Semiparametric Cross Entropy for rare-event simulation

The Cross Entropy method is a well-known adaptive importance sampling method for rare-event probability estimation, which requires estimating an optimal importance sampling density within a parametric class. In this article we estimate an optimal importance sampling density within a wider semiparametric class of distributions. We show that this semiparametric version of the Cross Entropy method frequently yields efficient estimators. We illustrate the excellent practical performance of the method with numerical experiments and show that for the problems we consider it typically outperforms alternative schemes by orders of magnitude

Integrating Randomized Placebo-Controlled Trial Data with External Controls: A Semiparametric Approach with Selective Borrowing

In recent years, real-world external controls (ECs) have grown in popularity as a tool to empower randomized placebo-controlled trials (RPCTs), particularly in rare diseases or cases where balanced randomization is unethical or impractical. However, as ECs are not always comparable to the RPCTs, direct borrowing ECs without scrutiny may heavily bias the treatment effect estimator. Our paper proposes a data-adaptive integrative framework capable of preventing unknown biases of ECs. The adaptive nature is achieved by dynamically sorting out a set of comparable ECs via bias penalization. Our proposed method can simultaneously achieve (a) the semiparametric efficiency bound when the ECs are comparable and (b) selective borrowing that mitigates the impact of the existence of incomparable ECs. Furthermore, we establish statistical guarantees, including consistency, asymptotic distribution, and inference, providing type-I error control and good power. Extensive simulations and two real-data applications show that the proposed method leads to improved performance over the RPCT-only estimator across various bias-generating scenarios

A note on maximum likelihood estimation of a Pareto mixture

In this paper we study Maximum Likelihood Estimation of the parameters of a Pareto mixture. Application of standard techniques to a mixture of Pareto is problematic. For this reason we develop two alternative algorithms. The first one is the Simulated Annealing and the second one is based on Cross-Entropy minimization. The Pareto distribution is a commonly used model for heavy-tailed data. It is a two-parameter distribution whose shape parameter determines the degree of heaviness of the tail, so that it can be adapted to data with different features. This work is motivated by an application in the operational risk measurement field: we fit a Pareto mixture to operational losses recorded by a bank in two different business lines. Losses below an unknown threshold are discarded, so that the observed data are truncated. The thresholds used in the two business lines are unknown. Thus, under the assumption that each population follows a Pareto distribution, the appropriate model is a mixture of Pareto where all the parameters have to be estimated.

A new approach to measure systemic risk:A bivariate copula model for dependent censored data

We propose a novel approach based on the Marshall-Olkin (MO) copula to estimate the impact of systematic and idiosyncratic components on cross-border systemic risk. To use the data on non-failed banks in the suggested method, we consider the time to bank failure as a censored variable. Therefore, we propose a pseudo-maximum likelihood estimation procedure for the MO copula for a Type I censored sample. We derive the log-likelihood function, the copula parameter estimator and the bootstrap confidence intervals. Empirical data on the banking system of three European countries (Germany, Italy and the UK) shows that the proposed censored model can accurately estimate the systematic component of cross-border systemic risk. (C) 2019 Elsevier B.V. All rights reserved

Collaborative Targeted Maximum Likelihood Estimation

Collaborative double robust targeted maximum likelihood estimators represent a fundamental further advance over standard targeted maximum likelihood estimators of causal inference and variable importance parameters. The targeted maximum likelihood approach involves fluctuating an initial density estimate, (Q), in order to make a bias/variance tradeoff targeted towards a specific parameter in a semi-parametric model. The fluctuation involves estimation of a nuisance parameter portion of the likelihood, g. TMLE and other double robust estimators have been shown to be consistent and asymptotically normally distributed (CAN) under regularity conditions, when either one of these two factors of the likelihood of the data is correctly specified. In this article we provide a template for applying collaborative targeted maximum likelihood estimation (C-TMLE) to the estimation of pathwise differentiable parameters in semi-parametric models. The procedure creates a sequence of candidate targeted maximum likelihood estimators based on an initial estimate for Q coupled with a succession of increasingly non-parametric estimates for g. In a departure from current state of the art nuisance parameter estimation, C-TMLE estimates of g are constructed based on a loss function for the relevant factor Q_0, instead of a loss function for the nuisance parameter itself. Likelihood-based cross-validation is used to select the best estimator among all candidate TMLE estimators in this sequence. A penalized-likelihood loss function for Q_0 is suggested when the parameter of interest is borderline-identifiable. We present theoretical results for collaborative double robustness, demonstrating that the collaborative targeted maximum likelihood estimator is CAN when Q and g are both mis-specified, providing that g solves a specified score equation implied by the difference between the Q and the true Q_0. This marks an improvement over the current definition of double robustness in the estimating equation literature. We also establish an asymptotic linearity theorem for the C-DR-TMLE of the target parameter, showing that the C-DR-TMLE is more adaptive to the truth, and, as a consequence, can even be super efficient if the first stage density estimator does an excellent job itself with respect to the target parameter. This research provides a template for targeted efficient and robust loss-based learning of a particular target feature of the probability distribution of the data within large (infinite dimensional) semi-parametric models, while still providing statistical inference in terms of confidence intervals and p-values. This research also breaks with a taboo (e.g., in the propensity score literature in the field of causal inference) on using the relevant part of likelihood to fine-tune the fitting of the nuisance parameter/censoring mechanism/treatment mechanism

ISBIS 2016: Meeting on Statistics in Business and Industry

This Book includes the abstracts of the talks presented at the 2016 International Symposium on Business and Industrial Statistics, held at Barcelona, June 8-10, 2016, hosted at the Universitat Politècnica de Catalunya - Barcelona TECH, by the Department of Statistics and Operations Research. The location of the meeting was at ETSEIB Building (Escola Tecnica Superior d'Enginyeria Industrial) at Avda Diagonal 647. The meeting organizers celebrated the continued success of ISBIS and ENBIS society, and the meeting draw together the international community of statisticians, both academics and industry professionals, who share the goal of making statistics the foundation for decision making in business and related applications. The Scientific Program Committee was constituted by: David Banks, Duke University Amílcar Oliveira, DCeT - Universidade Aberta and CEAUL Teresa A. Oliveira, DCeT - Universidade Aberta and CEAUL Nalini Ravishankar, University of Connecticut Xavier Tort Martorell, Universitat Politécnica de Catalunya, Barcelona TECH Martina Vandebroek, KU Leuven Vincenzo Esposito Vinzi, ESSEC Business Schoo

Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods

This book addresses contemporary statistical inference issues when no or minimal assumptions on the nature of studied phenomenon are imposed. Information theory methods play an important role in such scenarios. The approaches discussed include various high-dimensional regression problems, time series and dependence analyses

Moment Problems with Applications to Value-At-Risk and Portfolio Management

Moment Problems with Applications to Value-At-Risk and Portfolio Management By Ruilin Tian May 2008 Committee Chair: Dr. Samuel H. Cox Major Department: Risk Management and Insurance My dissertation provides new applications of moment theory and optimization to financial and insurance risk management. In the investment and managerial areas, one often needs to determine some measure of risk, especially the risk of extreme events. However, complete information of the underlying outcomes is usually unavailable; instead one has access to partial information such as the mean, variance, mode, or range. In Chapters 2 and 3, we find the semiparametric upper and lower bounds for the value-at-risk (VaR) with incomplete information, that is, moments of the underlying distribution. When a single variable is concerned, bounds on VaR are computed to obtain a 100% confidence interval. When the sample financial data have a global maximum, we show that unimodal assumption tightens the optimal bounds. Next we further analyze a function of two correlated random variables. Specifically, we find bounds on the probability of two joint extreme events. When three or more variables are involved, the multivariate problem can sometimes be converted to a single variable problem. In all cases, we use the physical measure rather than the commonly used equivalent pricing probability measure. In addition to solving these problems using the traditional approach based on the geometry of a moment problem, a more efficient method is proposed to solve a general class of moment bounds via semidefinite programming. In the last part of the thesis, we apply optimization techniques to improve financial portfolio risk management. Instead of considering VaR, we work with a coherent risk measure, the conditional VaR (CVaR). As an extension of Krokhmal et al. (2002), we impose CVaR-related functions to the portfolio selection problem. The CVaR approach sets a β-level CVaR as the objective function and maximizes the worst case on the tail of the distribution. The CVaR-like constraints approach adds a set of CVaR-like constraints to the traditional Markowitz problem, reshaping the portfolio distribution. Both methods greatly increase the skewness of portfolios, although the CVaR approach may lose control of the variance. This capability of increasing skewness is very attractive to the investors who may prefer higher probability of obtaining higher returns. We compare the CVaR-related approaches to some other popular portfolio optimization methods. Our numerical analysis provides empirical support for the superiority of the CVaR-like constraints approach in terms of portfolio efficiency

Efficient Estimation of Semiparametric Transformation Models for Two-Phase Cohort Studies

Under two-phase cohort designs, such as case-cohort and nested case-control sampling, information on observed event times, event indicators, and inexpensive covariates is collected in the first phase, and the first-phase information is used to select subjects for measurements of expensive covariates in the second phase; inexpensive covariates are also used in the data analysis to control for confounding and to evaluate interactions. This paper provides efficient estimation of semiparametric transformation models for such designs, accommodating both discrete and continuous covariates and allowing inexpensive and expensive covariates to be correlated. The estimation is based on the maximization of a modified nonparametric likelihood function through a generalization of the expectation-maximization algorithm. The resulting estimators are shown to be consistent, asymptotically normal and asymptotically efficient with easily estimated variances. Simulation studies demonstrate that the asymptotic approximations are accurate in practical situations. Empirical data from Wilms’ tumor studies and the Atherosclerosis Risk in Communities (ARIC) study are presented