in this paper, the estimation of M, the mean of the selected population. It is shown that the natural estimator is positively biased. We obtain the uniformly minimum variance unbiased estimator(UMVE) of M. We also consider certain subclasses of estikmators of the form c1x(1) +c1x(2) and derive admissible estimators in these classes. The minimazity of certain estimators of interest is investigated. Itis shown that p(p+1)-1x(1) is minimax and dominates the UMVUE. Also UMVUE is not minimax.

23 citations


Cited by
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TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

Book ChapterDOI
01 Jan 2011
TL;DR: Weakconvergence methods in metric spaces were studied in this article, with applications sufficient to show their power and utility, and the results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables.
Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

3,554 citations

Journal ArticleDOI
TL;DR: It is shown that, using an approximate stochastic weak solution to (linear) stochastically partial differential equations, some Gaussian fields in the Matérn class can provide an explicit link, for any triangulation of , between GFs and GMRFs, formulated as a basis function representation.
Abstract: Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere. (Less)

2,212 citations

Book
01 Jan 2013
TL;DR: In this paper, the authors consider the distributional properties of Levy processes and propose a potential theory for Levy processes, which is based on the Wiener-Hopf factorization.
Abstract: Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

1,957 citations

01 Jan 1996

1,282 citations