Density-based minimum divergence procedures represent popular techniques in parametric statistical inference. They combine strong robustness properties with high (sometimes full) asymptotic efficiency. Among density-based minimum distance procedures, the methods based on the Bregman-divergence have the attractive property that the empirical formulation of the divergence does not require the use of any non-parametric smoothing technique such as kernel density estimation. The methods based on the density power divergence (DPD) represent the current standard in this area of research. In this paper, we will present a more generalized divergence that subsumes the DPD as a special case and produces several new options providing better compromises between robustness and efficiency.
In this note we prove the dual representation formula of the divergence between two distributions in a parametric model. Resulting estimators for the divergence as for the parameter are derived. These estimators do not make use of any grouping nor smoothing. It is proved that all differentiable divergences induce the same estimator of the parameter on any regular exponential family, which is nothing else but the MLE.
This article is devoted to the study of overlap measures of densities of two exponential populations. Various Overlapping Coefficients, namely: Matusita's measure $ρ$, Morisita's measure $λ$ and Weitzman's measure $Δ$. A new overlap measure $Λ$ based on Kullback-Leibler measure is proposed. The invariance property and a method of statistical inference of these coefficients also are presented. Taylor series approximation are used to construct confidence intervals for the overlap measures. The bias and mean square error properties of the estimators are studied through a simulation study.
M-estimators offer simple robust alternatives to the maximum likelihood estimator. Much of the robustness literature, however, has focused on the problems of location, location-scale and regression estimation rather than on estimation of general parameters. The density power divergence (DPD) and the logarithmic density power divergence (LDPD) measures provide two classes of competitive M-estimators (obtained from divergences) in general parametric models which contain the MLE as a special case. In each of these families, the robustness of the estimator is achieved through a density power down-…
While robust divergence such as density power divergence and $γ$-divergence is helpful for robust statistical inference in the presence of outliers, the tuning parameter that controls the degree of robustness is chosen in a rule-of-thumb, which may lead to an inefficient inference. We here propose a selection criterion based on an asymptotic approximation of the Hyvarinen score applied to an unnormalized model defined by robust divergence. The proposed selection criterion only requires first and second-order partial derivatives of an assumed density function with respect to observations, which…
Minimum divergence methods are popular tools in a variety of statistical applications. We consider tubular model adequacy tests, and demonstrate that the new divergences that are generated in the process are very useful in robust statistical inference. In particular we show that the family of $S$-divergences can be alternatively developed using the tubular model adequacy tests; a further application of the paradigm generates a larger superfamily of divergences. We describe the properties of this larger class and its potential applications in robust inference. Along the way, the failure of the …
In information geometry, generalized exponential families and statistical manifolds with curvature are under active investigation in recent years. In this paper we consider the statistical manifold induced by a logarithmic $L^{(α)}$-divergence which generalizes the Bregman divergence. It is known that such a manifold is dually projectively flat with constant negative sectional curvature, and is closely related to the $\mathcal{F}^{(α)}$-family, a generalized exponential family introduced by the second author. Our main result constructs a dual foliation of the statistical manifold, i.e., an ort…
Statistical inference based on divergence measures have a long history. Recently, Maji, Ghosh and Basu (2014) have introduced a general family of divergences called the logarithmic super divergence (LSD) family. This family acts as a superfamily for both of the logarithmic power divergence (LPD) family (eg. Renyi, 1961) and the logarithmic density power divergence (LDPD)family introduced by Jones et al. (2001). In this paper we describe the asymptotic properties of the inference procedures resulting from this divergence in discrete models. The properties are well supported by real data example…
This paper introduces a new superfamily of divergences that is similar in spirit to the S-divergence family introduced by Ghosh et al. (2013). This new family serves as an umbrella that contains the logarithmic power divergence family (Renyi, 1961; Maji, Chakraborty and Basu 2014) and the logarithmic density power divergence family (Jones et al., 2001) as special cases. Various properties of this new family and the corresponding minimum distance procedures are discussed with particular emphasis on the robustness issue; these properties are demonstrated through simulation studies. In particular…
Recently Batsidis \textit{et al.} (2011) have presented a new procedure based on divergence measures for testing the hypothesis of the existence of a change point in exponential populations. A simulation study was carried out, in this paper, using the asymptotic critical points obtained from the asymptotic distribution of the new test statistics introduced there. The main purpose of this paper is to study the behavior of the test statistics introduced in the cited paper of Batsidis \textit{et al.} (2011), using simulated critical points.
Robust inference methods of diagnostic test accuracy meta-analysis for influential outlying studies via density power divergence
The Graduate Institute for Advanced Studies, The Graduate University for Advanced Studies（高级研究研究生院，高级研究院）；Human Biology Creation Hub, Deep Human Biology Learning, Eisai Co. Ltd.（人类生物学创造中心，深层人类生物学学习，卫材株式会社）；Department of Interdisciplinary Statistical Mathematics, The Institute of Statistical Mathematics（交叉统计数学系，统计数学研究所）；Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center—University of Freiburg（医学生物统计学与统计研究所，医学院及医学中心——弗赖堡大学）
Helsinki Institute of Information Technology (HIIT)（赫尔辛基信息技术研究所）；University of Helsinki（赫尔辛基大学）；University of Oslo（奥斯陆大学）；Wellcome Sanger Institute（惠康桑格研究所）；KTH Royal Institute of Technology（瑞典皇家理工学院）
Likelihood-free inference for simulator-based statistical models has recently attracted a surge of interest, both in the machine learning and statistics communities. The primary focus of these research fields has been to approximate the posterior distribution of model parameters, either by various types of Monte Carlo sampling algorithms or deep neural network -based surrogate models. Frequentist inference for simulator-based models has been given much less attention to date, despite that it would be particularly amenable to applications with big data where implicit asymptotic approximation of…
In this paper, we develop robust estimators and tests for one-shot device testing under proportional hazards assumption based on divergence measures. Through a detailed Monte Carlo simulation study and a numerical example, the developed inferential procedures are shown to be more robust than the classical procedures, based on maximum likelihood estimators.
Preserving the robustness of the procedure has, at the present time, become almost a default requirement for statistical data analysis. Since efficiency at the model and robustness under misspecification of the model are often in conflict, it is important to choose such inference procedures which provide the best compromise between these two concepts. Some minimum Bregman divergence estimators and related tests of hypothesis seem to be able to do well in this respect, with the procedures based on the density power divergence providing the existing standard. In this paper we propose a new famil…
Robust inference based on the minimization of statistical divergences has proved to be a useful alternative to the classical maximum likelihood based techniques. Recently Ghosh et al. (2013) proposed a general class of divergence measures for robust statistical inference, named the S-Divergence Family. Ghosh (2014) discussed its asymptotic properties for the discrete model of densities. In the present paper, we develop the asymptotic properties of the proposed minimum S-Divergence estimators under continuous models. Here we use the Basu-Lindsay approach (1994) of smoothing the model densities …
In this paper we are interested in testing whether there are any signals hidden in high dimensional noise data. Therefore we study the family of goodness-of-fit tests based on $Φ$-divergences including the test of Berk and Jones as well as Tukey's higher criticism test. The optimality of this family is already known for the heterogeneous normal mixture model. We now present a technique to transfer this optimality to more general models. For illustration we apply our results to dense signal and sparse signal models including the exponential-$χ^2$ mixture model and general exponential families a…
We study the problem of nonparametric estimation of density functions with a product form on the domain $\triangle=\{( x_1, \ldots, x_d)\in \mathbb{R}^d, 0\leq x_1\leq \dots \leq x_d \leq 1\}$. Such densities appear in the random truncation model as the joint density function of observations. They are also obtained as maximum entropy distributions of order statistics with given marginals. We propose an estimation method based on the approximation of the logarithm of the density by a carefully chosen family of basis functions. We show that the method achieves a fast convergence rate in probabil…
In the field of statistics, many kind of divergence functions have been studied as an amount which measures the discrepancy between two probability distributions. In the differential geometrical approach in statistics (information geometry), dually flat spaces play a key role. In a dually flat space, there exist dual affine coordinate systems and strictly convex functions called potential and a canonical divergence is naturally introduced as a function of the affine coordinates and potentials. The canonical divergence satisfies a relational expression called triangular relation. This can be re…
Divergence measures have a long association with statistical inference, machine learning and information theory. The density power divergence and related measures have produced many useful (and popular) statistical procedures, which provide a good balance between model efficiency on one hand and outlier stability or robustness on the other. The logarithmic density power divergence, a particular logarithmic transform of the density power divergence, has also been very successful in producing efficient and stable inference procedures; in addition it has also led to significant demonstrated appli…
A density ratio is defined by the ratio of two probability densities. We study the inference problem of density ratios and apply a semi-parametric density-ratio estimator to the two-sample homogeneity test. In the proposed test procedure, the f-divergence between two probability densities is estimated using a density-ratio estimator. The f-divergence estimator is then exploited for the two-sample homogeneity test. We derive the optimal estimator of f-divergence in the sense of the asymptotic variance, and then investigate the relation between the proposed test procedure and the existing score …
Measuring divergence between two distributions is essential in machine learning and statistics and has various applications including binary classification, change point detection, and two-sample test. Furthermore, in the era of big data, designing divergence measure that is interpretable and can handle high-dimensional and complex data becomes extremely important. In the paper, we propose a post selection inference (PSI) framework for divergence measure, which can select a set of statistically significant features that discriminate two distributions. Specifically, we employ an additive varian…
Minimum divergence procedures based on the density power divergence and the logarithmic density power divergence have been extremely popular and successful in generating inference procedures which combine a high degree of model efficiency with strong outlier stability. Such procedures are always preferable in practical situations over procedures which achieve their robustness at a major cost of efficiency or are highly efficient but have poor robustness properties. The density power divergence (DPD) family of Basu et al.(1998) and the logarithmic density power divergence (LDPD) family of Jones…
We consider on-line density estimation with a parameterized density from the exponential family. The on-line algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss which is the negative log-likelihood of the example w.r.t. the past parameter of the algorithm. An off-line algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the on-line algorithm over the total loss of the off-line algorithm. These relative loss boun…
In this paper a new family of minimum divergence estimators based on the Bregman divergence is proposed, where the defining convex function has an exponential nature. These estimators avoid the necessity of using an intermediate kernel density and many of them also have strong robustness properties. It is further demonstrated that the proposed approach can be extended to construct a class of generalized estimating equations, where the pool of the resultant estimators encompass a large variety of minimum divergence estimators and range from highly robust to fully efficient based on the choice o…
In any parametric inference problem, the robustness of the procedure is a real concern. A procedure which retains a high degree of efficiency under the model and simultaneously provides stable inference under data contamination is preferable in any practical situation over another procedure which achieves its efficiency at the cost of robustness or vice versa. The density power divergence family of Basu et al. (1998) provides a flexible class of divergences where the adjustment between efficiency and robustness is controlled by a single parameter $β$. In this paper we consider general tests of…
In this paper a new family of minimum divergence estimators based on the Bregman divergence is proposed. The popular density power divergence (DPD) class of estimators is a sub-class of Bregman divergences. We propose and study a new sub-class of Bregman divergences called the exponentially weighted divergence (EWD). Like the minimum DPD estimator, the minimum EWD estimator is recognised as an M-estimator. This characterisation is useful while discussing the asymptotic behaviour as well as the robustness properties of this class of estimators. Performances of the two classes are compared -- bo…
We introduce a methodology for robust Bayesian estimation with robust divergence (e.g., density power divergence or γ-divergence), indexed by a single tuning parameter. It is well known that the posterior density induced by robust divergence gives highly robust estimators against outliers if the tuning parameter is appropriately and carefully chosen. In a Bayesian framework, one way to find the optimal tuning parameter would be using evidence (marginal likelihood). However, we numerically illustrate that evidence induced by the density power divergence does not work to select the optimal tunin…
Contrastive divergence (CD) is a promising method of inference in high dimensional distributions with intractable normalizing constants, however, the theoretical foundations justifying its use are somewhat shaky. This document proposes a framework for understanding CD inference, how/when it works, and provides multiple justifications for the CD moment conditions, including framing them as a variational approximation. Algorithms for performing inference are discussed and are applied to social network data using an exponential-family random graph models (ERGM). The framework also provides guidan…
A stream of algorithmic advances has steadily increased the popularity of the Bayesian approach as an inference paradigm, both from the theoretical and applied perspective. Even with apparent successes in numerous application fields, a rising concern is the robustness of Bayesian inference in the presence of model misspecification, which may lead to undesirable extreme behavior of the posterior distributions for large sample sizes. Generalized belief updating with a loss function represents a central principle to making Bayesian inference more robust and less vulnerable to deviations from the …
The extreme value theory is very popular in applied sciences including Finance, economics, hydrology and many other disciplines. In univariate extreme value theory, we model the data by a suitable distribution from the general max-domain of attraction (MAD) characterized by its tail index; there are three broad classes of tails -- the Pareto type, the Weibull type and the Gumbel type. The simplest and most common estimator of the tail index is the Hill estimator that works only for Pareto type tails and has a high bias; it is also highly non-robust in presence of outliers with respect to the a…
In this Note we introduce a new methodology for Bayesian inference through the use of $ϕ$-divergences and the duality technique. The asymptotic laws of the estimates are established.
Harvard University（哈佛大学）；University of British Columbia（英属哥伦比亚大学）；University of Luxembourg（卢森堡大学）；MIT（麻省理工学院）
Gaussian processes (GPs) offer a flexible class of priors for nonparametric Bayesian regression, but popular GP posterior inference methods are typically prohibitively slow or lack desirable finite-data guarantees on quality. We develop an approach to scalable approximate GP regression with finite-data guarantees on the accuracy of pointwise posterior mean and variance estimates. Our main contribution is a novel objective for approximate inference in the nonparametric setting: the preconditioned Fisher (pF) divergence. We show that unlike the Kullback--Leibler divergence (used in variational i…
We consider the problem of robust inference under the generalized linear model (GLM) with stochastic covariates. We derive the properties of the minimum density power divergence estimator of the parameters in GLM with random design and use this estimator to propose robust Wald-type tests for testing any general composite null hypothesis about the GLM. The asymptotic and robustness properties of the proposed tests are also examined for the GLM with random design. Application of the proposed robust inference procedures to the popular Poisson regression model for analyzing count data is discussed…
In a first part, we present a mathematical analysis of a general methodology of a probabilistic learning inference that allows for estimating a posterior probability model for a stochastic boundary value problem from a prior probability model. The given targets are statistical moments for which the underlying realizations are not available. Under these conditions, the Kullback-Leibler divergence minimum principle is used for estimating the posterior probability measure. A statistical surrogate model of the implicit mapping, which represents the constraints, is introduced. The MCMC generator an…
Robust inference based on the minimization of statistical divergences has proved to be a useful alternative to the classical techniques based on maximum likelihood and related methods. Recently Ghosh et al. (2013) proposed a general class of divergence measures, namely the S-Divergence Family and discussed its usefulness in robust parametric estimation through some numerical illustrations. In this present paper, we develop the asymptotic properties of the proposed minimum S-Divergence estimators under discrete models.
