Author
Irene Crimaldi
Other affiliations: University of Bologna
Bio: Irene Crimaldi is an academic researcher from IMT Institute for Advanced Studies Lucca. The author has contributed to research in topics: Convergence of random variables & Central limit theorem. The author has an hindex of 12, co-authored 74 publications receiving 524 citations. Previous affiliations of Irene Crimaldi include University of Bologna.
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Papers
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01 Jan 2007
40 citations
TL;DR: In this article, a class of random sequences, called generalized species sampling sequences, is defined and a condition to have conditional identity in distribution is given, which generalizes the well-known notion of exchangeability.
Abstract: In this paper the theory of species sampling sequences is linked to the theory of conditionally identically distributed sequences in order to enlarge the set of species sampling sequences which are mathematically tractable. The conditional identity in distribution (see Berti, Pratelli and Rigo (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper a class of random sequences, called generalized species sampling sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, two types of generalized species sampling sequence that are conditionally identically distributed are introduced and studied: the generalized Poisson-Dirichlet sequence and the generalized Ottawa sequence. Some examples are discussed.
37 citations
TL;DR: In this article, a new version of the Central Limit Theorem for multivariate martingales is presented, which is based on the central limit theorem for multi-dimensional martingale.
34 citations
TL;DR: In this paper, the central limit theorem for urn problems was proved for multicolor randomly reinforced urns, and the latter was investigated by paying special attention to multicolored randomly reinforced IBEs.
Abstract: Let Xn be a sequence of integrable real random variables, adapted to a filtration (Gn). Define Cn = √{(1 / n)∑k=1nXk - E(Xn+1 | Gn)} and Dn = √n{E(Xn+1 | Gn) - Z}, where Z is the almost-sure limit of E(Xn+1 | Gn) (assumed to exist). Conditions for (Cn, Dn) → N(0, U) x N(0, V) stably are given, where U and V are certain random variables. In particular, under such conditions, we obtain √n{(1 / n)∑k=1nX_k - Z} = Cn + Dn → N(0, U + V) stably. This central limit theorem has natural applications to Bayesian statistics and urn problems. The latter are investigated, by paying special attention to multicolor randomly reinforced urns.
33 citations
01 Jan 2009
TL;DR: The authors proved an almost sure conditional convergence result toward a Gaussian kernel and applied it to a two-colors randomly reinforced urn and proved that it converges to a Gaussian kernel.
Abstract: We prove an almost sure conditional convergence result toward a Gaussian kernel and we apply it to a two-colors randomly reinforced urn. Mathematical Subject Classification: 60F15, 60G42
31 citations
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TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
5,689 citations
1,083 citations
TL;DR: In this article, the convergence of Distri butions of Likelihood Ratio has been discussed, and the authors propose a method to construct a set of limit laws for Likelihood Ratios.
Abstract: 1 Introduction.- 2 Experiments, Deficiencies, Distances v.- 2.1 Comparing Risk Functions.- 2.2 Deficiency and Distance between Experiments.- 2.3 Likelihood Ratios and Blackwell's Representation.- 2.4 Further Remarks on the Convergence of Distri butions of Likelihood Ratios.- 2.5 Historical Remarks.- 3 Contiguity - Hellinger Transforms.- 3.1 Contiguity.- 3.2 Hellinger Distances, Hellinger Transforms.- 3.3 Historical Remarks.- 4 Gaussian Shift and Poisson Experiments.- 4.1 Introduction.- 4.2 Gaussian Experiments.- 4.3 Poisson Experiments.- 4.4 Historical Remarks.- 5 Limit Laws for Likelihood Ratios.- 5.1 Introduction.- 5.2 Auxiliary Results.- 5.2.1 Lindeberg's Procedure.- 5.2.2 Levy Splittings.- 5.2.3 Paul Levy's Symmetrization Inequalities.- 5.2.4 Conditions for Shift-Compactness.- 5.2.5 A Central Limit Theorem for Infinitesimal Arrays.- 5.2.6 The Special Case of Gaussian Limits.- 5.2.7 Peano Differentiable Functions.- 5.3 Limits for Binary Experiments.- 5.4 Gaussian Limits.- 5.5 Historical Remarks.- 6 Local Asymptotic Normality.- 6.1 Introduction.- 6.2 Locally Asymptotically Quadratic Families.- 6.3 A Method of Construction of Estimates.- 6.4 Some Local Bayes Properties.- 6.5 Invariance and Regularity.- 6.6 The LAMN and LAN Conditions.- 6.7 Additional Remarks on the LAN Conditions.- 6.8 Wald's Tests and Confidence Ellipsoids.- 6.9 Possible Extensions.- 6.10 Historical Remarks.- 7 Independent, Identically Distributed Observations.- 7.1 Introduction.- 7.2 The Standard i.i.d. Case: Differentiability in Quadratic Mean.- 7.3 Some Examples.- 7.4 Some Nonparametric Considerations.- 7.5 Bounds on the Risk of Estimates.- 7.6 Some Cases Where the Number of Observations Is Random.- 7.7 Historical Remarks.- 8 On Bayes Procedures.- 8.1 Introduction.- 8.2 Bayes Procedures Behave Nicely.- 8.3 The Bernstein-von Mises Phenomenon.- 8.4 A Bernstein-von Mises Result for the i.i.d. Case.- 8.5 Bayes Procedures Behave Miserably.- 8.6 Historical Remarks.- Author Index.
483 citations
TL;DR: In this article, the authors derived the large sample distribution of propensity score matching estimators, taking into account that the propensity score is itself estimated in a first step, prior to matching, and they showed that first step estimation of the propensity scores affects the distribution of the matching estimator.
Abstract: Propensity score matching estimators (Rosenbaum and Rubin (1983)) are widely used in evaluation research to estimate average treatment effects. In this article, we derive the large sample distribution of propensity score matching estimators. Our derivations take into account that the propensity score is itself estimated in a first step, prior to matching. We prove that first step estimation of the propensity score affects the large sample distribution of propensity score matching estimators, and derive adjustments to the large sample variances of propensity score matching estimators of the average treatment effect (ATE) and the average treatment effect on the treated (ATET). The adjustment for the ATE estimator is negative (or zero in some special cases), implying that matching on the estimated propensity score is more efficient than matching on the true propensity score in large samples. However, for the ATET estimator, the sign of the adjustment term depends on the data generating process, and ignoring the estimation error in the propensity score may lead to confidence intervals that are either too large or too small.
427 citations
Journal Article•
TL;DR: A discrete time generative model for social network evolution that inherits the richness and flexibility of the class of exponential family random‐graph models and facilitates separable modelling of the tie duration distributions and the structural dynamics of tie formation is developed.
Abstract: Models of dynamic networks—networks that evolve over time—have manifold applications. We develop a discrete time generative model for social network evolution that inherits the richness and flexibility of the class of exponential family random‐graph models. The model—a separable temporal exponential family random‐graph model—facilitates separable modelling of the tie duration distributions and the structural dynamics of tie formation. We develop likelihood‐based inference for the model and provide computational algorithms for maximum likelihood estimation. We illustrate the interpretability of the model in analysing a longitudinal network of friendship ties within a school.
325 citations