Calyampudi Radhakrishna Rao

Model Selection with Data-Oriented Penalty.

We consider the problem of model (or variable) selection in the classical regression model using the GIC (general information criterion). In this method the maximum likelihood is used with a penalty function denoted by C n , depending on the sample size n and chosen to ensure consistency in the selection of the true model. There are various choices of C n suggested in the literature on model selection. In this paper we show that a particular choice of C n based on observed data, which makes it random, preserves the consistency property and provides improved performance over a fixed choice of C n .

The legacy of Alladi Ramakrishnan in the mathematical sciences

- Preface Part I. The Legacy of Alladi Ramakrishnan - Contributions of Alladi Ramakrishnan to the Mathematical Sciences. - Alladi Ramakrishnans Theoretical Physics Seminar. - Telegrams received for the MATSCIENCE inauguration. - The miracle has happened, speech given by Alladi Ramakrishnan at the inauguration of MATSCIENCE. - Overseas Trips of Alladi Ramakrishnan. - List of publications of Alladi Ramakrishnan. - List of PhD students of Alladi Ramakrishnan. Part II. Pure Mathematics - Inversion and invariance of characteristic terms - part I (Abhyankar). - Partitions with non-repeating odd parts and q-hypergeometric identities (Alladi). - q-Catalan identities (Andrews). - Completing Brahmaguptas extension of Ptolemys theorem (Askey). - A transformation formula involving the gamma and Riemann zeta functions in Ramanujans Lost Notebook (Berndt and Dixit). - Ternary quadratic forms, modular equations, and certain positivity conjectures (Berkovich and Jagy). - How often is n! a sum of three squares? (Deshouillers and Luca).- Crystal symmetry viewed as zeta symmetry - II (Kanemitsu and Tsukuda). - Eulerian polynomials: From Eulers time to the present (Foata). - Crystal symmetry viewed as zeta symmetry II (Kanemitsu and Tsukada). - Positive homogeneous minima for a system of linear forms (Raghavan). - The divisor matrix, Dirichlet series, and SL(2,Z)(Sin and Thompson). - Proof of a conjecture of Alladi Ramakrishnan on circulants (Waldschmidt). Part III. Probability and Statistics - Branching random walks (Athreya). - A commentary on the logistic distribution (Ghosh, Choi, and Li). - Entropy and cross entropy characterizations and applications (Rao). - Optimal weights for a class of rank tests for censored bivariate data (Rao, Raychaudhuri, and Wu). - Connections between Bernoulli strings and random permutations (Sethuraman and Sethuraman). - Storage models for a class of master equations with separable kernels (Vittal, Jayasankar, and Muralidhar). Part IV. Theoretical Physics and Applied Mathematics - Inverse consistent deformable image registration (Chen and Ye). - A statistical model for the quark structure of the nucleon (Devanathan and Karthiyayini). - On generalized Clifford algebras and their physical applications (Jagannathan). - (p,q)-Rogers-Szego polynomial and the (p,q)-oscillator (Jagannathan and Sridhar). - Rethinking renormalization (Klauder). - Magnetism, FeS celluloids, and origins of life (Mitra-Delmotte and Mitra). - The Ehrenfest theorem in quantum field theory (Parthasarathy).

Some unified characterization results on generalized Pareto distributions

In the present paper, we unify and extend various characterizations of exponential and geometric distributions, such as those based on order statistics, record values and the strong memoryless property to arrive at characterization results for the generalized Pareto distributions and their discrete versions. We introduce a new concept of extended neighbouring order statistics without restricting to distributions that are absolutely continuous with respect to Lebesgue measure.

Clustering social networks using ant colony optimization

Several e-marketing applications rely on the ability to understand the structure of social networks. Social networks can be represented as graphs with customers as nodes and their interactions as edges. Most real world social networks are known to contain extremely dense subgraphs (also called as communities) which often provide critical insights about the emergent properties of the social network. The communities, in most cases, correspond to the various segments in a social system. Such an observation led researchers to propose algorithms to detect communities in networks. A modularity measure representing the quality of a network division has been proposed which on maximization yields good partitions. The modularity maximization is a strongly NP-complete problem which renders mathematical programming based optimization intractable for large problem sizes. Many heuristics based on simulated annealing, genetic algorithms and extremal optimization have been used to maximize modularity but have lead to suboptimal solutions. In this paper, we propose an ant colony optimization (ACO) based approach to detect communities. To the best of our knowledge, this is the first application of ACO to community detection. We demonstrate that ACO based approach results in a significant improvement in modularity values as compared to existing heuristics in the literature. The reasons for this improvement when tested on real and synthetic data sets are discussed.

An efficient algorithm for estimating the parameters of superimposed exponential signals

An efficient computational algorithm is proposed for estimating the parameters of undamped exponential signals, when the parameters are complex valued. Such data arise in several areas of applications including telecommunications, radio location of objects, seismic signal processing and computer assisted medical diagnostics. It is observed that the proposed estimators are consistent and the dispersion matrix of these estimators is asymptotically the same as that of the least squares estimators. Moreover, the asymptotic variances of the proposed estimators attain the Cramer-Rao lower bounds, when the errors are Gaussian.

Models for Binary Response Variables

The test procedures in the linear regression model are based on the normal distribution of the error variable ∊ and thus on a normal distribution of the endogenous variable Y. However, in many fields of application this assumption may not be true. The response variable Y may be defined as a binary variable, or more generally, as a categorical variable. Thus Y has a binomial or a multinomial distribution.

Confidence limits to the distance of the true distribution from a misspecified family by bootstrap

In statistical practice, an estimated distribution function (d.f.) from a specified family is used for taking decisions. When the true d.f. from which samples are drawn does not belong to the specified family, it is of interest to know how close the true d.f. is to the specified family. In this paper, we use non-parametric bootstrap to obtain confidence limits to the difference between the true d.f. and a member of the specified family closest to it in the sense of Kullback-Leibler measure.

Approximation of the expected value of the harmonic mean and some applications

Significance The harmonic mean (HM) filter is better at removing positive outliers than the arithmetic mean (AM) filter. There are especially difficult issues when an accurate evaluation of expected HM is needed such as, for example, in image denoising and marginal likelihood evaluation. A major challenge is to develop a higher-order approximation of the expected HM when the central limit theorem is not applicable. A two-term approximation of the expected HM is derived in this paper. This approximation enables us to develop a new filtering procedure to denoise the noisy image with an improved performance, and construct a truncated HM estimator with a faster convergence rate in marginal likelihood evaluation. Although the harmonic mean (HM) is mentioned in textbooks along with the arithmetic mean (AM) and the geometric mean (GM) as three possible ways of summarizing the information in a set of observations, its appropriateness in some statistical applications is not mentioned in textbooks. During the last 10 y a number of papers were published giving some statistical applications where HM is appropriate and provides a better performance than AM. In the present paper some additional applications of HM are considered. The key result is to find a good approximation to E(Hn), the expectation of the harmonic mean of n observations from a probability distribution. In this paper a second-order approximation to E(Hn) is derived and applied to a number of problems.

An M-estimation-based criterion for simultaneous change point analysis and variable selection in a regression problem

In this paper, an M-estimation-based criterion is proposed for carrying out change point analysis and variable selection simultaneously in linear models with a possible change point, which includes the criterion proposed in Wu (2008) as its special case. Under certain mild conditions, this criterion is shown to be strongly consistent in the sense that with probability one, it chooses the smallest true model for large n. Its byproducts include strongly consistent estimates of the regression coefficients regardless if there is a change point. In case that there is a change point, its byproducts also include a strongly consistent estimate of the change point parameter. In addition, an algorithm is given in light of the algorithm in Wu (2008), which has significantly reduced the computation time needed by the proposed criterion for the same precision for a sample of large size. Data examples including a simulation study and a real data example are also provided.

Consistent and powerful graph-based change-point test for high-dimensional data

Significance Change-point detection in high-dimensional time series is necessary in many areas of science and engineering, including neuroscience, signal processing, network evolution, image analysis, and text analysis. In terms of a multivariate generalization of the Wald–Wolfowitz run test using the shortest Hamiltonian path, this paper proposes a distribution-free, consistent graph-based change-point detection for high-dimensional data. Once a change-point is detected, its location is estimated by using ratio cut. The test is very powerful against alternatives with a shift in mean or variance and is accurate in change-point estimation. Its applicability is demonstrated in the example of tracking cell division. A change-point detection is proposed by using a Bayesian-type statistic based on the shortest Hamiltonian path, and the change-point is estimated by using ratio cut. A permutation procedure is applied to approximate the significance of Bayesian-type statistics. The change-point test is proven to be consistent, and an error probability in change-point estimation is provided. The test is very powerful against alternatives with a shift in variance and is accurate in change-point estimation, as shown in simulation studies. Its applicability in tracking cell division is illustrated.