Victor M. Panaretos is Professor of Mathematical Statistics at the EPFL. He received his PhD in 2007 from UC Berkeley, advised by David Brillinger. Upon graduation he was appointed Assistant Professor at EPFL’s Mathematics Institute, where he rose the ranks to Full Professor, also serving as Institute Director. He has published widely on geometrical, functional, and nonparametric statistics and their interplay, and is the author of two books. He received the Erich Lehmann Award and an ERC Starting Grant Award. He is an Elected Member of the ISI, a Fellow of the IMS, and was Bernoulli Society Forum Lecturer in 2019. He has served on the Editorial Boards of the Annals of Statistics, the Annals of Applied Statistics, Biometrika, JASA (Theory & Methods), and EJS. He has also held several positions of service, notably that of President of the Bernoulli Society for Mathematical Statistics and Probability. Victor’s Medallion Lecture will be at JSM Nashville, August 2–8, 2025.

Near the Diagonal

The covariance kernel k(s, t) of a Gaussian process X(t) encapsulates all its fluctuation properties. Particularly the behaviour of k(s, t) near the diagonal (t, t) can reveal valuable information on the behaviour of the process, which in turn can be important as vehicle for or a target of statistical inference. Well-known examples include sample path regularity and effective dimensionality, and a more recent example pertains to Markov properties, via positive-definite extension from the vicinity of the diagonal. Alas, the covariance itself is seldom available. Rather, it often needs to be estimated from noise-corrupted discrete observations on sample paths—and the effects of noise are most perturbative precisely near the diagonal.

In the tradition of functional data analysis, these effects are mitigated by smoothing the “raw covariance” corresponding to the noise-corrupted discrete data. Yet smoothing over the diagonal introduces biases that obfuscate properties such as those listed above. Path regularity will now be dictated by the choice of smoother; dimensionality will be influenced by the smoother’s effective degrees of freedom; and Markov properties will be distorted by the smoother’s bandwidth.

We will see how completion-inspired methods can be effective in circumventing the noise problem and allowing progress on some long-standing problems in functional data analysis, that ultimately hinge on the covariance structure near the diagonal. Time permitting, we will touch on statistical problems including inferring the dimensionality of a random process [1, 3], analyzing rough functional data [5, 4], continuum graphical modelling [8], extrapolating correlation [7, 6], and handling non-separable random fields [2].

References

[1] A. Chakraborty and V. M. Panaretos. Testing for the rank of a covariance operator. The Annals of Statistics, 50(6):3510–3537, 2022.

[2] T. Masak and V. M. Panaretos. Random surface covariance estimation by shifted partial tracing. Journal of the American Statistical Association, 118(544):2562–2574, 2023.

[3] N. Mohammadi and V. M. Panaretos. Detecting whether a stochastic process is finitely expressed in a basis. Applied and Computational Harmonic Analysis, 67:101578, 2023.

[4] N. Mohammadi and V. M. Panaretos. Functional data analysis with rough sample paths? Journal of Nonparametric Statistics, 36(1):4–22, 2024.

[5] N. Mohammadi, L. V. Santoro, and V. M. Panaretos. Nonparametric estimation for SDE with sparsely sampled paths: An FDA perspective. Stochastic Processes and their Applications, 167:104239, 2024.

[6] K. Waghmare and V. M. Panaretos. The positive-definite completion problem. Transactions of the American Mathematical Society, 377(09):6549–6594, 2024.

[7] K. G. Waghmare and V. M. Panaretos. The completion of covariance kernels. The Annals of Statistics, 50(6):3281–3306, 2022.

[8] K. G. Waghmare and V. M. Panaretos. Continuously indexed graphical models. Journal of the Royal Statistical Society Series B: Statistical Methodology, 87(1): 211–231, 2024.