Probability and Statistics Seminar Past sessions

A LASSO-type model for the bulk and tail of a heavy-tailed response

As widely known, in an extreme value framework, interest focuses on modelling the most extreme observations — disregarding the central part of the distribution; commonly, the effort centers on modelling the tail of the distribution by the generalized Pareto distribution, in a Peaks over threshold framework. Yet, in most practical situations it would be desirable to model both the bulk of the data along with the extreme values. In this talk, I will introduce a novel regression model for the bulk and the tail of a heavy-tailed response. Our regression model builds over the extended generalized Pareto distribution, as recently proposed by Naveau et al (2016). The proposed model allows us to learn the effect of covariates on a heavy-tailed response via a LASSO-type specification conducted via a Lagrangian restriction. The performance of the proposed approach will be assessed through a simulation study, and the method will be applied to a real data set.

First Come, First Served Queues with Two Classes of Impatient Customers

We study systems with two classes of impatient customers who differ across the classes in their distribution of service times and patience times. The customers are served on a first-come, first served basis (FCFS), regardless of their class. Such systems are common in customer call centers, which often segment their arrivals into classes of callers whose requests may differ greatly in their complexity and criticality. We first consider an $M/G/1 + M$ queue and then analyze the $M/M/k + M$ case. Analyzing these systems using a queue length process proves intractable as it would require us to keep track of the class of each customer at each position in queue. Consequently, we introduce a virtual waiting time process where the service times of customers who will eventually abandon the system are not considered. We analyze this process to obtain performance characteristics such as the percentage of customers who receive service in each class, the expected waiting times of customers in each class, and the average number of customers waiting in queue. We use our characterization of the system to perform a numerical analysis of the $M/M/k + M$ system, and find several managerial implications of administering a FCFS system with multiple classes of impatient customers. Finally, we compare the performance a system based on data from a call center with the steady-state performance measures of a comparable $M/M/k + M$ system. We find that the performance measures of the $M/M/k + M$ system serve as good approximations of the system based on real data.

Joint work with:

Ivo Adan, Eindhoven University of Technology, the Netherlands,

and

Brett Hathaway, Kenan-Flagler School of Business, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA.

Optimal reinsurance of dependent risks

The talk will focus on the optimal reinsurance problem for two dependent risks, from the point of view of the ceding insurance company. We aim at maximizing the expected utility or the adjustment coefficient of the insurer wealth. The insurer buys reinsurance on each risk separately. By risk we mean a line of business, a portfolio of policies or a policy. We assume a generic known dependence structure, so that the optimal solution depends on the joint distribution. Due to dependencies, the optimal level of reinsurance for each risk involves a trade-off between the reinsurance premia of both risks. We study the shape of this trade-off and characterize the optimal treaties. We show that an optimal solution exists and provide an optimality condition. Unfortunately, explicit optimal treaties are not easy to compute from this condition. We discuss some strategies to obtain numerical approximations for the optimal treaties and discuss some aspects of the structure of the optimal strategy. Numerical results are presented assuming that the two risks are dependent by means of a copula structure and that the reinsurance treaty consists of a combination of quota-share and stop-loss. Sensitivity of the optimal reinsurance strategy is analyzed numerically to several factors, including the dependence structure, through the copula chosen, and the dependence strength, by means of the dependence parameter, corresponding to different values of the Kendall’s tau. A variety of reinsurance premium calculation principles are also considered.

Improving the ARL profile of the Poisson EWMA chart

The Poisson exponentially weighted moving average (PEWMA) chart was proposed by Borror et al. (1998) to monitor the mean of counts of nonconformities. This chart regrettably fails to have an in-control average run length (ARL) larger than any out-of-control ARL, i.e., the PEWMA chart is ARL-biased. Moreover, due to the discrete character of its control statistic the PEWMA it is difficult to set the control limits in such way that the in-control takes a desired value, say ARL0. In this paper, we propose an ARL-unbiased counterpart of the PEWMA chart and use the R statistical software to provide gripping illustrations of this chart with a decidedly improved ARL profile and an in-control ARL equal to ARL0. We also compare the ARL performance of the proposed chart with the one of a few competing control charts for the mean of i.i.d. Poisson counts.

Joint work with Sven Knoth (Department of Mathematics and Statistics — Faculty of Economics and Social Sciences — Helmut Schmidt University, Hamburg, Germany)

The coupling method in extreme value theory

One of the main goal of extreme value theory is to infer probabilities of extreme events for which only limited observations are available and require extrapolation of the tail distribution of the observations. One major result is Balkema-de Haan-Pickands theorem that provides an approximation of the distribution of exceedances above high threshold by a Generalized Pareto distribution. We revisit these results with coupling arguments and provide quantitative estimates for the Wasserstein distance between the empirical distribution of exceedances and the limit Pareto model. In a second part of the talk, we extend the results to the analysis of a proportional tail model for quantile regression closely related to the heteroscedastic extremes framework developed by Einmahl et al. (JRSSB 2016). We introduce coupling arguments relying on total variation and Wasserstein distances for the analysis of the asymptotic behavior of estimators of the extreme value index and integrated skedasis function.

Joint work with B. Bobbia and D. Varron (Université de Franche Comté).

Geostatistical analysis of sardine eggs data — a Bayesian approach

Understanding the distribution of animals over space, as well as how that distribution is influenced by environmental covariates, is a fundamental requirement for the effective management of animal populations. This is especially the case for populations which are harvested. The sardine is one of the most important fisheries species, both for its economic, sociologic, antropologic and cultural values.

Here we intend to understand the spatial distribution of the average number of sardine eggs by $m^3$. Our main objectives are to identify the environmental variables that better explain the spatial variation in sardine eggs density and to make predictions in spatial points that were not observed.

The data structure presents an excess of zeros and extreme values. To deal with this, we propose a point-referenced zero-inflated model to model the probability of presence together with the positive sardine eggs density and a point-referenced generalized Pareto model for the extremes. Finally, we combine the results of these two models to get the spatial predictions of the variable of interest. We follow a Bayesian approach and the inference is made using the package R-INLA in the software R.

Distributed Learning Algorithms for Big Data

Modern datasets are increasingly collected by teams of agents that are spatially distributed: sensor networks, networks of cameras, and teams of robots. To extract information in a scalable manner from those distributed datasets, we need distributed learning. In the vision of distributed learning, no central node exists; the spatially distributed agents are linked by a sparse communication network and exchange short messages between themselves to directly solve the learning problem. To work in the real-world, a distributed learning algorithm must cope with several challenges, e.g., correlated data, failures in the communication network, and minimal knowledge of the network topology. In this talk, we present some recent distributed learning algorithms that can cope with such challenges. Although our algorithms are simple extensions of known ones, these extensions require new mathematical proofs that elicit interesting applications of probability theory tools, namely, ergodic theory.

Contributions for the detection of multivariate outliers

The detection of outliers in multivariate models is always a dicult matter, but the subject is even more complex when dealing with dependent structures, as it is the case with the Simultaneous Equation Model (SEM). Unlike other models dened by systems of equations, such as the multivariate regression, the SEM assumes that the response variable in each equation can be stated as an explanatory variable in the rest of the system, meaning that explanatory variables can be correlated with the error terms. We present a method of outlier detection that bypasses those diculties using the asymptotic distribution of adequate robust Mahalanobis distances. The process identies anomalous data points as outliers of the SEM in simple steps and it provides a clear visualization. We illustrate this procedure with a real econometric data set.

Robust logistic regression with sparse predictor variables

Nowadays, dealing with high-dimensional data is a recurrent problem that cuts across modern statistics. One main feature of high dimensional data is that the dimension $p$, that is, the number of covariates, is high, while the sample size $n$ is relatively small. In this circumstance, the bet on sparsity principle suggests to proceed under the assumption that most of the effects are not significant. Sparse covariates are frequent in the classification problem and in this situation the task of variable selection may be also of interest. We focus on the logistic regression model and our aim is to address robust and sparse estimators of the regression parameter in order to perform estimation and variable selection at the same time.For this purpose, we introduce a family of penalized M-type estimators for the logistic regression parameter that are stable against atypical data. We explore different penalizations functions and we introduce the so-called sign penalization. This new penalty has the advantage that it does not shrink the estimated coefficients to $0$ and that it depends only on one parameter.We will discuss the variable selection capability of the proposal as well as its asymptotic behaviour. Through a numerical study, we compare the finite sample performance of the proposal with different penalized estimators either robust or classical, under different scenarios.

A Comprehensive Methodology to Analyse Topic Difficulties in Educational Programmes

We propose a comprehensive Learning Analytics methodology to investigate the level of understanding students achieve in the learning process. The goals of such methodology are

1. To identify topics in which students experience difficulties on;
2. To assess whether these difficulties are recurrent along semesters;
3. To decide if there are conceptual associations between topics in which students experience difficulties on; and, more generally,
4. To discover statistically significant groups of topics in which students show similar performance.

The proposed methodology uses statistics and data visualization techniques to address the first and the second goals, frequent itemset mining to tackle the third goal, and biclustering is proposed to find relationships within educational data, revealing meaningful and statistically significant patterns of students’ performance.

We illustrate the application of the methodology to a Computer Science course.

Working towards a typology of indices of agreement for clustering evaluation

Indices of agreement (IA) are commonly used to evaluate stability of a clustering solution or its agreement with ground truth – internal and external validation of the same solution, respectively.

IA provide different measures of the accordance between two partitions of the same data set, being based on contingency table data. Despite their frequent use in clustering evaluation, there are still open issues regarding the specific thresholds for each index to conclude about the degree of agreement between the partitions.

To acquire new insights on the indices behavior that may help improve clustering evaluation, 14 paired indices of indices are analyzed within diverse experimental scenarios - with balanced or unbalanced clusters and poorly, moderately or well separated ones. The paired indices’ observed values are all based on a cross-classification table of counts of pairs of observations both partitions agree to join and/or separate in the clusters. The IADJUST method is used to learn about the behavior of the indices under the hypothesis of agreement between partitions occurring by chance (H0). It relies on the generation of contingency tables under H0, being a simulation based procedure that enables to correct any index of agreement by deducting agreement by chance, overcoming previous limitations of analytical or approximate approaches – (Amorim and Cardoso, 2015).

The results suggest a preliminary typology of paired indices of agreement based on their distributional characteristics under H0. Inter-scenarios symbolic data referring to location, dispersion and shape measures of IA distributions under H0 are used to build this typology.

Reference

Amorim, M. J., & Cardoso, M. G. (2015). Comparing clustering solutions: The use of adjusted paired indices. Intelligent Data Analysis, 19(6), 1275-1296.

Joint work with Maria José Amorim (Department of Mathematics of ISEL, Lisbon, Portugal).

Feed-in Tariff Contract Schemes and Regulatory Uncertainty

This paper presents a novel analysis of four finite feed-in tariff (FIT) schemes, namely fixed-price, fixed-premium, minimum price guarantee and sliding premium with a cap and a floor, under market and regulatory uncertainty. Using an analytical real options framework, we derive the project value, the optimal investment threshold and the value of the investment opportunity for the four FIT schemes. Regulatory uncertainty is modeled allowing the tariff to be reduced before the signature of the contract. While market uncertainty defers investment, a higher and more likely tariff reduction accelerates investment. We also present several findings that are aimed at policymaking decisions, regarding namely the choice, level and duration of the FIT. For instance, the investment threshold of the sliding premium with a cap and a floor is lower than the minimum price guarantee, which suggests that the first regime is a better policy than the latter because it accelerates the investment while avoiding overcompensation.

Selecting differentially expressed genes in samples subgroups on microarray data

A common task in analysing microarray data is to determine which genes are differentially expressed under two (or more) kinds of tissue samples or samples submitted under different experimental conditions. It is well known that biological samples are heterogeneous due to factors such as molecular subtypes or genetic background, which are often unknown to the investigator. For instance, in experiments which involve molecular classification of tumours it is important to identify significant subtypes of cancer. Bimodal or multimodal distributions often reflect the presence of subsamples mixtures.

Consequently, truly differentially expressed genes on sample subgroups may be lost if usual statistical approaches are used. In this work it is proposed a graphical tool which identifies genes with up and down regulation, as well as genes with differential expression which revels hidden subclasses, that are usually missed if current statistical methods are used.

Optimal investment decision under switching regimes of subsidy support

We address the problem of making a managerial decision when the investment project is subsidized, which results in the resolution of an infinite-horizon optimal stopping problem of a switching diffusion driven by either a homogeneous or an inhomogeneous continuous-time Markov chain. We provide a characterization of the value function (and optimal strategy) of the optimal stopping problem. On the one hand, broadly, we can prove that the value function is the unique viscosity solution to a system of HJB equations. On the other hand, when the Markov chain is homogeneous and the switching diffusion is one-dimensional, we obtain stronger results: the value function is the difference between two convex functions

Monitoring Non-Stationary Processes

In nearly all papers on statistical process control for time-dependent data it is assumed that the underlying process is stationary. However, in finance and economics we are often faced with situations where the process is close to non-stationarity or it is even non-stationary.

In this talk the target process is modeled by a multivariate state-space model which may be non-stationary. Our aim is to monitor its mean behavior. The likelihood ratio method, the sequential probability ratio test, and the Shiryaev-Roberts procedure are applied to derive control charts signaling a change from the supposed mean structure. These procedures depend on certain reference values which have to be chosen by the practitioner in advance. The corresponding generalized approaches are considered as well, and generalized control charts are determined for state-space processes. These schemes do not have further design parameters. In an extensive simulation study the behavior of the introduced schemes is compared with each other using various performance criteria as the average run length, the average delay, the probability of a successful detection, and the probability of a false detection.

Literature

• Lazariv T. and Schmid W. (2018). Surveillance of non-stationary processes. AStA - Advances in Statistical Analysis, https://doi.org/10.1007/s10182-018-00330-4 .
• Lazariv T. and Schmid W. (2018). Challenges in monitoring non-stationary time series. In Frontiers in Statistical Process Control, Vol. 12, pp. 257-275. Berlin: Springer.

Joint work with Taras Lazariv (European University Viadrina, Department of Statistics, Germany).

A thinning-based EWMA chart to monitor counts: some preliminary results

Shewhart control charts are known to be somewhat insensitive to shifts of small and moderate size. Expectedly, alternative control schemes such as the cumulative sum (CUSUM) and the exponentially weighted moving average (EWMA) charts have been proposed to speed up the detection of such shifts.

The novel chart we propose relies on a EWMA control statistic where the usual scalar product is replaced by what we call a fractional binomial thinning to avoid the typical over smoothing ascribable to ceiling, rounding, and flooring operations. The properties of this discrete statistic are, to a moderate extent, similar to the ones of its continuous EWMA counterpart and the run length (RL) performance of the associated chart can be computed exactly using the Markov chain approach for independent and identically distributed (i.i.d.) counts. Moreover, this chart is set in such way that: the average run length (ARL) curve attains a maximum in the in-control situation, i.e., the chart is ARL- unbiased; and the in-control ARL is equal to a pre-specified value.

We use the R statistical software to provide compelling illustrations of this unconventional EWMA chart and to compare its RL performance with the ones of a few competing control charts for the mean of i.i.d. Poisson counts.

Keywords

Average run length; Exponentially weighted moving average; Fractional binomial thinning; Statistical process control.

Distributed learning in large scale networks: from GPS-denied localization to MAP inference

Big Data can elicit greater insight, but storage or computational limitations — or even privacy concerns — challenge learning from massive data sets. The distributed paradigm fits such problems just right: such algorithms work on partial data and fuse intermediate results within local neighborhoods, over a distributed network of computing nodes. In this talk we will take a tour starting on GPS-denied localization and culminating on a general distributed MAP inference algorithm for graphical models.

Estimation of the drift of a $2n$-dimension OU process

A $2n$-dimension Ornstein-Uhlenbeck (OU) process for which the diffusion matrix is singular is considered. This process is used as a model for the dynamic behavior of vibrating engineering structures such as bridges, buildings, dams, among others. We study the problem of estimating the vibration frequencies of the structure or, equivalently, the parameters of the stochastic differential equation (SDE) that governs the OU process.

Firstly, it is considered the case where the OU process is perturbed by an independent wiener process. The maximum likelihood estimator of the drift matrix is obtained and the properties of the estimator are established. The local asymptotic normality of the estimator is analyzed in detail. Since general regularity conditions do not hold in this case (the diffusion matrix is singular), theoretical results from the classic literature on the subject do not immediately apply and an alternative approach based on the Laplace transform is used.

Secondly, it is considered the case where the OU process is perturbed by two independent fractional brownian motions. Models involving fractional noises have not been widely used in engineering. However, many problems in engineering involve processes exhibiting long memory. For this reason, the estimation of the parameters of multidimensional state space linear models, described by SDEs and disturbed by fractional Brownian motion, has a potential application in different areas of engineering. We analyze the problem of estimating the drift parameters of a $2$- dimension linear stochastic differential equation perturbed by two independent fractional Brownian motions with the same Hurst parameter belonging to $(1/2,1)$. The maximum likelihood estimator of the drift parameters is obtained after a transformation of the original model and making use of the so called fundamental martingale.

In both cases, a simulation study is presented in the context of a real world situation that illustrates the asymptotic behavior of the maximum likelihood estimator of the drift matrix.

Multiple-valued symbolic data clustering: heuristic and model-based approaches

Symbolic data analysis (SDA) has been developed as an extension of the data analysis to handle more complex data structures. In this general framework the pair observation/variable is characterized by more than one value: from two (e.g., interval-value data defined by minimum and maximum values) to multiple-valued variables (e.g., frequencies or proportions).

This research discusses the clustering of multiple-valued symbolic data. First, we discuss an extension of heuristic clustering based on the symmetric Kullback-Leibler distance combined with a complete-linkage rule within the hierarchical clustering framework. Then, we propose a new model-based clustering framework. These new family of models based on the Dirichlet distribution includes mixture of regression/expert models. Results are illustrated with synthetic and demographic (population pyramids) data.

Market Risk Measurement — Theory and Practice

Topics that will be covered in this talk:

• Value-at-Risk (VaR)
• Expected Shortfall (ES)
• VaR/ES Measurement
• Historical Simulation
• Model Building Approach
• Monte Carlo Simulation Approach
• VaR Backtesting

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