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We use information theoretic information recovery methods, on a 2005 sample of household income data from the Chinese InterCensus, to estimate the income distribution for China and each of its 31 provinces and to obtain corresponding measures of income inequality. Using entropy divergence methods, we seek a probability density function solution that is as close to a uniform probability distribution of income (with the least inequality), as the data will permit. These entropy measures of income inequality reflect how the allocation and distribution systems are performing, and we show the advantages of investigating province variation in income inequality using entropy measures rather than Gini coefficients. Finally, we use a sample of data from the China Family Panel Study to recover an estimate of the 2010 and the 2016 to investigate possible directions of inequality changes using these different additional data sources, given that the 2015 Inter-Census is not yet available.

Since Benford’s law is an empirical phenomenon that occurs in a range of data sets, this raises the question as to whether or not the same thing might be true in terms of the Chinese income distribution data. We focus on the first significant digit (FSD) distribution of Chinese micro income data from the 2005 Inter-Census sample, which corresponds to 1% of Chinese population and other micro income data from the China family panel studies (CFPS) and Chinese General Social Survey (CGSS). We use information theoretic-entropy based methods to investigate the degree to which Benford’s FSD law is consistent with the FSD of Chinese income data and our findings suggest consistency between the Chinese FSD income distribution and Benford’s distribution. The close connection between the two distributions has implications for the quality of the sample of Chinese micro data.

This paper is concerned with the problem of information recovery and measuring evidence that involves uncontrolled indirect noisy effects data and stochastic ill posed inverse problems in economics-econometrics. Information theoretic methods based on a multi parametric family of power divergence measures are suggested as an estimation and inference framework for dealing with these problems, analyzing questions of a causal nature and learning about hidden dynamic economic processes and systems that may or may not be in equilibrium. The paper concludes with some comments on the implications for information recovery of continuing to use traditional economic-econometric models and methods.

With an eye to providing a methodology for tracking the dynamic integrity of prices for important market indicators, in this paper we use Benford second digit reference distribution to track the daily London Interbank Offered Rate (Libor) over the period 2005-2008. This reference, known as Benford’s law, is present in many naturally occurring numerical data sets as well as in several financial data sets. We find that in two recent periods Libor rates depart significantly from the expected Benford reference distribution. This raises potential concerns relative to the unbiased nature of the signals coming from the sixteen banks from which the Libor is computed and the usefulness of the Libor as a major economic indicator.

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.

Despite the productive efforts of economists, the disequilibrium nature of the economic system and imprecise predictions persist. One reason for this outcome is that traditional econometric models and estimation and inference methods cannot provide the necessary quantitative information for the causal influence-dynamic micro and macro questions we need to ask given the noisy indirect effects data we use. To  move economics in the direction of a probabilistic and causal based predictive science, in this paper information theoretic estimation and inference methods are suggested as a basis for  understanding and making predictions about dynamic micro and macro economic processes and systems.

Empirical Likelihood (EL) and other methods that operate within the Empirical Estimating Equations (E3) approach to estimation and inference are challenged by the Empty Set Problem (ESP). ESP concerns the possibility that a model set, which is data-dependent, may be empty for some data sets. To avoid ESP we return from E3 back to the Estimating Equations, and explore the Bayesian infinite-dimensional Maximum A-posteriori Probability (MAP) method. The Bayesian MAP with Dirichlet prior motivates a Revised EL (ReEL) method. ReEL i) avoids ESP as well as the convex hull restriction, ii) attains the same basic asymptotic properties as EL, and iii) its computation complexity is comparable to that of EL.

Methods, like Maximum Empirical Likelihood (MEL), that operate within the Empirical Estimating Equations (E3) approach to estimation and inference are challenged by the Empty Set Problem (ESP). We propose to return from E3 back to the Estimating Equations, and to use the Maximum Likelihood method. In the discrete case the Maximum Likelihood with Estimating Equations (MLEE) method avoids ESP. In the continuous case, how to make ML-EE operational is an open question. Instead of it, we propose a Patched Empirical Likelihood, and demonstrate that it avoids ESP. The methods enjoy, in general, the same asymptotic properties as MEL.

In the Empirical Estimating Equations (E^3) approach to estimation and inference estimating equations are replaced by their data-dependent empirical counterparts. It is odd but with E^3 there are models where the E^3-based estimator does not exist for some data set, and does exist for others. This depends on whether or not a set of data-supported probability mass functions that satisfy the empirical estimating equations is empty for the data set. In a finite sample context, this unnoted feature invalidates methods of estimation and inference, such as the Maximum Empirical Likelihood, that operate within E^3. The empty set problem of E^3 is illustrated by several examples and possible remedies are discussed.

In 1881, Newcomb conjectured that the first significant digits (FSDs) of numbers in statistical tables would follow a logarithmic distribution with the digit “1” occurring most often. However, because Newcomb’s proposal was not presented with a theoretical basis, it was not given much attention. Fifty-seven years later, Benford argued for the same principle and showed it was relevant to a large range of data sets, and the logarithmic FSD distribution became known as “Benford’s Law.” In the mid-1940s, Stigler claimed Benford’s Law contained a theoretical inconsistency and supplied an alternative derivation for the distribution of FSDs. In this paper, we examine the theoretical basis of the Stigler distribution and extend his reasoning by incorporating FSD first moment information and information-theoretic methods.

Clinical data serve as a necessary basis for medical decisions. Consequently, the importance of methods that help officials quickly identify human tampering of data cannot be underestimated. In this paper, we suggest Benford’s Law as a basis for objectively identifying the presence of experimenter distortions in the outcome of clinical research data. We test this tool on a clinical data set that contains falsified data and discuss the implications of using this and information-theoretic methods as a basis for identifying data manipulation and fraud.

The Cressie-Read (CR) family of power divergence measures is used to identify a new class of statistical models and estimators for competing explanations of the data in binary choice models. A large flexible class of cumulative distribution functions and associated probability density functions emerge that subsumes the conventional logit model, and forms the basis for a large set of estimation alternatives to traditional logit and probit methods. Asymptotic properties of estimators are identified, and sampling experiments are used to provide a basis for gauging the finite sample performance of the estimators in this new class of statistical models.

Using a large deviations approach, Maximum A-Posteriori Probability (MAP) and Empirical Likelihood (EL) are shown to possess, under misspecification, an exclusive property of Bayesian consistency. Under conditions of consistency, regardless of prior the MAP estimator asymptotically coincides with EL. The consistency property is also studied for sampling processes other than iid.

The Bayesian Sanov Theorem (BST) identifies, under both correct and incorrect specification of infinite dimensional model, the points of concentration of the posterior measure. Utilizing this insight in the context of Polya urn sampling, Bayesian nonparametric consistency is established. Polya BST is also used to provide an extension of Maximum Non-parametric Likelihood and Empirical Likelihood methods to the Polya case.

In this paper we demonstrate, in a parametric Estimating Equations setting, that the Empirical Likelihood (EL) method is an asymptotic instance of the Bayesian non-parametric Maximum-A-Posteriori approach. The resulting probabilistic interpretation and justifcation of EL rests on Bayesian non-parametric consistency in L-divergence.

Criterion choice is such a hard problem in information recovery and in estimation and inference. In the case of inverse problems with noise, can probabilistic laws provide a basis for empirical estimator choice? That is the problem we investigate in this paper. Large Deviations Theory is used to evaluate the choice of estimator in the case of two fundamental situations-problems in modelling data. The probabilistic laws developed demonstrate that each problem has a unique solution-empirical estimator. Whether other members of the empirical estimator family can be associated a particular problem and conditional limit theorem, is an open question.

This paper introduces a new class of estimators based on minimization of the Cressie-Read (CR)power divergence measure for binary choice models, where neither a parameterized distribution nor a parameterization of the mean is specified explicitly in the statistical model. By incorporating sample information in the form of conditional moment conditions and estimating choice probabilities by optimizing a member of the set of divergence measures in the CR family, a new class of nonparametric estimators evolves that requires less a priori model structure than conventional parametric estimators such as probit or logit. Asymptotic properties are derived under general regularity conditions and finite sampling properties are illustrated by Monte Carlo sampling experiments. Except for some special cases in which the general regularity conditions do not hold, the estimators have asymptotic normal distributions, similar to conventional parametric estimators of the binary choice model. The sampling experiments focus on the mean square errors in the choice probability predictions and the probability derivatives with respect to the response variable values. The simulation results suggest that estimators within the CR class are more robust than conventional methods of estimation across varying probability distributions underlying the Bernoulli process. The size and power of test statistics based on the asymptotics of the CR-based estimators exhibit behavior similar to those based on conventional parametric methods. Overall, the new class of nonparametric estimators for the binary response model is a promising and potentially more robust alternative to the arametric methods often used in empirical practice.

This paper considers estimation and inference for the multinomial response model in the case where endogenous variables are arguments of the unknown link function. Semiparametric estimators are proposed that avoid the parametric assumptions underlying the likelihood approach as well as the loss of precision when using nonparametric estimation. A data based shrinkage estimator that seeks an optimal combination of estimators and results in superior risk performance under quadratic loss is also developed.

This paper presents empirical evidence concerning the finite sample performance of conventional and generalized empirical likelihood-type estimators that utilize instruments in the context of linear structural models characterized by endogenous explanatory variables. There are suggestions in the literature that traditional and non-traditional asymptotically efficient estimators based on moment equations may, for the relatively small sample sizes usually encountered in econometric practice, have relatively large biases and/or variances and provide an inadequate basis for estimation and inference. Given this uncertainty we use a range of data sampling processes and Monte Carlo sampling procedures to accumulate finite sample empirical evidence concerning these questions for a family of generalized empirical likelihood-type estimators in comparison to conventional 2SLS and GMM estimators. Solutions to EL-type empirical momentconstrained optimization problems present formidable numerical challenges. We identify effective optimization algorithms for meeting these challenges.

When there is uncertainty concerning the appropriate statistical model to use in representing the data sampling process and corresponding estimators, we consider a basis for optimally combining estimation problems. In the context of the multivariate linear statistical model, we consider a semi-parametric Stein-like (SPSL) estimator, B( ), that shrinks to a random data-dependent vector and, under quadratic loss, has superior performance relative to the conventional least squares estimator. The relationship of the SPSL estimator to the family of Stein estimators is noted and risk dominance extensions between correlated estimators are demonstrated. As an application we consider the problem of a possibly ill-conditioned design matrix and devise a corresponding SPSL estimator. Asymptotic and analytic finite sample risk properties of the estimator are demonstrated. An extensive sampling experiment is used to investigate finite sample performance over a wide range of data sampling processes to illustrate the robustness of the estimator for an array of symmetric and skewed distributions. Bootstrapping procedures are used to develop confidence sets and a basis for inference.

We seek to identify the impact of data measurement error problems in the context of ecological inference applications. We explore the statistical and substantive implications of using inaccurate proxy variables in the estimation and inference process. The focus of our analysis is on applications of ecological inference in cases involving the Voting Rights Act. We demonstrate our findings with a unique data set on racial registration and turnout in Louisiana and South Carolina.

Using a maximum entropy technique, we estimate the market shares of each firm in an industry using the available government summary statistics such as the four-firm concentration ratio (C4) and the Herfindahl-Hirschmann Index (HHI). We show that our technique is very effective in estimating the distribution of market shares in 20 industries. Our results provide support for the recent practice of using HHI rather than C4 as the key explanatory variable in many market power studies, if only one measure is to be used.

Good quality data is paramount for applied economic research. If the data are distorted, corresponding conclusions may be incorrect. We demonstrate how Benford's law, the distribution that first digits of numbers in certain data sets should follow, can be used to test for data abnormalities. We conduct an analysis of nine commonly used data sets and find that much data from developing countries is of poor quality while data from the United States seems to be of better quality. Female and male respondents give data of similar quality. (Contains 10 tables and 10 footnotes.)