IPDF Progress Seminar
Speaker: Dr. Utsav Dewan, IIT Bombay
Host: Mayukh Mukherjee
Title: Boundary behaviour of eigenfunctions and superharmonic functions on harmonic manifolds of purely exponential volume growth
Time, day and date: 3:00:00 PM – 4:00:00 PM, Wednesday, September 16
Venue: Ramanujan Hall
Abstract: On X, a non-positively curved harmonic manifold of purely exponential volume growth, of dimension at least 3, we study certain quantitative aspects of the boundary behaviour of eigenfunctions and superharmonic functions. We first focus on complex-valued eigenfunctions with eigenvalues lying outside the L^2-spectrum of the Laplace-Beltrami operator and obtain the almost everywhere existence of weighted non-tangential limits, sharp Hausdorff dimension and Hausdorff measure estimates of the boundary exceptional sets for radial limits. Then in the second part, we shift our attention to non-tangential and tangential boundary behaviour of positive superharmonic functions. Most of our results are new even for the homogeneous setting of rank one Riemannian symmetric spaces of non-compact type and Damek-Ricci spaces. Our arguments are based on potential theory adapted to the intrinsic Gromov hyperbolic geometry of X.

Seminar
Speaker: Dr. Smaranika Bera, Department of Mathematics Indian Institute of Engineering Science and Technology, Shibpur
Host: Siuli Mukhopadhyay
Title: Some Problems Related to Nonparametric Inference and Stochastic Ordering
Time, day and date: 4:30:00 PM – 5:30:00 PM, Wednesday, September 16
Venue: Online (https://meet.google.com/ywy-cbhx-eeg)
Abstract: My research work can be broadly divided into two parts. The first part focuses on the inferential aspects of nonparametric ageing classes of life distributions. Particularly in this side I have worked on hypothesis testing problems related to testing exponentiality against different non-monotonic ageing alternatives and on goodness of-fit assessment. The second part deals with selected probabilistic problems in reliability theory. This side mainly involves stochastic ordering, characterization of ageing classes and related probabilistic methods. In this presentation, I shall discuss two selected problems, one from each of these two areas. The first problem is from the inferential side and concerns a two-sample nonparametric testing problem for Second-Order Stochastic Dominance (SSD). We develop a large-sample testing procedure based on U-statistic theory. Since conventional U statistic methods can involve considerable computational effort, we consider alternative approaches based on Jackknife Empirical Likelihood (JEL), Adjusted JEL (AdJEL), and Balanced Adjusted JEL (BaJEL). The performance of these methods is studied through simulation experiments and their practical usefulness is demonstrated using real-life data.
The second problem comes from the probabilistic side and concerns stochastic ordering of order statistics from heterogeneous generalized Gompertz distributions. Here, we consider independent but non-identically distributed generalized Gompertz random variables. One of the main advantages of this distribution is its flexibility in describing different ageing behaviours, including increasing, decreasing and bathtub-shaped hazard rates. Using vector majorization techniques, we establish a variety of stochastic ordering results for order statistics from heterogeneous generalized Gompertz populations.
Although the main focus is on extreme order statistics, the results are also extended to general kth order statistics. We further study multiple-outlier models and obtain stochastic ordering results for extreme order statistics from two independent collections of generalized Gompertz random variables. Stronger ordering results are also obtained for second-order statistics.
Through these two problems, the presentation highlights how both nonparametric inference and stochastic ordering can be used to address interesting and practically relevant questions in reliability theory and survival analysis.