# Seminarier i matematisk statistik

Seminarier i matematisk statistik utgör en del av seminarieserien Seminarier i statistik och matematisk statistik, gemensamt organiserad av Matematiska institutionen (MAI) och Institutionen för datavetenskap (IDA). Ämnet för seminarierna är sannolikhetslära, stokastiska processer, statistisk teori och tillämpad statistik.

## Tid och lokal

Seminarietiden är vanligtvis på en tisdag kl 15:15 i Hopningspunkten som ligger i B-huset, ingång 23, plan 2, Campus Valla i Linköping.

## Kommande seminarier

Inga seminarier är inplanerade just nu.

## Tidigare seminarierVisa/dölj innehåll

### Tisdag 15 oktober 2019, Kristoffer Lindensjö, Uppsala universitet

Titel: Moment constrained optimal dividends: precommitment & consistent planning

Sammanfattning: A moment constraint that limits the number of dividends in the optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz' consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

### Tisdag 17 september 2019, Jimmy Olsson, KTH

Titel: Bayesian learning of weakly structural Markov graph laws using sequential Monte Carlo methods

Sammanfattning: We shall discuss a sequential Monte Carlo-based approach to approximation of weakly structural Markov graph laws on spaces of decomposable graphs, or, more generally, spaces of junction (clique) trees associated with such graphs. In particular, we apply a particle Gibbs version of the algorithm to Bayesian structure learning in decomposable graphical models, where the target distribution is a junction tree posterior distribution. Moreover, we use the proposed algorithm for exploring certain fundamental combinatorial properties of decomposable graphs, e.g. clique size distributions. Our approach requires the design of a family of proposal kernels, so-called junction tree expanders, which expand junction trees by connecting randomly new nodes to the underlying graphs. The performance of the estimators is illustrated through a collection of numerical examples demonstrating the feasibility of the suggested approach in high-dimensional domains.

### Tisdag 7 maj 2019, Chun-Biu Li, Stockholms universitet

Titel: Statistical Learning as a Compression Problem from the Information Theory Perspective

Sammanfattning: Although it was introduced in the context of communication theory, modern information theory provides us with a nonparametric probabilistic framework for statistical learning free from a priori assumption on the underlying statistical model. In this talk, I will discuss some of the information theory based methods for unsupervised and supervised learning. In particular, the soft (fuzzy) clustering problem in unsupervised learning can be viewed as a tradeoff between data compression and minimizing the distortion of the data. Similarly, modeling in supervised learning can be treated as a tradeoff between compression of the predictor variables and retaining the relevant information about the response variable. To illustrate the usage of these methods, some applications in biophysical problems and time series analysis will be addressed in the talk.

### Tisdag 5 mars 2019, Peter Olofsson, Högskolan i Jönköping

Titel: Muller's Ratchet in Populations Doomed to Extinction

Sammanfattning: Muller's ratchet is the process by which asexual populations accumulate deleterious mutations in an irreversible manner. Most mathematical models have been of the Wright-Fisher type with fixed population size and relative fitness. In contrast, we use a branching process model with absolute fitness, leading to unavoidable extinction. Individuals are divided into classes depending on how many mutations they have accumulated, and we give results for the rate of the ratchet and the size of the fittest class.

### Tisdag 6 november 2018, Ingemar Kaj, Uppsala universitet

Titel: Fractional limit processes in shot noise models

Sammanfattning: A wide variety of random processes and spatial random fields arise naturally as Poisson shot noise models, with shots of random location and size. Such models with power-law size intensities, display a range of limit processes under aggregation and suitable scaling of parameters. We discuss the various scaling regimes and their limits, which include fractional Brownian motion, fractional Poisson type motions, and stable processes, Allowing for a type of dependence between shots, yet another hybrid Gaussian-Poisson model appears in the limit.

### Tisdag 9 oktober 2018, Johan Tykesson, Chalmers Tekniska Högskola

Titel: Generalized Divide and Color models

Sammanfattning: In this talk, we consider the following model: one starts with a finite or countable set V, a random partition of V and a parameter p in [0,1]. The corresponding Generalized Divide and Color Model is the {0,1}-valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p, assigning all the elements of the partition element the value 1, and with probability 1-p, assigning all the elements of the partition element the value 0.

A very special interesting case of this is the "Divide and Color Model'' (which motivates the name we use) introduced and studied by Olle Häggström. A number of quite varied well-studied processes actually fit into this context such as the Ising model, the stationary distributions for the Voter Model and random walk in random scenery.

Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have? In the talk, we will focus most attention to the case when V is a finite set.

The talk is based on joint work with Jeff Steif.

### Tisdag 28 augusti 2018, Chencheng Hao, Shanghai University of International Business and Economics

Titel: Estimation of Kronecker structured covariance based on modified Cholesky decomposition

Sammanfattning: This paper is to study covariance estimation problems for high dimensional matrix-valued data. We propose a covariance estimator for the matrix-valued data from penalized matrix normal likelihood. Modified Cholesky decomposition of covariance matrix is utilized to construct positive definite estimators. The method is applied for identify parsimony and for producing a statistically efficient estimator of a large covariance matrix of matrix-valued data. Simulation results are illustrated.

### Tisdag 12 juni 2018, Jean-Claude Zambrini, Department of Mathematics, University of Lisbon, Portugal

Titel: Stochastic Deformation of Classical Integrability

Sammanfattning: Is it possible to deform, along quantum-like trajectories, one of the deepest notions of ODE's theory, the one of integrable systems?

We shall start from a classical example, then summarize the method of Stochastic Deformation. It will provide a way to deform Jacobi's strategy to reach this goal in the classical, deterministic, case. This talk is founded on a joint work with C. Léonard (Paris-Ouest Nanterre).

### Tisdag 5 juni 2018, Vlad Bally, Université Paris-Est Marne-la-Vallée, Frankrike

Titel: Asymptotic integration by parts formula and regularity of probability laws

Sammanfattning: We consider a sequence of random variables $F_{n}\sim p_{n}(x)dx$ which converge to a random variable $F.$ If we know that $p_{n}\rightarrow p$ in some sweated sense, then we obtain $F\sim p(x)dx.$ But in many interesting situations $p_{n}$ blows up as $n\rightarrow \infty .$ Our aim is to give a criterion which says that, if there is a "good equilibrium" between $\left\Vert F-F_{n}\right\Vert _{1}\rightarrow 0$ and $\left\Vert p_{n}\right\Vert \uparrow \infty$ then we are still able to obtain the absolute continuity of the law of $F$ and to study the regularity of the density $p.$ Moreover we get some upper bounds for $p.$ The blow up of $p_{n}$ is characterized in terms of integration by parts formulae.

We give two examples. The first one is about diffusion processes with Hölder coefficients. The second one concerns the solution $f_{t}(dv)$ of the two dimensional homogeneous Boltzmann equation. We prove that, under some conditions on the parameters of the equation, we have $f_{t}(dv)=f_{t}(v)dv.$ The initial distribution $f_{0}(dv)$ is a general measure (except a Dirac mass) so our result says that a regularization effect is at work; moreover, if the initial distribution has exponential moments $\int e^{\left\vert v\right\vert ^{\lambda }}f_{0}(dv)<\infty ,$ then we prove that $f_{t}(v)\leq Ct^{-\eta }e^{-\left\vert v\right\vert ^{\lambda ^{\prime }}}$ for every $\lambda ^{\prime }<\lambda .$ So we have exponential upper bounds in space and at most polynomial blow up in time.

### Onsdag 2 maj 2018, Katarzyna Filipiak, Poznań University of Technology, Polen

Titel: Testing hypotheses about covariance structures under multi-level multivariate models using Rao score

Sammanfattning: Modern experimental techniques allow to collect and store multi- level multivariate data in almost all fields such as agriculture, biology, biomedical, medical, environmental and engineering areas, where the observations are collected on more than one response variable at different locations, repeatedly over time, and at different ”depths”, etc. Before any statistical analysis it is vital to test the appropriate mean and variance-covariance structures on the multi-level multivariate observations.

In this talk the Rao’s score test (RS) statistic for testing the hypotheses about variance-covariance structures, such as e.g. separable structures with one component structured or exchangeable structures, is presented. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data. Monte Carlo simulations are then used to study the behavior of the empirical type I error as well as the empirical null distribution of the RST statistic with respect to the sample size. It is shown that RST outperforms the commonly used likelihood ratio test in all considered areas.

References:

[1] Roy, A., K. Filipiak, and D. Klein (2018). Testing a block exchangeable covariance matrix. Statistics 52(2), 393–408.

[2] Filipiak, K., D. Klein, and A. Roy (2017). A comparison of likelihood ratio tests and Rao’s score test for three separable covariance matrix structures. Biometrical Journal 59, 192–215.

[3] Filipiak, K., D. Klein, and A. Roy (2016). Score test for a separable covariance structure with the first component as compound symmetric correlation matrix. Journal of Multivariate Analysis 150, 105–124.

### Tisdag 20 mars 2018, Béatrice Byukusenge, Matematiska institutionen, Linköpings universitet

Titel: Estimation and residual analysis in the GMANOVA-MANOVA model

Sammanfattning: In this talk we will consider the GMANOVA-MANOVA model, which is a special case of an extended growth curve model, with no assumption of the nested subspace condition. We derive two residuals, establish their properties and give interpretation. Finally, a numerical example on a data set from a study that was conducted to investigate two treatments for patients suffering from multiple sclerosis is performed to validate the theoretical results.

### Tisdag 13 mars 2018, Silvelyn Zwanzig, Uppsala universitet

Titel: On high dimensional data analysis under errors in variables

Sammanfattning: Errors in variables induce additional complications already in models with p<n. The total least squares estimator is theoretically the best estimator in this case. In literature methods are presented for high dimensional sparse models with errors in variables. In the talk I will study the behavior of total least squares estimator for non sparse models with n << p and propose a generalized version of it.

### Tisdag 13 februari 2018, Kevin Schnelli, KTH

Titel: Local law of addition of random matrices on optimal scale

Sammanfattning: Describing the eigenvalue distribution of the sum of two general Hermitian matrices is basic question going back to Weyl. If the matrices have high dimensionality and are in general position in the sense that one of them is conjugated by a random Haar unitary matrix, the eigenvalue distribution of the sum is given by the free additive convolution of the respective spectral distributions. This result was obtained by Voiculescu on the macroscopic scale. In this talk, I show that it holds on the microscopic scale all the way down to the eigenvalue spacing. This shows a remarkable rigidity phenomenon for the eigenvalues.

## OrganisatörVisa/dölj innehåll

### Torkel Erhardsson

Universitetslektor