Past Seminars

In this talk, we discuss briefly minimum distance Cramer-Von Mises estimators of underlying parameters in the one and two sample location models, linear regression models and autoregressive models.

In this talk, we discuss Shannon-type sampling formulas, Kadec-type theorems, and Beurling-Landau-type theorems for shift-invariant spaces. We provide the sharp maximum gap condition for a sampling set in the shift-invariant space generated by Meyer scaling functions. This is an open problem (sampling density condition for shift invariant spaces) proposed by Aldroubi and Grochenig in 2000. Finally, we discuss a new sampling density condition for shift-invariant spaces.

Mechanisation of Mathematics refers to use of computers to generate or to check mathematical proofs. An automated theorem prover is a software tool which generates proofs and counter examples, while an interactive theorem prover is a software tool which partly automates and checks such 'proofs' by human-machine collaboration. The process of mechanization of mathematics refers to the use of one of the above-mentioned tools. The impact of recent developments in automated and interactive theorem proving on the practice of mathematical research ranges from their use in verification of controversial and experimental mathematical proofs, to a renewed interest in logical foundations of mathematics.
In this talk, I will discuss my work about mechanisation of knot theory. This work includes includes, thought is not restricted to, my earlier work on formalization of knot theory in a higher order logic based interactive theorem prover- Isabelle. I will also describe a recent work with D. Kulkarni about use of first order logic based automated theorem provers to classify a class of knots of importance in differential geometry called Legendrian knots, which led to discovery of a new invariant of Legendrian knots.
No prerequisites in Knot theory or Automated/Interactive Theorem Proving, shall be assumed.

We consider complex systems composed of independent components and consider issues such as importance of components, models for the failure of load sharing systems and bounds for the reiability of systems when the component life times exhibit positive ageing. After introducing reliability concepts we discuss some recent results in this area.

A fractal set is a union of many smaller copy of itself and it has a highly irregular structure. Using Hutchinsons operator, Barnsley, introduced Fractal Interpolation Function (FIF) via certain Iterated Function System (IFS). The FIF is continuous and self-a ne in nature. By de ning IFS suitably, one can construct various form of fractal functions including non-self-a ne and partially self-a ne (and partially non-self-a ne) FIFs. For any continuous function f, the corresponding fractal analogue f is non-self-a ne, continuous, nowhere di erentiable function and it was de ned by Navascuess. The function f , known as-fractal interpolation function, generates a family of continuous functions corresponding to the original function f for di erent scale vector = ( 1 2 N) ( 11)N. Thefractal function f interpolates and approximates f. Kolomogorov, introduced the notion of fractal dimension which later demonstrated by Mandelbrot to quantify irregular patterns. The fractal dimension or box dimension measures the complexity or irregularity of sets. It is seen in the literature that the graphs of FIFs have non-integer Hausdor-Besicovitch dimensions. The present talk is mainly devoted to the study pertaining to fractal dimensions and approximations of-fractal interpolation functions.

MDS matrices allow to build optimal diffusion layers in the design of block ciphers and hash functions. A recursive MDS matrix is an MDS matrix which can be expressed as a power of some companion matrix. The advantage of such a matrix is that it can be implemented by a single LFSR clocking several times. Such matrices are suitable for the design of diffusion layer in light-weight cryptographic applications. In this talk we will discuss some constructions of recursive MDS matrices.

We conducted an experiment where participants played a perfect-information game against a computer, which was programmed to deviate often from its backward induction strategy right at the beginning of the game. Participants knew that in each game, the computer was nevertheless optimizing against some belief about the participant's future strategy. Based on the findings of the experiment, which were not conclusive, we redesigned the games, the main change being 'incorporating new payoff structures'. We conducted a new experiment under a similar setting, and recorded and analysed our findings.

The idea of quantum mechanical interference led, nearly sixty years ago, the physicist Philip W. Anderson to predict the phenomenon of localization of quantum mechanical particles in a random potential, (Nobel Prize in Physics, 1977) a phenomenon absent in the classical limit i.e., when one considers only particles moving in a "random" environment, and not the wave-particle duality that is a part of our microscopic world. But quantum particles interact with each other, and the theory of localization could not account for inter-particle interactions. In this talk, I will give an introduction to this open problem in quantum statistical mechanics, and show some results from our research.

In ISI we learn probability as a vehicle for describing/modelling stochasticity. In one of the early demonstrations, second to be precise, of application of probability to combinatorics Erdos obtained lower bounds to Ramsey numbers. We wish to discuss that. If time permits we will discuss the first application of probability to combinatorics. As the title suggests this talk is accessible to M.Stat students.

We look at the classical Cramer-Lundberg risk process using the Skorokhod problem, which arises in the study of reflected processes. This leads to some interesting observations. The talk is basically a version of the chapter `Random walks in the real line' in Vol.2 of Feller's classic.

Large dimensional random matrices have been studied and applied in several branches of science such as statistical physics, wireless communication, mathematics and statistics. This lecture will introduce some spectral properties of a few basic random matrices and indicate some uses in the emerging applications of random matrices to high dimensional statistics problems.

ffi ?n f D={z?C: |z|<1} f(0)=0=f?(0)?1 log(f(z)/z)=2? n=1?nzn 0 |?n| n=1,2,3 f 0 0 |?n| n=1,2,3 f

Network or graphical representations of multivariate processes have assumed prominence in the recent era of multivariate time-series analysis. The primary support for this imminent paradigm shift stems from rapid developments in the reconstruction of causal (directed) networks that mainly draws on the rich ideas in econometrics, social sciences and neuroscience. These combined developments have, in turn, propelled and nucleated the ideas of causality analysis in data-driven process engineering over the last decade. The first part of this talk presents an overview of the developments in causality analysis centred around the notion of Granger causality (GC). The second part of the talk presents results from our research in this field. A systematic methodology for hierarchical reconstruction of Granger-causal graphs using spectral measures of GC is presented. Finally, two novel scalar correlation function measures for vector time-series modelling of multivariate processes towards efficient estimation of spectral GC measures are presented. Simulation and real-life case studies are portrayed as illustrations of the associated theoretical concepts.

In survival studies, subject under consideration is observed at particular time point and asked to recall the time to event of interest, if it has already occurred. In this talk, we proposed a new technique to analyze such recall based time to event data. After discussing the basic concepts in survival analysis I will explain the proposed method in detail.

A major concern for the regulators and owners of financial institutions is the market risk of a portfolio and the adequacy of capital to meet such risk. Market risk is the risk of losses in positions arising from the movements in market prices. A risk measure is used to determine the amount of capital to be kept in reserve. Some of the well known measures of market risk are: Value-at-Risk (VaR), Median Shortfall (MS) and Expected Shortfall (ES). VaR is an extreme quantile of the marginal loss distribution. Its use was recommended by the Basel Committee on Banking Supervision in 1996. MS is the median loss when the loss in the investment exceeds the VaR level. ES is the mean of the conditional loss distribution, given the event that the loss exceeds the VaR. We study the problems of estimation of these risk measures based on asset or portfolio return data, and their applications in comparing the market risk of a wide variety of Indian mutual funds. For each fund, the data on the daily and monthly net asset value (or closing price for an exchange traded fund) from 1 April 2007 to 31 March 2015 is considered, during which the Indian equity market experienced extreme volatility due to global recession and subsequent recovery. Our future work consists of two more risk measures, viz. the Entropic Value-at-Risk(EVaR) and Spectral Risk Measures (SRMs). EVaRis acoherent risk measure proposed by Ahmedi-Javid in 2012. Ahmedi-Javid showed that EVaR is more risk-averse as compared to the ES at the same confidence level. SRMs proposed by Acerbi in 2002 is the most promising development in the financial risk area. It belongs to the family of coherent risk measures and hence inherit the properties of such measures. Our aim is to study the problems of estimation of these risk measures based on asset or portfolio return data, and their applications.

In this paper, we develop a simple non parametric test for testing the independence of time to failure and cause of failure in competing risks set up. We generalize the test to the situation where the data contain right censored failure times. Asymptotic properties of the test statistics in both uncensored and censored situations are studied. The efficiency loss due to censoring is studied using Pitman efficiency. The performance of the proposed test is studied through simulations. Finally we illustrated our test procedure using real data sets.

In various strategies for solving primal-dual conic linear problems or cone complementarity problems, one tries to write the optimality/complementarity conditions in the form of a square system by replacing the complementarity constraints by linearly independent bilinear relations (which come from the so-called Lyapunov-like transformations). In order to identify proper cones where this can be achieved, we de ne the Lyapunov rank of a proper cone as the maximal number of linearly independent Lyapunov-like transformations on the cone, or equivalently, as the dimension of the Lie algebra of the automorphism group of that cone. In this talk, we present some rank results for various cones of interest (polyhedral cones, symmetric cones, and completely positive cones). In particular, we describe a recent rank result on permutation invariant proper polyhedral cones.

Accurate modeling of patterns in inter-event durations is of considerable interest in high-frequency financial data analysis. The class of logarithmic autoregressive conditional duration (Log ACD) models provides a rich framework for analyzing durations, and recent research is focused on developing fast and accurate methods for fitting these models to long time series of durations under least restrictive assumptions. This talk describes a semi-parametric modeling approach using Godambe-Durbin martingale estimating functions. This approach has wide applicability to several classes of linear and nonlinear time series. It only requires assumptions on the first few conditional moments of the process and does not require specification of its probability distribution. Three approaches for parameter estimation are discussed: solution of nonlinear estimating equations, recursive formulas for the vectorvalued parameter estimates, and iterated component-wise scalar recursions. Effective starting values from an approximating time series model increase the accuracy of the final estimates. Our approach is illustrated via a simulation study and a real data example based on high-frequency transaction level data on several stocks. Using this approach for structural break detection in a retrospective and an online way is discussed.

A brief overview of the data analyses of socio-economic variables – income, wealth, consumption, will be presented. It has been found empirically that the distributions of income, wealth and consumption possess fairly robust features: the bulk is log-normally distributed, followed by a power law tail. The mechanisms leading to such inequalities and invariant features for the distributions of the socioeconomic variables are not well-understood. We will also present some simple models inspired from physics (which studies matter and its interactions) and demonstrate how the models can be adapted to study socio-economic ‘complex’ systems, or explain some of the empirical findings and their consequences.