Contributed 04

## Stochastic Processes and Related Topics

Conference
10:30 PM — 11:00 PM KST
Local
Jul 22 Thu, 9:30 AM — 10:00 AM EDT

### Parameter estimation for weakly interacting particle systems and stochastic McKean-Vlasov processes

Louis Sharrock (Imperial College London)

4
In this presentation, we consider the problem of parameter estimation for a fully observed McKean-Vlasov stochastic differential equation (MVSDE), and the associated system of weakly interacting particles. We begin by establishing consistency and asymptotic normality of the offline maximum likelihood estimator (MLE) of the interacting particle system (IPS) in the limit as the number of particles (N) tends to infinity. We then propose a recursive MLE for the MVSDE, which evolves according to a stochastic gradient ascent algorithm on the asymptotic log-likelihood of the IPS. Under suitable assumptions which guarantee exponential ergodicity and uniform-in-time propagation of chaos for the MVSDE and the IPS, we prove that this estimator converges in L1 to the stationary points of the asymptotic log-likelihood of the MVSDE in the joint limit as N and t tend to infinity. Under the additional assumption of global strong concavity, we also demonstrate that our estimator converges in L2 to the unique maximiser of the asymptotic log-likelihood of the MVSDE, and establish an L2 convergence rate. Our results are demonstrated via several numerical examples of practical interest, including a linear mean field model, and a stochastic opinion dynamics model.

### CLT for cyclic long-memory processes

Andriy Olenko (La Trobe University)

3
Cyclic long-memory stochastic processes are studied. Cyclic long-memory behavior attracted increasing attention in recent years due to its importance in finance, hydrology, cosmology, internet modelling, and other applications to data with non-seasonal cyclicity. However, various functionals of cyclic long-memory processes have complex asymptotic behavior that has not yet been fully understood and investigated. Spectral singularities at non-zero frequencies play an important role in investigating cyclic processes. The publication [1] introduced the generalized filtered method-of-moments approach to simultaneously estimate singularity location and long-memory parameters. The law of large numbers for the proposed estimators was proved. This talk discusses the central limit theorem for these simultaneous estimators. A wide class of Gegenbauer-type semi-parametric models is considered. Asymptotic normality of several functionals of the cyclic long-memory processes is proved. For the case when values of the functionals are outside the feasible region, we propose new adjusted estimators and investigate their properties. It is shown that they have same asymptotic distributions as the corresponding ones in [1], but are computationally simpler. The methodology includes wavelet transformations as a particular case.

This presentation is based on recent joint results in [2] with A.Ayache, M.Fradon (University of Lille, France) and R. Nanayakkara (La Trobe University, Australia).

[1] Alomari, H.M., Ayache, A., Fradon, M., Olenko, A. Estimation of cyclic long-memory parameters. Scand. J. Statist., 47, 1, 2020, 104-133.

[2] Ayache, A., Fradon, M., Nanayakkara, R., Olenko, A. Asymptotic normality of simultaneous estimators of cyclic long-memory processes. submitted, 1-30, https://arxiv.org/abs/2011.06229.

### Q&A for Contributed Session 04

0
This talk does not have an abstract.

###### Session Chair

Yeonwoo Rho (Michigan Technology University)

Contributed 17

## Various Limit Theorems

Conference
10:30 PM — 11:00 PM KST
Local
Jul 22 Thu, 9:30 AM — 10:00 AM EDT

### Limit theorems for non-stationary strongly mixing random fields

Cristina Tone (University of Louisville)

3
In applications of statistics to data indexed by location, there is often an apparent lack of both stationarity and independence, but with a reasonable indication of “weak dependence” between data whose locations are “far apart”. This has motivated a large amount of research on the theoretical question of to what extent central limit theorems hold for non-stationary random fields. Bradley and Tone examined this theoretical question for “arrays of (non- stationary) random fields” under certain mixing assumptions. Our main result presents a central limit theorem for sequences of random fields that satisfy a Lindeberg condition and uniformly satisfy both strong mixing and an upper bound less than 1 on $\rho^{\prime}(\cdot, 1)$, in the absence of stationarity. There is no requirement of either a mixing rate assumption or the existence of moments of order higher than two. The additional assumption of a uniform upper bound less than 1 for $\rho^{\prime}(\cdot, 1)$ cannot simply be deleted altogether from the theorem, even in the case of strict stationarity.

### On the law of the iterated logarithm and strong invariance principles in stochastic geometry

Johannes Krebs (Heidelberg University)

3

### Functional limit theorems for U-statistics

Mikolaj Kasprzak (University of Luxembourg)

4
I will discuss a number of results obtained through three pieces of joined work with Christian Doebler and Giovanni Peccati. Firstly, I will talk about sequences of U-processes based on symmetric kernels of a fixed order that may depend on the sample size and present analytic sufficient conditions under which they converge to a linear combination of time-changed Brownian Motions. I will show how these conditions may be applied to deduce functional convergence of quadratic estimators in certain non-parametric models. Secondly, I will present quantitative bounds on the rate of functional convergence of vectors of weighted degenerate U-statistics to time-changed Brownian Motion, obtained via Stein’s method of exchangeable pairs. Finally, I will discuss a multivariate functional version of de Jong's CLT, yielding that, given a sequence of vectors of degenerate U-statistics, the corresponding empirical processes on [0,1] weakly converge in the Skorohod space as soon as their fourth cumulants in t=1 vanish asymptotically and a certain strengthening of the Lindeberg-type condition is verified.

### Proving Liggett's FCLT via Stein's method

Wasamon Jantai (Oregon State University)

3
In 1990, A. D. Barbour extended Stein's method for approximation by Gaussian processes, including Brownian bridge. In this work, we rederive Liggett's functional central limit theorem (FCLT) using Barbour's approach.

### Q&A for Contributed Session 17

0
This talk does not have an abstract.

###### Session Chair

Chi Tim Ng (Hang Seng University of Hong Kong)

Contributed 32

## Statistical Modeling and Prediction

Conference
10:30 PM — 11:00 PM KST
Local
Jul 22 Thu, 9:30 AM — 10:00 AM EDT

### An evolution of the beta regression for non-monotone relations

Gloria Gheno (Ronin Institute)

2
The beta regression is based on the beta distribution or its reparameterizations, which are used to obtain a regression structure on the mean which is much easier to analyze and interpret. This method analyzes data whose value is within the range (0.1), such as rates, proportions or percentage and is useful for studying how the explanatory variables affect them. For the mean of the beta regression scholars continued to use the traditional link functions used for binary regressions, i.e. the logit, the probit and the complementary log-log. In this paper I propose a new link function for the parameter mean of a beta regression which has as its particular cases the logit, representing a traditional symmetric link function, and the gev (Generalized extreme value), introduced precisely because of its asymmetry. In the simplest form of the beta regression, the inverse of the link function, called response function, represents the link between the mean and the explanatory variables. In this paper the response function can become non-monotone and therefore the link function is calculated for intervals. This particularity has never been proposed so far in the literature although some scholars have found non-monotone relationships between the response variable and its explanatory variables. The parameters of the function are estimated by maximizing the likelihood function, using my modified version of the genetic algorithm. I compare my method with the one proposed by Cribari-Neto, in which the link function is decided a priori, using simulated data, so as to be able to compare which of the two methods is closest to the true values. My method is better because it is able to correctly determine the link function with which the data are simulated and to estimate the parameters with less error.

### Robust censored regression with l1-norm regularization

Jad Beyhum (ORSTAT, Katholieke Universiteit Leuven)

2
This paper considers inference in a linear regression model with random right-censoring and outliers. The number of outliers can grow with sample size while their proportion goes to 0. We propose to penalize the estimator of Stute (1993) by the l1-norm. We derive rates of convergence and establish asymptotic normality. Our estimator has the same asymptotic variance as Stute's estimator in the censored linear model without outliers. Tests and confidence sets can therefore rely on the theory developed by Stute. The outlined procedure is also computationally advantageous, it amounts to solving a convex optimization program. We also propose a second estimator which uses the l1-norm penalized Stute estimator as a first step to detect outliers. It has similar theoretical properties but better performances in finite samples as assessed by simulations.

### SPLVC modal regression with error-prone linear covariate

Tao Wang (University of California, Riverside)

2
To broaden the scope of existing modal regressions, we in this paper propose two procedures, called B-splines-based procedure and stepwise-based procedure, to retrieve the estimates for a semiparametric partially linear varying coefficient (SPLVC) modal regression with error-prone linear covariate in which a linear covariate is not observed, but an ancillary variable is available. With B-splines-based procedure, varying coefficients are approximated through B-splines, and a deconvoluting kernel-based objective function is constructed straightly. For the stepwise-based procedure, by defining restricted regression mode via imposing a constrictive condition on model format, a two-step method is developed in which the varying coefficients are concentrated out by applying the "correction for attenuation" methodology in mean regression to alter the original model to a reduced parametric modal regression. Consistency and asymptotic properties of the estimators for these two newly proposed procedures are investigated under mild conditions according to the tail behavior of the characteristic function of the error distribution, either ordinary smooth distribution or super smooth distribution. Bandwidth selection in theory and practice are explored. For comparison, we also develop the asymptotic theorems for the SPLVC modal estimators with B-splines approximation without covariate measurement error. Monte Carlo simulations are conducted to examine the finite sample performance of the estimators and a pseudo data analysis is presented to further illustrate the proposed estimation procedures.

### Regularized double machine learning in partially linear models with unobserved confounding

Corinne Emmenegger (Swiss Federal Institute of Technology Zürich)

4
Double machine learning (DML) can be used to estimate the linear coefficient in a partially linear model with confounding variables. However, the standard DML estimator has a two-stage least squares interpretation and may yield overly wide confidence intervals. To address this issue, we present the regularization-selection regsDML method that leads to narrower confidence intervals but preserves coverage guarantees. We rely on DML to estimate nuisance parameters with arbitrary machine learning algorithms and combine it with a regularization and selection scheme. Our regsDML method is fully data driven and optimizes the estimated asymptotic mean squared error of the coefficient estimate. The regsDML estimator can be expected to converge at the parametric rate and to follow an asymptotic Gaussian distribution. Empirical examples demonstrate our theoretical and methodological developments. Software code for the regsDML method is available in the R-package dmlalg.

### Q&A for Contributed Session 32

0
This talk does not have an abstract.

###### Session Chair

Eun Jeong Min (Catholic University)