Aida Toma

Dual divergence estimators and tests: Robustness results

The class of dual @f-divergence estimators (introduced in Broniatowski and Keziou (2009) [5]) is explored with respect to robustness through the influence function approach. For scale and location models, this class is investigated in terms of robustness and asymptotic relative efficiency. Some hypothesis tests based on dual divergence criteria are proposed and their robustness properties are studied. The empirical performances of these estimators and tests are illustrated by Monte Carlo simulation for both non-contaminated and contaminated data.

Robust tests based on dual divergence estimators and saddlepoint approximations

This paper is devoted to robust hypothesis testing based on saddlepoint approximations in the framework of general parametric models. As is known, two main problems can arise when using classical tests. First, the models are approximations of reality and slight deviations from them can lead to unreliable results when using classical tests based on these models. Then, even if a model is correctly chosen, the classical tests are based on first order asymptotic theory. This can lead to inaccurate p-values when the sample size is moderate or small. To overcome these problems, robust tests based on dual divergence estimators and saddlepoint approximations, with good performances in small samples, are proposed.

Quantitative Techniques for Financial Risk Assessment: A Comparative Approach Using Different Risk Measures and Estimation Methods

Abstract The aim of this paper is to highlight and illustrate the use of some quantitative techniques for risk estimation in finance and insurance. The first component involved in risk assessment concerns the risk measure used and the second one is based on the estimation technique. We will study the theoretical properties, the accuracy of modeling the economic phenomena and the computational performances of the risk measures Value-at-Risk, Conditional Tail Expectation, Conditional Value-at-Risk and Limited Value-at-Risk in the case of logistic distribution. We also investigate the most important statistical estimation methods for risk measure evaluation and we will compare their theoretical and empirical behavior. The quality of the risk estimation process corresponding to the quantitative techniques discussed will be tested for both real and simulated data. Numerical results will be provided.

Model Selection Criteria Using Divergences

In this note we introduce some divergence-based model selection criteria. These criteria are defined by estimators of the expected overall discrepancy between the true unknown model and the candidate model, using dual representations of divergences and associated minimum divergence estimators. It is shown that the proposed criteria are asymptotically unbiased. The influence functions of these criteria are also derived and some comments on robustness are provided.

Optimal robust M-estimators using Rényi pseudodistances

Using Renyi pseudodistances, new robustness and efficiency measures are defined. On the basis of these measures, new optimal robust M-estimators for multidimensional parameters, called optimal BR@a-robust M-estimators, are derived using the Hampels infinitesimal approach. The classical optimal Bi-robust estimator is particularly obtained. It is shown that the new optimal estimators are characterized by equivariance properties: equivariance with respect to reparametrizations, as well as equivariance with respect to transformations of the data set when the model is generated by a group of transformations. The performance of these estimators is illustrated by Monte Carlo simulations in the case of the Weibull distribution, as well as on the basis of real data.

Robust Estimators for the Parameters of Multivariate Lognormal Distribution

Abstract In this paper, we introduce a class of location and scale estimators for the p-variate lognormal distribution. These estimators are obtained by applying a log transform to the data, computing robust Fisher consistent estimators for the obtained Gaussian data and transforming those estimators for the lognormal using the relationship between the parameters of both distributions. We prove some of the properties of these estimators, such as Fisher consistency, robustness and asymptotic normality.

Robust Portfolio Optimization Using Pseudodistances

The presence of outliers in financial asset returns is a frequently occurring phenomenon which may lead to unreliable mean-variance optimized portfolios. This fact is due to the unbounded influence that outliers can have on the mean returns and covariance estimators that are inputs in the optimization procedure. In this paper we present robust estimators of mean and covariance matrix obtained by minimizing an empirical version of a pseudodistance between the assumed model and the true model underlying the data. We prove and discuss theoretical properties of these estimators, such as affine equivariance, B-robustness, asymptotic normality and asymptotic relative efficiency. These estimators can be easily used in place of the classical estimators, thereby providing robust optimized portfolios. A Monte Carlo simulation study and applications to real data show the advantages of the proposed approach. We study both in-sample and out-of-sample performance of the proposed robust portfolios comparing them with some other portfolios known in literature.

An Integrated Risk Measure and Information Theory Approach for Modeling Financial Data and Solving Decision Making Problems

Abstract In this paper we build some integrated techniques for modeling financial data and solving decision making problems, based on risk theory and information theory. Several risk measures and entropy measures are investigated and compared with respect to their analytical properties and effectiveness in solving real problems. Some criteria for portfolio selection are derived combining the classical risk measure approach with the information theory approach. We analyze the performance of the methods proposed in case of some financial applications.Computational results are provided.

Robust Estimation for the Single Index Model Using Pseudodistances

For portfolios with a large number of assets, the single index model allows for expressing the large number of covariances between individual asset returns through a significantly smaller number of parameters. This avoids the constraint of having very large samples to estimate the mean and the covariance matrix of the asset returns, which practically would be unrealistic given the dynamic of market conditions. The traditional way to estimate the regression parameters in the single index model is the maximum likelihood method. Although the maximum likelihood estimators have desirable theoretical properties when the model is exactly satisfied, they may give completely erroneous results when outliers are present in the data set. In this paper, we define minimum pseudodistance estimators for the parameters of the single index model and using them we construct new robust optimal portfolios. We prove theoretical properties of the estimators, such as consistency, asymptotic normality, equivariance, robustness, and illustrate the benefits of the new portfolio optimization method for real financial data.