Cressie Read Family Research Articles

A general similarity measure for simple correspondence analysis

This article presents a general similarity measure for comparing different variants of simple correspondence analysis when analysing the association using the Cressie-Read family of divergence statistics (Beh and Lombardo 2024, International Statistical Review). It includes, as special cases, the similarity measures that have been proposed in the correspondence analysis literature for assessing the similarities and differences between the traditional approach to simple correspondence analysis and new approaches like the log-ratio analysis, and the Hellinger distance method. This article describes six further properties that show how the proposed general similarity measure can be expanded upon.

Correspondence Analysis for Assessing Departures from Perfect Symmetry Using the Cressie–Read Family of Divergence Statistics

Recently, Beh and Lombardo (2022, Symmetry, 14, 1103) showed how to perform a correspondence analysis on a two-way contingency table where Bowker’s statistic lies at the numerical heart of this analysis. Thus, we showed how this statistic could be used to visually identify departures from perfect symmetry. Interestingly, Bowker’s statistic is a special case of the symmetry version of the Cressie–Read family of divergence statistics. Therefore, this paper presents a new framework for visually assessing departures from perfect symmetry using a second-order Taylor series approximation of the Cressie–Read family of divergence statistics.

Large-Scale Non-convex Stochastic Constrained Distributionally Robust Optimization

Distributionally robust optimization (DRO) is a powerful framework for training robust models against data distribution shifts. This paper focuses on constrained DRO, which has an explicit characterization of the robustness level. Existing studies on constrained DRO mostly focus on convex loss function, and exclude the practical and challenging case with non-convex loss function, e.g., neural network. This paper develops a stochastic algorithm and its performance analysis for non-convex constrained DRO. The computational complexity of our stochastic algorithm at each iteration is independent of the overall dataset size, and thus is suitable for large-scale applications. We focus on the general Cressie-Read family divergence defined uncertainty set which includes chi^2-divergences as a special case. We prove that our algorithm finds an epsilon-stationary point with an improved computational complexity than existing methods. Our method also applies to the smoothed conditional value at risk (CVaR) DRO.

On the P-value for Members of the Cressie-Read Family of Divergence Statistics

A few years ago a paper appeared in this Journal that proposed a closed-form approximation of the p-value for Pearson's chi-squared statistic. Extensive empirical and simulation studies were performed and it was shown that the approximation provides very accurate p-values when compared with what we considered to be the ``true'' p-value (obtained using a base R function). It is important to note, however, that Pearson's chi-squared statistic is a special case of the Cressie-Read family of divergence statistics as is, for example, the log-likelihood ratio statistic, the Freeman-Tukey statistic and the Cressie-Read statistic. Therefore, this paper adapts the previously published closed form approximation of the p-value by demonstrating its applicability to any member of the Cressie-Read family of divergence statistics. We also give two further closed form approximations and assess their accuracy by analysing three contingency tables of varying sample size, degrees of freedom and statistical significance of the association.

Correspondence Analysis Using the Cressie–Read Family of Divergence Statistics

SummaryThe foundations of correspondence analysis rests with Pearson's chi‐squared statistic. More recently, it has been shown that the Freeman–Tukey statistic plays an important role in correspondence analysis and confirmed the advantages of the Hellinger distance that have long been advocated in the literature. Pearson's and the Freeman–Tukey statistics are two of five commonly used special cases of the Cressie–Read family of divergence statistics. Therefore, this paper explores the features of correspondence analysis where its foundations lie with this family and shows that log‐ratio analysis (an approach that has gained increasing attention in the correspondence analysis and compositional data analysis literature) and the method based on the Hellinger distance are special cases of this new framework.

Nonparametric Option Pricing with Generalized Entropic Estimators

We propose a family of nonparametric estimators for an option price that require only the use of underlying return data, but can also easily incorporate information from observed option prices. Each estimator comes from a risk-neutral measure minimizing generalized entropy according to a different Cressie–Read discrepancy. We apply our method to price S&P 500 options and the cross-section of individual equity options, using distinct amounts of option data in the estimation. Estimators incorporating mild nonlinearities produce optimal pricing accuracy within the Cressie–Read family and outperform several benchmarks such as Black–Scholes and different GARCH option pricing models. Overall, we provide a powerful option pricing technique suitable for scenarios of limited option data availability.

A unified entropic pricing framework of option: Using Cressie-Read family of divergences

The entropy valuation of option (Stutzer, 1996) provides a risk-neutral probability distribution (RND) as the pricing measure by minimizing the Kullback–Leibler (KL) divergence between the empirical probability distribution and its risk-neutral counterpart. This article establishes a unified entropic framework by developing a class of generalized entropy pricing models based upon Cressie-Read (CR) family of divergences. The main contributions of this study are: (1) this unified framework can readily incorporate a set of informative risk-neutral moments (RNMs) of underlying return extracted from the option market which accurately captures the characteristics of the underlying distribution; (2) the classical KL-based entropy pricing model is extended to a unified entropic pricing framework upon a family of CR divergences. For each of the proposed models under the unified framework, the optimal RND is derived by employing the dual method. Simulations show that, compared to the true price, each model of the proposed family can produce high accuracy for option pricing. Meanwhile, the pricing biases among the models are different, and we hence conduct theoretical analysis and experimental investigations to explore the driving causes.

An SDF Approach to Hedge Funds' Tail Risk:Evidence from Brazilian Funds

The main purpose of this paper is to propose a methodology to obtain a hedge fund tail risk measure. Our measure builds on the methodologies proposed by \citet*{ag15} and \citet*{aagvg15}, which rely in solving dual minimization problems of Cressie Read discrepancy functions in spaces of probability measures. Due to the recently documented robustness of the Hellinger estimator (Kitamura et al., 2013), we adopt within the Cressie Read family, this specific discrepancy as loss function. From this choice, we derive a minimum Hellinger risk-neutral measure that correctly prices an observed panel of hedge fund returns. The estimated risk-neutral measure is used to construct our tail risk measure by pricing synthetic out-of-the-money put options on hedge fund returns of ten specific categories. We provide a detailed description of our methodology, extract a Tail risk hedge fund factor for Brazilian funds, and relate it to a commonly adopted market volatility measure.

Non-parametric American option valuation using Cressie–Read divergences

In this paper we build on the possibility that the use of the Cressie–Read family with the non-parametic method for valuing European option might be extended to non-parametric valuation of American options. We derive a suite of non-parametric methods to price and hedge American-style options, utilising the Cressie-Read family of divergences. We test the efficacy of these methods using a large sample of traded American-style options struck on the S&P100 index. We find that in general, our suite of non-parametric valuation schemes generate more accurate price estimates than traditional parametric schemes, especially for longer-dated options.

Non-Parametric American Option Valuation Using Cressie-Read Divergences

Haley & Walker (2010) suggest that their use of Cressie-Read family within Stutzer's (1996) non-parametric method for valuing European option might be extended to Alcock & Carmichael's (2008) non-parametric valuation of American options. We derive this suite of non-parametric methods to price and hedge American-style options. We test their efficacy using a large sample of traded American style options struck on the S&P100 index. We find that in general, the suite of non-parametric valuation schemes generate more accurate price estimates than traditional parametric schemes, especially for longer-dated options.

Information recovery in behavioral networks.

In the context of agent based modeling and network theory, we focus on the problem of recovering behavior-related choice information from origin-destination type data, a topic also known under the name of network tomography. As a basis for predicting agents' choices we emphasize the connection between adaptive intelligent behavior, causal entropy maximization, and self-organized behavior in an open dynamic system. We cast this problem in the form of binary and weighted networks and suggest information theoretic entropy-driven methods to recover estimates of the unknown behavioral flow parameters. Our objective is to recover the unknown behavioral values across the ensemble analytically, without explicitly sampling the configuration space. In order to do so, we consider the Cressie-Read family of entropic functionals, enlarging the set of estimators commonly employed to make optimal use of the available information. More specifically, we explicitly work out two cases of particular interest: Shannon functional and the likelihood functional. We then employ them for the analysis of both univariate and bivariate data sets, comparing their accuracy in reproducing the observed trends.

Generalized relative entropies in the classical limit

Our protagonists are (i) the Cressie–Read family of divergences (characterized by the parameter γ), (ii) Tsallis’ generalized relative entropies (characterized by the q one), and, as a particular instance of both, (iii) the Kullback–Leibler (KL) relative entropy. In their normalized versions, we ascertain the equivalence between (i) and (ii). Additionally, we employ these three entropic quantifiers in order to provide a statistical investigation of the classical limit of a semiclassical model, whose properties are well known from a purely dynamic viewpoint. This places us in a good position to assess the appropriateness of our statistical quantifiers for describing involved systems. We compare the behaviour of (i), (ii), and (iii) as one proceeds towards the classical limit. We determine optimal ranges for γ and/or q. It is shown the Tsallis-quantifier is better than KL’s for 1.5<q<2.5.

On Bartlett correctability of empirical likelihood in generalized power divergence family

Baggerly (1998) showed that empirical likelihood is the only member in the Cressie–Read power divergence family to be Bartlett correctable. This paper strengthens Baggerly’s result by showing that in a generalized class of the power divergence family, which includes the Cressie–Read family and other nonparametric likelihood such as Schennach’s (2005, 2007) exponentially tilted empirical likelihood, empirical likelihood is still the only member to be Bartlett correctable.

Book Reviews

Provides a conceptual and empirical understanding of basic information theoretic econometric models and methods. Discusses formulation and analysis of parametric and semiparametric linear models; method of moments, generalized method of moments, and estimating equations; a stochastic-empirical likelihood inverse problem—formulation and estimation; a stochastic empirical likelihood inverse problem—estimation and inference; Kullback–Leibler information and the maximum empirical exponential likelihood; the Cressie–Read family of divergence measures and empirical maximum likelihood functions; Cressie–Read minimum power divergence (MPD) type estimators in practice—Monte Carlo evidence of estimation and inference sampling performance; family of MPD distribution functions for the binary response-choice model; estimation and inference for the binary response model based on the MPD family of distributions; and choosing the optimal divergence under quadratic loss. Judge is a professor at the University of California, Berkeley. Mittelhammer is Regents Professor of Economic Sciences and Statistics at Washington State University.

Implications of the Cressie-Read Family of Additive Divergences for Information Recovery

To address the unknown nature of probability-sampling models, in this paper we use information theoretic concepts and the Cressie-Read (CR) family of information divergence measures to produce a flexible family of probability distributions, likelihood functions, estimators, and inference procedures. The usual case in statistical modeling is that the noisy indirect data are observed and known and the sampling model-error distribution-probability space, consistent with the data, is unknown. To address the unknown sampling process underlying the data, we consider a convex combination of two or more estimators derived from members of the flexible CR family of divergence measures and optimize that combination to select an estimator that minimizes expected quadratic loss. Sampling experiments are used to illustrate the finite sample properties of the resulting estimator and the nature of the recovered sampling distribution.

Implications of the Cressie-Read Family of Additive Divergences for Information Recovery

To address the unknown nature of probability-­sampling models, in this paper we use information theoretic concepts and the Cressie-Read (CR) family of information divergence measures to produce a flexible family of probability distributions, likelihood function relationships, estimators, and inference procedures. The usual case in statistical modeling is that the noisy indirect data are observed and known and the sampling model-­error distribution-­probability space, consistent with the data, is unknown. To address the unknown sampling process underlying the data, we consider a convex combination of two or more estimators derived from members of the flexible CR family of divergence measures and optimize that combination to select an estimator that minimizes expected quadratic loss. Sampling experiments are used to illustrate the finite sample properties of the resulting estimator and the nature of the recovered sampling distribution.

Assessing misspecified asset pricing models with empirical likelihood estimators

Hansen and Jagannathan (1997) compare misspecified asset pricing models based on least-square projections on a family of admissible stochastic discount factors. We extend their fundamental contribution by considering Minimum Discrepancy projections where misspecification is measured by a family of convex functions that take into account higher moments of asset returns. The Minimum Discrepancy problems are solved on dual spaces producing a family of estimators that captures the least-square problem as a particular case. We derive the asymptotic distributions of the estimators for the Cressie–Read family of discrepancies, and illustrate their use with an assessment of the Consumption Asset Pricing Model.

On Bartlett Correctability of Empirical Likelihood in Generalized Power Divergence Family

Baggerly (1998) showed that empirical likelihood is the only member in the Cressie-Read power divergence family to be Bartlett correctable. This paper strengthens Baggerly's result by showing that in a generalized class of the power divergence family, which includes the Cressie-Read family and other nonparametric likelihood such as Schennach's (2005, 2007) exponentially tilted empirical likelihood, empirical likelihood is still the only member to be Bartlett correctable.

Assessing Misspecifications in Asset Pricing Models with Nonlinear Projections of Pricing Kernels

We develop a new approach to identify model misspecifications based on Minimum Discrepancy (MD) projections that correct asset pricing models with the use of nonlinear functions of basis assets returns. These nonlinear corrections make our method more effective than the Hansen and Jagannathan distance in detecting sources of model specifications, especially in economies where nonlinear risk is priced. A theoretical example illustrates our point, with an economy where the true SDF prices coskewness risk with respect to the market portfolio (Kraus and Litzemberger (1976)). We suggest diagnosing the CAPM model under this economy, and show that while the HJ distance can not identify the exact source of model misspecification (a quadratic term in the market return), there are nonlinear projections in the class of MD problems that correctly capture this term. We also derive the asymptotic distributions of the estimators for the Cressie Read family of discrepancies, and illustrate their use with an empirical assessment of the CCAPM.

Alternative tilts for nonparametric option pricing

AbstractThis study generalizes the nonparametric approach to option pricing of Stutzer, M. (1996) by demonstrating that the canonical valuation methodology introduced therein is one member of the Cressie–Read family of divergence measures. Alhough the limiting distribution of the alternative measures is identical to the canonical measure, the finite sample properties are quite different. We assess the ability of the alternative divergence measures to price European call options by approximating the risk‐neutral, equivalent martingale measure from an empirical distribution of the underlying asset. A simulation study of the finite sample properties of the alternative measure changes reveals that the optimal divergence measure depends upon how accurately the empirical distribution of the underlying asset is estimated. In a simple Black–Scholes model, the optimal measure change is contingent upon the number of outliers observed, whereas the optimal measure change is a function of time to expiration in the stochastic volatility model of Heston, S. L. (1993). Our extension of Stutzer's technique preserves the clean analytic structure of imposing moment restrictions to price options, yet demonstrates that the nonparametric approach is even more general in pricing options than originally believed. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:983–1006, 2010