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This paper deals with a new design methodology for stratified comparative experiments based on interacting reinforced urn systems. The key idea is to model the interaction between urns for borrowing information across strata and to use it in the design phase in order to i) enhance the information exchange at the beginning of the study, when only few subjects have been enrolled and the stratum-specific information on treatments' efficacy could be scarce, ii) let the information sharing adaptively evolves via a reinforcement mechanism based on the observed outcomes, for skewing at each step the allocations towards the stratum-specific most promising treatment and iii) make the contribution of the strata with different treatment efficacy vanishing as the stratum information grows. In particular, we introduce the Interacting Urns Design, namely a new Covariate-Adjusted Response-Adaptive procedure, that randomizes the treatment allocations according to the evolution of the urn system. The theoretical properties of this proposal are described and the corresponding asymptotic inference is provided. Moreover, by a functional central limit theorem, we obtain the asymptotic joint distribution of the Wald-type sequential test statistics, which allows to sequentially monitor the suggested design in the clinical practice.

In this work, we introduce an extremely general model for a collection of innovation processes in order to model and analyze the interaction among them. We provide theoretical results, analytically proven, and we show how the proposed model fits the behaviors observed in some real data sets (from Reddit and Gutenberg). It is worth mentioning that the given applications are only examples of the potentialities of the proposed model and related results: due to its abstractness and generality, it can be applied to many interacting innovation processes., Comment: 6 figures

In a network of reinforced stochastic processes, for certain values of the parameters, all the agents' inclinations synchronize and converge almost surely toward a certain random variable. The present work aims at clarifying when the agents can asymptotically polarize, i.e. when the common limit inclination can take the extreme values, 0 or 1, with probability zero, strictly positive, or equal to one. Moreover, we present a suitable technique to estimate this probability that, along with the theoretical results, has been framed in the more general setting of a class of martingales taking values in [0, 1] and following a specific dynamics., Comment: second part of arXiv:2206.07514, 22 pages (appendix 3 pages), 2 figures

We consider a finite collection of reinforced stochastic processes with a general network-based interaction among them. We provide sufficient and necessary conditions in order to have some form of almost sure asymptotic synchronization, which could be roughly defined as the almost sure long-run uniformization of the behavior of interacting processes. Specifically, we detect a regime of complete synchronization, where all the processes converge toward the same random variable, a second regime where the system almost surely converges, but there exists no form of almost sure asymptotic synchronization, and another regime where the system does not converge with a strictly positive probability. In this latter case, partitioning the system in cyclic classes according to the period of the interaction matrix, we have an almost sure asymptotic synchronization within the cyclic classes, and, with a strictly positive probability, an asymptotic periodic behavior of these classes., Comment: 28 pages + supplementary material (13 pages), 2 figures

This work deals with a system of interacting reinforced stochastic processes, where each process $X^j=(X_{n,j})_n$ is located at a vertex $j$ of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent $j$ of the network. The interaction among the dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal "inclination" $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h\neq j$, of the other agents according to the entries of $W$. The best known example of reinforced stochastic process is the Polya urn. The present paper characterizes the asymptotic behavior of the weighted empirical means $N_{n,j}=\sum_{k=1}^n q_{n,k} X_{k,j}$, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. By means of a more sophisticated decomposition of the considered processes adopted here, these findings complete and improve some asymptotic results for the personal inclinations $Z^j=(Z_{n,j})_n$ and for the empirical means $\overline{X}^j=(\sum_{k=1}^n X_{k,j}/n)_n$ given in recent papers (e.g. [arXiv:1705.02126, Bernoulli, Forth.]; [arXiv:1607.08514, Ann. Appl. Probab., 27(6):3787-3844, 2017]; [arXiv:1602.06217, Stochastic Process. Appl., 129(1):70-101, 2019]). Our work is motivated by the aim to understand how the different rates of convergence of the involved stochastic processes combine and, from an applicative point of view, by the construction of confidence intervals for the common limit inclination of the agents and of a test statistics to make inference on the matrix $W$, based on the weighted empirical means. In particular, we answer a research question posed in [arXiv:1705.02126, Bernoulli, Forth.]

This paper proposes a clustering procedure for samples of multivariate functions in $(L^2(I))^{J}$, with $J\geq1$. This method is based on a k-means algorithm in which the distance between the curves is measured with a metrics that generalizes the Mahalanobis distance in Hilbert spaces, considering the correlation and the variability along all the components of the functional data. The proposed procedure has been studied in simulation and compared with the k-means based on other distances typically adopted for clustering multivariate functional data. In these simulations, it is shown that the k-means algorithm with the generalized Mahalanobis distance provides the best clustering performances, both in terms of mean and standard deviation of the number of misclassified curves. Finally, the proposed method has been applied to two real cases studies, concerning ECG signals and growth curves, where the results obtained in simulation are confirmed and strengthened., Comment: 22 pages

This work deals with systems of interacting reinforced stochastic processes, where each process $X^j=(X_{n,j})_n$ is located at a vertex $j$ of a finite weighted direct graph, and it can be interpreted as the sequence of "actions" adopted by an agent $j$ of the network. The interaction among the evolving dynamics of these processes depends on the weighted adjacency matrix $W$ associated to the underlying graph: indeed, the probability that an agent $j$ chooses a certain action depends on its personal "inclination" $Z_{n,j}$ and on the inclinations $Z_{n,h}$, with $h\neq j$, of the other agents according to the elements of $W$. Asymptotic results for the stochastic processes of the personal inclinations $Z^j=(Z_{n,j})_n$ have been subject of studies in recent papers (e.g. Aletti, Crimaldi, and Ghiglietti [arXiv:1607.08514, Ann. Appl. Probab., 27(6):3787-3844, 2017]; Crimaldi, Dai Pra, Louis, and Minelli [arXiv:1602.06217, Forthcoming in Stochastic Process. Appl.]); while the asymptotic behavior of quantities based on the stochastic processes $X^j$ of the actions has never been studied yet. In this paper, we fill this gap by characterizing the asymptotic behavior of the empirical means $N_{n,j}=\sum_{k=1}^n X_{k,j}/n$, proving their almost sure synchronization and some central limit theorems in the sense of stable convergence. Moreover, we discuss some statistical applications of these convergence results concerning confidence intervals for the random limit toward which all the processes of the system converge and tools to make inference on the matrix $W$.

In this paper we propose a general class of covariate-adjusted response-adaptive (CARA) designs based on a new functional urn model. We prove strong consistency concerning the functional urn proportion and the proportion of subjects assigned to the treatment groups, in the whole study and for each covariate profile, allowing the distribution of the responses conditioned on covariates to be estimated nonparametrically. In addition, we establish joint central limit theorems for the above quantities and the sufficient statistics of features of interest, which allow to construct procedures to make inference on the conditional response distributions. These results are then applied to typical situations concerning Gaussian and binary responses., Comment: to Appear in The Annals of Statistics http://www.imstat.org/aos/future_papers.html, 27 pages + 18 Supplementary materials

Randomly evolving systems composed by elements which interact among each other have always been of great interest in several scientific fields. This work deals with the synchronization phenomenon, that could be roughly defined as the tendency of different components to adopt a common behavior. We continue the study of a model of interacting stochastic processes with reinforcement, that recently has been introduced in Crimaldi et al. (2016, arXiv:1602.06217). Generally speaking, by reinforcement we mean any mechanism for which the probability that a given event occurs has an increasing dependence on the number of times that events of the same type occurred in the past. The particularity of systems of such stochastic processes is that synchronization is induced along time by the reinforcement mechanism itself and does not require a large-scale limit. We focus on the relationship between the topology of the network of the interactions and the long-time synchronization phenomenon. After proving the almost sure synchronization, we provide some CLTs in the sense of stable convergence that establish the convergence rates and the asymptotic distributions for both convergence to the common limit and synchronization. The obtained results lead to the construction of asymptotic confidence intervals for the limit random variable and of statistical tests to make inference on the topology of the network given the observation of the reinforced stochastic processes positioned at the vertices., Comment: 40 pages

We consider systems of interacting Generalized Friedman's Urns (GFUs) having irreducible mean replacement matrices. The interaction is modeled through the probability to sample the colors from each urn, that is defined as convex combination of the urn proportions in the system. From the weights of these combinations we individuate subsystems of urns evolving with different behaviors. We provide a complete description of the asymptotic properties of urn proportions in each subsystem by establishing limiting proportions, convergence rates and Central Limit Theorems. The main proofs are based on a detailed eigenanalysis and stochastic approximation techniques.

Adaptive randomly reinforced urn (ARRU) is a two-color urn model where the updating process is defined by a sequence of non-negative random vectors $\{(D_{1,n}, D_{2,n});n\geq1\}$ and randomly evolving thresholds which utilize accruing statistical information for the updates. Let $m_1=E[D_{1,n}]$ and $m_2=E[D_{2,n}]$. Motivated by applications, in this paper we undertake a detailed study of the dynamics of the ARRU model. First, for the case $m_1 \neq m_2$, we establish $L_1$ bounds on the increments of the urn proportion at fixed and increasing times under very weak assumptions on the random threshold sequence. As a consequence, we deduce weak consistency of the evolving urn proportions. Second, under slightly stronger conditions, we establish the strong consistency of the urn proportions for all finite values of $m_1$ and $m_2$. Specifically, we show that when $m_1=m_2$ the proportion converges to a non-degenerate random variable. Third, we establish the asymptotic distribution, after appropriate centering and scaling, of the proportion of sampled balls in the case $m_1=m_2$. In the process, we settle the issue of asymptotic distribution of the number of sampled balls for a randomly reinforced urn (RRU). To address the technical issues, we establish results on the harmonic moments of the total number of balls in the urn at different times under very weak conditions, which is of independent interest., Comment: arXiv admin note: text overlap with arXiv:1502.06130

The generalized P\`olya urn (GPU) models and their variants have been investigated in several disciplines. However, typical assumptions made with respect to the GPU do not include urn models with diagonal replacement matrix, which arise in several applications, specifically in clinical trials. To facilitate mathematical analyses of models in these applications, we introduce an adaptive randomly reinforced urn model that uses accruing statistical information to adaptively skew the urn proportion toward specific targets. We study several probabilistic aspects that are important in implementing the urn model in practice. Specifically, we establish the law of large numbers and a central limit theorem for the number of sampled balls. To establish these results, we develop new techniques involving last exit times and crossing time analyses of the proportion of balls in the urn. To obtain precise estimates in these techniques, we establish results on the harmonic moments of the total number of balls in the urn. Finally, we describe our main results in the context an application to response-adaptive randomization in clinical trials. Our simulation experiments in this context demonstrate the ease and scope of our model.

In functional linear regression, the parameters estimation involves solving a non necessarily well-posed problem and it has points of contact with a range of methodologies, including statistical smoothing, deconvolution and projection on finite-dimensional subspaces. We discuss the standard approach based explicitly on functional principal components analysis, nevertheless the choice of the number of basis components remains something subjective and not always properly discussed and justified. In this work we discuss inferential properties of least square estimation in this context with different choices of projection subspaces, as well as we study asymptotic behaviour increasing the dimension of subspaces.