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A Celebration in Memory of Ross Leadbetter

A Celebration in Memory
of Ross Leadbetter

Extremes, Dependence, and More

The Department of Statistics and Operations Research is organizing a meeting on Saturday, February 25, 2023.

Speakers

Richard Davis (Columbia University)
Ivette Gomes (Universidade de Lisboa)
Tailen Hsing (University of Michigan)
Steve Marron (University of North Carolina at Chapel Hill)
Thomas Mikosch (University of Copenhagen)
Susan Murphy (Harvard University)
Vladas Pipiras (University of North Carolina at Chapel Hill)
Sid Resnick (Cornell University)
Holger Rootzen (Chalmers University of Technology)
Gennady Samorodnitsky (Cornell University)

Talks

This talk provides three results for SVARs under the assumption that the primitive shocks are mutually independent. First, a framework is proposed to accommodate a disaster-type variable with infinite variance into a SVAR. We show that the least squares estimates of the SVAR are consistent but have non-standard asymptotics. Second, the disaster shock is identified as the component with the largest kurtosis. An estimator that is robust to infinite variance is used to recover the mutually independent components. Third, an independence test on the residuals pre-whitened by the Choleski decomposition is proposed to test the restrictions imposed on a SVAR. The test can be applied whether the data have fat or thin tails, and to over as well as exactly identified models. Three applications are considered. In the first, the independence test is used to shed light on the conflicting evidence regarding the role of uncertainty in economic fluctuations. In the second, disaster shocks are shown to have short term economic impact arising mostly from feedback dynamics. The third uses the framework to study the dynamic effects of economic shocks post-covid. (This is joint work with Serena Ng.)
Being both “extremists”, my main scientific connection with Ross is obviously related to dependence conditions and the extremal index (Leadbetter, 1974, 1983; Leadbetter, Lindgren & Rootzen, 1983), a field where I have worked only sporadically, despite of my high interest in the theme. I was acquainted with Ross’ work since the mid- seventies, but the first time I met Ross was in Vimeiro, 1983, already almost 40 years ago, and it happened what we can call ‘friendship at first sight’. In 1983, we organized in Portugal an event currently recognized as a milestone in the affirmation of the area of Extremes and in the launch of what I dare to call the ‘Portuguese School of Extremes and Applications’ (PORTSEA), due to the work of Portuguese “extremists” that Ross Leadbetter, with his sense of humour, used to call the “Portuguese Gang” (see, Gomes, 2021). We have never co-authored any paper, but Ross was a mentor of some of my PhD students. And some of them still work deeply in the aforementioned interesting area, where, without doubt, Ross is a King. After recalling a few moments spent with Ross, and if I have time, I will move to a scientific topic in which I still began developing unpublished research together with Ross, the use of resampling methodologies in the reliable estimation of parameters of extreme events, like the extreme value index (EVI) and the extremal index (EI). After a brief reference to some estimators of the aforementioned parameters and their asymptotic properties, we present, essentially on the basis of the bootstrap and the jackknife, results produced by members of the PORTSEA for an adaptive estimation of the EVI and the EI. Going back to my personal feelings: I am indeed missing Ross deeply. But he is in my mind and his work will always be relevant and available for present and future generations.

References:
Gomes MI (2021). The ‘Portuguese School of Extremes and Applications’ (PORTSEA). Notas e Comunicaçōes CEAUL 01/2021.
Leadbetter, M.R. (1974). On extreme values in stationary sequences. Z. Wahrsch. und Verw. Gebiete 28, 289–303.
Leadbetter, M.R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65:2, 291–306.
Leadbetter, M.R., Lindgren, G. and Rootz ́en, H. (1983). Extremes and Related Properties of Random Sequences and Series. Springer-Verlag, New York.

This talk studies the local structure of continuous random fields taking values in a complete separable linear metric space. Extending the work of Falconer, we show that the generalized k-th order increment tangent fields are self-similar and almost everywhere intrinsically stationary in the sense of Matheron. We establish the spectral characterization of all second-order intrinsic random functions, extending the classical Matheron theory. (This talk is based on joint work with Jinqi Shen and Stilian Stoev.)
Much will be said at this meeting about the important work that Ross did on the probability of extreme values, particularly in the challenging dependent case. But earlier in his career, Ross made major contributions to two other areas. The first of these was in the early days of no-parametric curve estimation, where among several innovations he wrote the seminal paper that started the whole area of non-parametric hazard function estimation. The second was his pivotal book on stationary Gaussian processes with Harald Cramer.
In his 1983 paper, Ross Leadbetter introduced the extremal index of a stationary sequence in a rigorous way. It is often interpreted as the reciprocal of the expected value of extremal clusters above high thresholds. For a regularly varying stationary sequence, i.e., any lagged vector from this time series has a multiviariate regularly varying distribution; see Davis and Hsing (1995), it is possible to give various other interpretations of the extremal index. It is closely related to large deviation probabilities for the maxima of a stationary sequence, and it can be used as a benchmark for analogous indices e.g. for sum processes.
Adaptive sampling methods, such as reinforcement learning (RL) and bandit algorithms, are increasingly used for the real-time personalization of interventions in digital applications like mobile health and education. As a result, there is a need to be able to use the resulting adaptively collected user data to address a variety of inferential questions, including questions about time-varying causal effects. However, current methods for statistical inference on such data (a) make strong assumptions regarding the environment dynamics, e.g., assume the longitudinal data follows a Markovian process, or (b) require data to be collected with one adaptive sampling algorithm per user, which excludes algorithms that learn to select actions using data collected from multiple users. These are major obstacles preventing the use of adaptive sampling algorithms more widely in practice. In this work, we proved statistical inference for the common Z-estimator based on adaptively sampled data. The inference is valid even when observations are non-stationary and highly dependent over time, and (b) allow the online adaptive sampling algorithm to learn using the data of all users. Furthermore, our inference method is robust to miss-specification of the reward models used by the adaptive sampling algorithm. This work is motivated by our work in designing the Oralytics oral health clinical trial in which an RL adaptive sampling algorithm will be used to select treatments, yet valid statistical inference is essential for conducting primary data analyses after the trial is over.
About 10 years ago, I started collaborating with US Navy researchers at the NSWC Carderock base. I was put in touch with them by Ross Leadbetter, who by that time, had worked with the Navy on a range of projects and was a highly respected figure on the base. In this talk, I will describe some extreme value research problems of interest to the Navy that I have been involved with over the years, continuing the work started by Ross Leadbetter.
Standard preferential attachment often alarmingly induces small reciprocity. Adding a reciprocity feature to the model fixes this but dramatically changes the asymptotic heavy tail limit measure for frequency of nodes with specified (in, out)-degree. Without reciprocity, the limit measure is spread out in the first quadrant. With reciprocity, the standardized limit measure concentrates on a ray from the origin and there is even hidden regular variation depending on the sophistication of the reciprocity feature. (The work on reciprocity was led by Tiandong Wang, Fudan University, Shanghai.)
In 1974 Ross Leadbetter developed extreme value limit theory for stationary sequences using much weaker conditions than those needed for proving central limit theorems for sums. One important part of this was his introduction of the Extremal Index which connects this theory to earlier results for extremes of independent sequences. In the first part of this talk I will give a brief account of this theory and some glimpses into the later very large literature which builds on his work. The second part will discuss two problems left open by Leadbetter: how to find a counterpart to the Extremal Index for unordered random variables, and how to describe the asymptotic distribution of not just maxima, but of all extreme order statistics.

We propose kernel PCA as a method for analyzing the dependence structure of multivariate extremes and demonstrate that it can be a powerful tool for clustering and dimension reduction.

Organizers

Tailen Hsing
Steve Marron
Vladas Pipiras
Holger Rootzen

More about Ross Leadbetter

Schedule

8:45am – 9:00am Opening, including remarks by Chris Clemens, the Provost and Chief Academic Officer.
9:00am – 9:35am Talk: Steve Marron (UNC-CH)
9:35am – 10:10am Talk: Ivette Gomes (U de Lisboa)
10:10am – 10:20am Coffee break
10:20am – 10:55am Talk: Holger Rootzen (Chalmers U of Technology)
10:55am – 11:30am Talk: Thomas Mikosch (U of Copenhagen)
11:30am – 12:05pm Talk: Vladas Pipiras (UNC-CH)
12:05pm – 1:30pm Lunch
1:30pm – 2:05pm Talk: Gennady Samorodnitsky (Cornell)
2:05pm – 2:40pm Talk: Richard Davis (Columbia)
2:40pm – 2:50pm Coffee break
2:50pm – 3:25pm Talk: Tailen Hsing (U of Michigan)
3:25pm – 4:00pm Talk: Susan Murphy (Harvard)
4:00pm – 4:35pm Talk: Sid Resnick (Cornell)
4:35pm – 4:50pm Closing remarks

Location

The currently reserved rooms are in Frank Porter Graham Student Union (Carolina Union):

  • Room 3408 (Talks)
  • Room 3409 (Coffee, Lunch)

(If there is a change due to the number of attendees, this will be communicated.)

The map of the third floor of the building can be found here: https://carolinaunion.unc.edu/about-us/building-maps/

Note that the rooms are at the one far end of the building. The building itself can be seen on this map, with a route from the STOR department building included for reference:

Contact

If you have any questions please email Steve Marron (marron@unc.edu) or Vladas Pipiras (pipiras@email.unc.edu).

Registration

Please tell us that you plan to attend by completing this brief survey

Local Information

Parking: Best bet for parking is on the big Chapel Hill town Parking decks on Franklin street including the Rosemary/Columbia parking lot, and the Wallace parking deck. There is also (paid) street parking. See the town parking website for more info on the various parking options near Franklin street.

Coffee: Closest walkable good cafe is Epilogue books chocolate brews. Further from the university, Caffe Driade and Grey Squirrel coffee are phenomenal but probably require a car.

Food: Franklin Street has a good collection of restaurants both for sandwiches, Mexican food, Indian food, etc.

Hotels:

Support

The financial support of the College of Arts and Sciences at UNC-CH for this event is greatly appreciated.
The event website was developed by Prof. Nicolas Fraiman at STOR.

Statistics & Operations Research
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