Abstracts

 

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  1. A class of latent Markov models for multivariate categorical panel data fit through birth and death MCMC

    Alessio Farcomeni
    Università di Roma “La Sapienza”

    We propose a class of models for the analysis of multivariate categorical longitudinal data, based on a marginal parametrization. We model the conditional distribution of each vector of response variables given the covariates, the lagged response variables, and a set of subject-specific intercepts to handle unobserved heterogeneity. A suitable multivariate link allows to conveniently handle any number and any kind of categorical responses. The random effects are assumed to arise from a discrete distribution, and to follow a first-order Markov chain. In this way distributional assumptions on the random effects, such as normality, are avoided; and the true latent distribution is somehow approximated through the finite mixture. The model, which is presented in Bartolucci and Farcomeni (2009), gives also some advantages with respect to interpretability. While Bartolucci and Farcomeni (2009) derive the maximum likelihood estimates through EM, in this work we cast the model in a Bayesian framework, which provides many advantages over the frequentist framework. In fact, apart from the usual advantages (i.e., prior information can be easily embedded) the Bayesian approach allows to (i) easily handle missing data and drop-outs, (ii) handle much larger data sets, (iii) handle smaller data sets with much less troubles from multi-modality of the likelihood, and (iv) coherently derive inference on the number of latent states. Further, in presence of high uncertainty in model selection and/or on the number of latent states, model averaging is seen to be straightforward. For approximating the posterior distribution we set up a continuous time birth and death MCMC, which is argued to be more convenient with respect to a more classical reversible jump approach. Transdimensionality (with respect to the number of latent states) is handled through a continuous birth and death process as in Stephens (2000), which has beed extended to the hidden Markov setting by Shi et al. (2002). The latent indicators are block updated at each iteration through a forward-backward algorithm, which dramatically speeds up convergence. The remaining parameters are updated through appropriate Metropolis steps. We illustrate the approach through an application on a data set which derives from the Panel Study on Income Dynamics, and concerns fertility and female participation to the labor market in USA.

  2. Bayes Factors for multiple testing

    F. Bertolino, S. Cabras, M.E. Castellanos(*), W. Racugno
    Università di Cagliari
    (*) Università Rey Juan Carlos (Madrid)

    Multiple hypothesis testing collects a serie of techniques usually based on p-values as a summary of the available evidence from many statistical tests. In hypothesis testing, under a Bayesian perspective, the use of Bayes Factors is usually preferred to that of p-values, because they provide a more choerent summary of the available evidence. In this work we approach the multiple hypothesis testing analysis as a multiple model selection problem throught the use of Bayes Factors defined with default priors. These priors are typically improper and full Bayes Factors cannot be used in single hypothesis testing because of the undetermined ratio of prior pseudo-constants. We show here that these undetermined ratio seems not to cause problems in multiple hypotheses testing because the proposed undetermined Bayes Factors are used whithin a comparative scheme. Using partial information from the p-values corresponding to the same tests, we are able to achieve a satisfactory approximation of the sampling null distribution of the undetermined Bayes Factors and use them within the Efron’s multiple testing procedure. In order to show the advantages of this approach with respect to the use of p-values alone, we carry out a simulation study and two applied analysis to microarray experiments. This comparison favors undetermined Bayes Factors in terms of a reduced number of false discoveries and false negatives.

  3. Mixtures of prior distributions for predictive Bayesian sample size
    calculations in clinical trials

    Pierpaolo Brutti *, Fulvio De Santis**, Stefania Gubbiotti**
    * LUISS Guido Carli
    ** Sapienza Università di Roma

    In this work we propose a predictive Bayesian approach to sample size
    determination and re-estimation in clinical trials, in the presence of multiple sources of prior information. The method we suggest is based on the use of mixtures of prior
    distributions for the unknown quantity of interest, typically a treatment effect or an effects-difference. Methodologies are developed using normal models with mixtures of
    conjugate priors. In particular we extend the sample size determination analysis of
    Gajewski and Mayo (Statist. Med. 2006; 25: 2554-66) and the sample size re-estimation technique of Wang (Biometrical Journ. 2006; 48, 5: 1-13).

  4. Measurement Error Correction in Exploiting Gene-Environment Independence in Family-Based Case-Control Studies

    Annamaria Guolo
    Università di Verona

    Family-based case-control studies are usually adopted to evaluate the influence of genetic susceptibility and environmental exposure on the etiology of diseases. In many cases, such as, for example in case of external environmental exposure, it is reasonable to assume gene-environment being independent within families in the source population. Within this framework, we consider the common situation of measurement error affecting the assessment of the environmental exposure. We propose to correct for measurement error through a likelihood approach which exploits he conditional likelihood of Chatterjee et al. (2005, Genetic Epidemiology). Simulation studies show that the method is preferable to that based on the traditional logistic regression in terms of properties of the parameters estimators.

  5. Marginal Regression Models with Correlated Normal Errors

    Cristiano Varin
    Ca’ Foscari University – Venice.

    In this talk I will present a class of models for regression analysis of non-normal dependent observations. The proposed class provides a natural extension of traditional linear regression models with correlated errors. Likelihood inference will be discussed in detail and compared with non-likelihood methods such as generalized estimating equations. Illustration include analysis of two famous data set: trends in U.S. polio incidence in 1970s and salamander mating.

    Based on joint work with Guido Masarotto, University of Padova.

  6. Local Depth

    C. Agostinelli, M. Romanazzi
    Ca’ Foscari University – Venice.

    Keywords: Cluster analysis, Geometrical depth functions, Local features, Mixture Models, Multimodality

    Data depth is a distribution-free statistical methodology for graphical/analytical investigation of multivariate distributions and data sets. The main applications are a center-outward ordering of multivariate observations, location estimators and some graphical presentations (scale curve, DD-plot).
    By definition, a depth function provides a measure of centralness which is monotonically decreasing along any given ray from the deepest point. As a consequence, it is unable to account for multimodality and mixture distributions. To overcome this problem we introduced the notion of Local Depth which generalizes the bacis notion of depth. The Local Depth evaluates the centrality of a point conditional on a bounded neighborhood. For example, the local version of simplicial depth is the ordinary simplicial depth, conditional on random simplices whose volume is not greater than a prescribed threshold. These generalized depth functions are able to record local fluctuations of the density function and are very useful in mode detection, identification of the components in a mixture model and in the definition of ”nonparametric” distances in cluster analysis. We provide theoretical results on the behavior of the Local Depth and discuss the computational problems involved by its evaluation. Several illustrations are enclosed.
    References
    C. Agostinelli and M. Romanazzi. Local depth of univariate distributions. submitted, 2008a.
    C. Agostinelli and M. Romanazzi. Multivariate local depth. submitted, 2008b.
    C. Agostinelli and M. Romanazzi. Advances in data depth. submitted, 2009.
    R.Y. Liu, J.M. Parelius, and K. Singh. Multivariate analysis by data depth: Descriptive statistics, graphics and inference. The Annals of Statistics, 27:783–858, 1999.

  7. A weighted strategy to handle likelihood uncertainty in Bayesian inference.

    L. Greco(1), C. Agostinelli(2)
    (1) Università di Sannio
    (2) Università Ca’ Foscari University – Venezia.

    The sensitivity of posterior measures of interest with respect to uncertainty in the likelihood function is an important topic in robust Bayesian analysis (Shyamalkumar, 2000). In robustness studies, the assumed sampling model can be considered as an initial guess; hence, even in the Bayesian framework, the assumed model can be thought to be only an approximation to reality, not dissimilarly from the classical robust theory.
    In the robust Bayesian analysis, likelihood uncertainty is represented by allowing the assumed model to vary over a certain class of distributions and studying the stability of the posterior distribution and the variations of the related quantities. For instance, this class may be a discrete set of models, a larger parametric family which includes the original model as a special case, a semi–parametric family.
    Likelihood uncertainty, essentially, comes from the presence of outliers and misspecification’s problems: the theory of robustness provides statistical procedures that are resistant with respect to the occurrence of outliers in the sample and stable with respect to small departures of the data from the assumed parametric model.
    Clearly, deviations from the model assumptions will heavily influence the shape of the posterior distribution through the likelihood, by keeping the prior distribution fixed.
    For instance, one single outlier can modify the likelihood drastically and, therefore, the information on the data support over the parameter space can be seriously misleading and invalidate the updating mechanism of our initial knowledge, summarized by the prior distribution. Hence, it is reasonable to look for a posterior distribution which is stable under small deviation from model assumptions and posterior measures which are robust with respect to model and data inadequacies.
    One recent approach to obtain a robust posterior distribution has been outlined in Greco et al. (2008). The authors investigate the use of pseudo–likelihoods with robustness properties in place of the genuine likelihood function and prove the validity of the resulting posterior distributions for Bayesian inference (see also Lazar, 2003). The main gain of the method is that it avoids the introduction a large family of models to take into account likelihood uncertainty. The only requirement is a set of opportune estimating functions which defines robust estimators.
    The main drawback relies in the construction of the selected pseudo–likelihood, which may become difficult in multidimensional problems, when some conditions are not met, even in the simple case of location-scale families.
    Here, we aim at discussing a more general strategy based on the use of a weighted likelihood function, that is a likelihood function whose single term components are opportunely down-weighted by a set of fixed weights. This methodology has the great advantage to lead to posterior distributions belonging to the same family of those one would obtain by using the genuine likelihood function and, at the same time, to robust posterior summaries.
    Actually, when the weights are introduced in order to regulate the effect of departures of the data from the assumed model, their effect is supposed to be that of down-weighting the contributes to the full likelihood of those observations which deviate from the model assumptions.

  8. Una Nuova Tecnica di Previsione Stocastica delle Strutture di Popolazioni

    Salvatore Bertino, Eugenio Sonnino
    Università di Roma “La Sapienza”

    Parole Chiave: Processo di Poisson, Simulazione, Tempi di attesa, Composizione di processi di punti, Tasso istantaneo di realizzazione di eventi, Quozienti di fecondità, Quozienti di mortalità, Quozienti di emigratorietà.

    L’evoluzione della struttura di una popolazione è determinata dalla successione di eventi di nascita, morte, immigrazione e emigrazione. Questa successione di eventi può essere pensata come una realizzazione di un processo stocastico di punti. Analiticamente è difficile studiare un tale processo stocastico. Al contrario è possibile simulare delle realizzazioni del processo e quindi l’evoluzione della popolazione allo studio.
    L’ipotesi di base è che la successione di eventi che determinano l’evoluzione della popolazione è generata da un processo di punti che è la composizione di più processi di Poisson indipendenti: nascite, morti, immigrazioni e emigrazioni.
    Ogni processo di Poisson è caratterizzato da un tasso istantaneo di realizzazione degli eventi. Componendo più processi di Poisson indipendenti si ottiene ancora un processo di Poisson con tasso pari alla somma dei tassi di realizzazione dei processi componenti. D’altra parte, in un processo di Poisson, il tempo di attesa per un evento, a partire da un tempo iniziale, o dal tempo in cui si è verificato l’ultimo evento, è distribuito secondo una legge esponenziale negativa.
    Questi ed altri risultati teorici ci permettono di simulare, per ciascun anno di studio, gli istanti in cui si verificano gli eventi e il tipo di evento occorso in ciascun istante e quindi, anno per anno, l’evoluzione della popolazione.
    La procedura di simulazione proposta fornisce le stime dei valori medi e della deviazione standard di tutti i parametri caratteristici della popolazione in ciascun anno del periodo di studio. In tal modo si ottiene una importante informazione sulla precisione delle proiezioni e possono essere utilizzate tecniche di stima per intervalli.

  9. Adjustments of profile likelihood and predictive densities

    Luigi Pace(1), Alessandra Salvan and Laura Ventura
    (1) University of Udine
    University of Padova

    A second-order link between adjusted profile likelihoods and refinements of the estimative predictive density is shown.
    The result provides a new interpretation for modified profile likelihoods, that complements results in the literature. Moreover, it suggests how to construct adjusted profile likelihoods using accurate predictive densities.

  10. Directional test for contingency tables

    N. Sartori
    Ca’ Foscari University – Venice

    Recent likelihood theory has produced highly accurate third order p-value for assessing scalar interest parameters of regular continuous models. This has been extended to discrete models (Davison et al., 2006) but with a reduction to second order accuracy due to the discreteness. We develop p-values for assessing a vector interest parameter, for both discrete and continuous models. For this we follow Davison et al. (2006) and use the observed loglikelihood and the observed likelihood gradient, where the gradient is a sample space gradient calculated conditional on an appropriate approximate ancillary. We then assess the vector parameter by means of a directional test, following Fraser & Massam (1985), Skovgaard (1988) but using one dimensional numerical integration rather than a gamma kernel inversion. Here we concentrate on the particular case of testing nested loglinear models for contingency tables.

    Joint work with D.A.S. Fraser, N. Reid (University of Toronto) and A.C. Davison (EPFL)

    References
    – Davison, A.C., Fraser, D.A.S. and Reid, N. (2006). Improved likelihood inference for discrete data. JRSSB.
    – Fraser, D.A.S. and Massam, H. (1985). Conical tests: observed levels of significance and confidence regions. Statistical Papers.
    – Skovgaard, I.M. (1988). Saddlepoint expansions for directional tests. JRSSB.

  11. Correlation models for paired comparison data

    Manuela Cattelan
    University of Padova

    Traditional models developed for the analysis of paired comparison data assume independence among all observations. In many instances, this assumption may be unrealistic. We propose new extensions of the traditional models that describe the structure of cross-correlation between paired comparisons with an object in common. The difficulties encountered in applying likelihood inference methods are presented. To overcome these problems a composite likelihood inference approach is suggested. The models are illustrated through an application on sports data.

    Based on joint work with Cristiano Varin, Ca’ Foscari University – Venice, and David Firth, University of Warwick.

  12. Inference for competing risks in presence of time-dependent covariates

    Giuliana Cortese
    University of Padova

    Time-dependent covariates are frequently encountered in regression analysis for event history data and competing risks. They are often essential predictors which can not be substituted by time-fixed covariates.
    Time-dependent covariates have been classified as external and internal (random) covariates, but their role and limitations in survival models, and more specifically in the competing risks setting, have not been deeply investigated. The main difference with an internal covariate is that it carries
    information about failure times of individuals.
    When different causes of an event are acting simultaneously, the main interest is in estimating quantities such as cause-specific hazards, cumulative incidences or marginal survival probabilities.
    If internal time-dependent covariates are included in the modelling process, then it is still possible to estimate cause-specific hazard functions, but prediction of the cumulative incidences and survival probabilities based on these functions is no longer feasible. This limitation is also encountered in the direct regression approach for cumulative incidences based on “subdistribution hazards”.
    For the estimation of hazard functions inference can still be based on the partial likelihood. However, when the interest is on the cumulative incidence probabilities, inference can not only be based on the partial likelihood but would also need parameters from the marginal model of the internal covariate, which then would need to be identified.
    The work aims at providing three possible strategies for dealing with these prediction problems in case of categorical internal covariates. Some approaches for estimating cumulative incidences and survival probabilities in presence of random time-dependent covariates, are presented under a multi-state framework. They are based on extensions of the original competing risks model and on the so-called “landmark analysis”, which enables us to study cumulative incidences at different sub-intervals of the entire study period.
    An illustrative example based on bone marrow transplant data is presented in order to compare the different methods.

  13. Prior distributions from pseudo-likelihoods in the presence of nuisance parameters

    L. Ventura(1), S. Cabras, W. Racugno
    (1) University of Padova
    University of Cagliari

    Consider a model parametrized by theta=(psi,lambda), where psi is the parameter of interest. The problem of eliminating the nuisance parameter lambda can be tackled by resorting to a pseudo-likelihood function L*(psi) for psi, namely a function of psi only and the data y with properties similar to those of a likelihood function. If one treats L*(psi) as a true likelihood, the posterior distribution pi*(psi|y) is proportional to pi(psi)L* (psi) can be considered, where pi(psi) is a prior distribution on psi. The goal of this paper is to construct probability matching priors for a scalar parameter of interest only, i.e. priors for which Bayesian and frequentist inference agree to some order of approximation, to be used in pi*(psi|y). When L*(psi) is a marginal, a conditional or a modification of the profile likelihood, we show that pi(psi) is simply proportional to the square root of the inverse of the asymptotic variance of the pseudo-maximum likelihood estimator. The proposed priors are compared to the reference or Jeffreys’ priors in four examples.

  14. Weight of evidence forecasting in indirect identification

    Fabio Corradi
    University of Florence

    This work deals with indirect identification derived from DNA evidence. Indirect identification refers to cases involving the identification of an individual as a well defined member of a family. “Paternity tests” and familial reconstructions are significant examples of such identification activity.
    Results are usually provided in terms of a weight of evidence, which, in forensic science, indicates the ratio between the probability of the observed evidence evaluated conditionally to two alternative end exhaustive hypotheses. This measure is well known in Bayesian statistics as the Bayes factor, a tool used for models selection, here considered to evaluate alternative familial structures.
    Often the weight of evidence is employed to update the prior identification probability to the its posterior, being these values defined by the authority who decides about the identification.
    A valid identification tool should be able to produce, with a defined (and conservative) probability, identification, if the alleged relative is, actually, a specific member of the family, or no-identification if that person is an unrelated member of a population.
    In this work we want to show how to reach these two goals by assessing, in each specific case the DNA-based identification potentiality. The result is obtained by studying the distribution of the possible weights of evidence arising in the specific identification procedure. To avoid misleading conclusions, this analysis must be performed before the DNA evidence of the alleged relative is obtained
    Finally we illustrate what actions may be set up to improve the performances of an identification procedure by selecting and adding loci in the analysis.

  15. Marginal likelihood for phylogenetic model: the IDR approach

    S. Arima, L. Tardella
    Università di Roma “La Sapienza”

    Phylogenetic studies aim at constructing a phylogenetic tree which describes the evolutionary relationships between species. Several methods for phylogenetic tree reconstruction have been suggested in literature: we will deal with phylogeny reconstruction methods based on stochastic models in a Bayesian framework. These methods model the change of nucleotide in the DNA sequences as a Markov process with four state (Adenine, Guanine, Cytosine and Thymine): a
    transition matrix, called substitution matrix, defines the probability change from the nucleotide i to j and completely specifies the process.
    Bayesian phylogenetic methods are generating noticeable enthusiasm in the field of molecular systematics. Much of this interest stems from the methods’ ability to quantify uncertainty in comparing complex evolutionary hypotheses using Bayesian model selection. Examples of such comparisons range from detecting positive selection through selection of appropriate nucleotide substitution models [6] to testing molecular clock assumptions [5]. Within the framework of
    Bayesian model selection, phylogenetic researchers are exploring several different approaches. These include developing measures of model fit using predictive distributions and performance-based criteria [2] and comparing the posterior probabilities of hypotheses using Bayes factors [6]. The Bayes factor [1] quantifies the relative support of two competing hypotheses given the observed data and is the Bayesian analogue of the likelihood ratio test (LRT). However, the computation of Bayes Factor and of the marginal likelihood for such complex models is a challenging problem, because of the complexity of the parameter space: in fact, the parameter space in phylogenetic models consists of discrete parameters (the so-called topology) and continuous parameters (the branch lengths and the parameters of the substitution matrix). This mixed structure of the parameter space makes the computation of the Bayes factor more complex. Several methods have been proposed in literature in order to overcome this problem: [6] introduce efficient methods using the Savage-Dickey ratio [7] to calculate the Bayes factor comparing nested evolutionary hypotheses when selecting an appropriate nucleotide substitution model or when testing for the presence of a molecular clock. Bayes factor approaches can be naturally extended to compare non-nested hypotheses, such as competing beliefs about the inferred evolutionary tree. This can be accomplished by generating a posterior sample over the joint space
    of all possible trees using standard Bayesian phylogenetic reconstruction software. We focus on the approach proposed in [4], hereafter named IDR (Inflated Density Ratio), which recicles simulated values coming from MCMC algorithm. The IDR approach consists of an alternative formalization of the Generalized Harmonic Mean (GHM) method in [3]. In particular, the IDR method relies on a different choice of the importance function, defined as a paramet-
    rically inflated version of the target density. After discussing the IDR method, we will focus on some particular challenging settings not discussed in the original paper, such as the case of asymmetric and bimodal distribution. In particular, we propose some transformations of the target distribution which reduce the variance of inflated density ratio estimator. Then we apply the improved method to phylogenetic data: using simulated data, we compare the IDR estimates with those obtained with the GHM method, which is the most widely used model comparison tool since it is automatically implemented in most phylogenetic softwares. For a fixed topology,
    the IDR method produces more precise and robust estimates of the GHM method. Moreover, the flexibility of the approach proposed in [4] allows to extend the IDR estimator in order to estimate the marginal likelihood when both substitution model parameters and topology are
    not fixed.

    References
    [1] R.E. Kass and A. Raftery. Bayes factor. Journal of American Statistical Association, 90:773–795, 1995.
    [2] V.N. Minin, Z. Abdo, P. Joyce, and J. Sullivan. Performance-based selection of likelihood models for phylogeny estimation. Systematic Biology, 52(5):674–683, 2003.
    [3] M.A. Newton and A. Raftery. Approximate Bayesian inference by the weighted likelihood bootstrap. Journal of Royal Statistical Society (Series B), 56:3–48 (with discussion), 1994.
    [4] G. Petris and L. Tardella. New perspectives for Estimating Normalizing Constants via Posterior Simulation (Technical Report). DSPSA, Sapienza Università di Roma, 2007.
    [5] M.A. Suchard. Stochastic Mode for Horizontal Gene Transfer: Taking a Random Walk Through Tree Space. Genetics, 170:419–431, 2005.
    [6] Weiss R.E. Suchard, M.A. and J.S. Sinsheimer. Bayesian selection of continuous-time Markov chain evolutionary models. Molecular Biology and Evolution, 18(6):1001–1013, 2001.
    [7] I. Verdinelli and L. Wasserman. Computing Bayes Factor using a generalization of the Savage-Dickey density ratio. Journal of American Statistical Association, 90:614–618, 1995.

  16. A Review of Various Matching Priors

    Gauri Sankar Datta
    University of Georgia

    Objective priors have played an important role in the recent surge of Bayesian approach to statistics. Although Jeffreys’ prior is suitable as an objective prior for single-parameter models, this prior suffers from some undesirable consequences in multi-parameter models. To rectify some of the deficiencies, various methods have been advocated for development of objective priors in multi-parameter models. The notion of probability matching prior has played a significant role in the derivation of objective priors. Roughly speaking, a matching prior is a prior distribution under which the posterior probabilities of certain credible sets coincide with their frequentist coverage probabilities, either exactly or approximately. Use of such a prior will ensure exact or approximate frequentist validity of Bayesian credible regions. Probability matching priors have been of interest for many years but there has been a resurgence of interest over the last twenty years. In this talk, we survey the main developments in probability matching priors, which have been derived for various types of parametric and predictive region. We also briefly discuss matching priors obtained from matching other performance measure of a procedure.

  17. Predictive densities and prediction limits based on predictive likelihoods

    Paolo Vidoni
    University of Udine

    The notion of predictive likelihood stems from the fact that in the prediction problem there are two unknown quantities to deal with: the future observation and the model parameter.
    Since, according to the likelihood principle, all the evidence is contained in the joint likelihood function, a predictive likelihood for the future observation is obtained by eliminating the nuisance quantity, namely the unknown model parameter. This paper focuses on the profile predictive likelihood and on some modified versions obtained by mimicking the solutions proposed to improve the profile (parametric) likelihood.
    These predictive likelihoods are evaluated by studying how well they generate prediction intervals.
    In particular, we find that, at least in some specific applications, these solution usually improve those ones based on the plug-in procedure. However, the associated predictive densities and prediction limits do not correspond to the optimal frequentist solutions already known in the literature.

  18. Nonparametric methods for filament detection

    Marco Perone Pacifico, Isa Verdinelli
    Università di Roma “La Sapienza”

    Filaments are sets of one-dimensional curves embedded in a point
    process or random field. This talk presents two different nonparametric methods for estimating filamentary structures. The first procedure is based on the mean shift algorithm and on a curve clustering technique, the second on a classification step followed by local averaging. Even if this work is motivated by a cosmological application, the results are of interest for applications in several fields.
    The material presented is part of joint papers with Chris Genovese and Larry Wasserman.

  19. An alternative Monte Carlo approach for estimating rare event probabilities

    Serena Arima(1), Giovanni Petris(2), Luca Tardella(1)
    (1) Università di Roma “La Sapienza”
    (2) University of Arkansas

    There are numerous fields where it is of interest the evauation of the probability of an event which is rare. Of course it is not possible to define how small the probability of an event must be in order to be considered rare regardless of the problem at hand. Nonetheless there are many real contexts such as civil aircraft catastrofic failures, ruin probability for insurance company and overflow of memory buffers in telecommunication systems where a convenient threshold can be set in the order of $10^{-9}$. The problematic aspect of rare event probability estimation is that the complexity of the system often makes the analytic determination of so small numbers infeasible. Hence one of the possible approximation strategy relies on the simulation of the complex system. However, crude Monte Carlo estimates lead to unacceptable error control and an interesting area of research has then been developed to build up reliable estimates based on simulations. Overviews of the rare event simulation theory and tools are contained in Heidelberger (1995); Juneja and Shahabuddin (2006); Blanchet and
    Mandjes (2007) and some recent books (Bucklew, 2004; Rubino and
    Tuffin, 2009) are now available for a more comprehensive account. One of the most used approach relies on importance sampling techniques based on simulating from a distribution which differs from the original one in that rare events have a larger probability. In this preliminary investigation we initially focus on a very simple problem: the estimation of the tail probability of a random variable $Pr(X>c)=\rho$ where $X$ has continuous density $f_X$. We show how it is possible to solve the original problem with a reformulation of the problem and an easy-to-implement simulation which consists of a simple deterministic transform of the random sample drawn from the original distribution. This novel approach is shown to yield strongly efficient estimators with distributions with different tail behavior of $f_X$. The approach is then generalized and extended to the sum of i.i.d. copies of $X$ in the particular case of non negative random variables and its actual performance verified in practice.

  20. Once were wavelets

    Pierpaolo Brutti
    LUISS Guido Carli

    In recent years, a large body of literature has focussed on building wavelet-like systems to analyze (scalar) functions defined on rather general manifolds. One of such successful constructions, called needlets (Narcowich et al., 2006), produces a tight frame (very close to an orthonormal basis), composed by atoms that are compactly
    supported in frequency and show excellent localization properties in space with quasi-exponentially decaying tails. Needlets have been defined over various spaces (e.g. S2 , the unit sphere of R3 equipped with the uniform measure; L^2([−1, 1], μ) where dμ(x) ∝ (1 − x)^α(1 + x)^β and α, β > −1/2), and applied to a variety of statistical problems (e.g. gaussianity test on the sphere, Baldi et al. 2006; inverse problems, Kerkyacharian et al. 2007; density estimation on the sphere, Baldi et al. 2009). After a brief review
    of the state-of-the-art, in this talk I will present two extensions to the original setting that allow us to deal with:
    – Unknown underlying measure More often than not we need to infer the underlying measure μ from the data (e.g. nonparametric regression with random design). For this reason it can be handy to have at our disposal an easy-to-construct needlet system adapted to the data empirical measure or to any other consistent estimator.
    – Matrix-valued data Matrix fields have recently gained significant importance in areas like medical imaging (diffusion tensor magnetic resonance imaging), astrophysics (CMB polarization data), geology and solid mechanics. Given this wide spectrum of applications, it seems worthwhile to develop an appropriate needlet-like tool for restoration/processing and testing specific for this type of structured data.

    References
    [1] Narcowich F., Petrushev P. and Ward J. (2006). Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal., 238, 530–564.
    [2] Baldi P., Kerkyacharian G., Marinucci D. and Picard D. (2006). Asymptotics for spherical needlets. Annals of Statistics.
    [3] Kerkyacharian G., Petrushev P., Picard D. and Willer T. (2007). Needlet algorithms for estimation in inverse problems. Electronic Journal of Statistics, 1, 30–76.
    [4] Baldi P., Kerkyacharian G., Marinucci D. and Picard D. (2009). Adaptive density estimation for directional data using needlets. Annals of Statistics (in press).

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