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41
Stratified exponential families: Graphical models and model selection
 ANNALS OF STATISTICS
, 2001
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Marginal models for categorical data
, 1997
"... Statistical models defined by imposing restrictions on marginal distributions of contingency tables have received considerable attention recently. This paper introduces a general definition of marginal loglinear parameters and describes conditions for a marginal loglinear parameter to be a smoot ..."
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Cited by 30 (8 self)
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Statistical models defined by imposing restrictions on marginal distributions of contingency tables have received considerable attention recently. This paper introduces a general definition of marginal loglinear parameters and describes conditions for a marginal loglinear parameter to be a smooth parameterization of the distribution, and to be variation independent. Statistical models defined by imposing affine restrictions on the marginal loglinear parameters are investigated. These models generalize ordinary loglinear and multivariate logistic models. Sufficient conditions for a logaffine marginal model to be nonempty, and to be a curved exponential family are given. Standard large sample theory is shown to apply to maximum likelihood estimation of logaffine marginal models for a variety of sampling procedures.
P.: 2006, On the simple economics of advertising, marketing, and product design
 American Economic Review
"... We propose a framework for analyzing transformations of the demand facing a monopolist. Our approach is based on the observation that such transformations frequently stem from changes in the dispersion of consumers ’ valuations, which lead to rotations of the demand curve. In a wide variety of setti ..."
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Cited by 15 (0 self)
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We propose a framework for analyzing transformations of the demand facing a monopolist. Our approach is based on the observation that such transformations frequently stem from changes in the dispersion of consumers ’ valuations, which lead to rotations of the demand curve. In a wide variety of settings, profits are a Ushaped function of dispersion. The monopolist will adopt a massmarket posture when dispersion is low, and a niche posture when dispersion is high. We investigate numerous applications of our framework, including product design and development; advertising, marketing and sales advice; and the construction of qualitydifferentiated product lines. We also suggest a new taxonomy of advertising, distinguishing between hype, which shifts demand, and real information, which rotates demand. 1. Shaping Consumer Demand The shape of the demand curve for a product is fundamental to the behavior and profitability of a supplier. Demand is influenced by many factors, some of which are exogenous, such as consumers ’ incomes and preferences, and others of which are endogenous. For instance, advertising, marketing, and product design are all activities that shape demand. Here we
Aster Models for Life History Analysis By
"... We present a new class of statistical models designed for life history analysis of plants and animals. They allow joint analysis of data on survival and reproduction over multiple years, allow for variables having different statistical distributions, and correctly account for the dependence of varia ..."
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Cited by 11 (10 self)
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We present a new class of statistical models designed for life history analysis of plants and animals. They allow joint analysis of data on survival and reproduction over multiple years, allow for variables having different statistical distributions, and correctly account for the dependence of variables on earlier variables (for example, that a dead individual stays dead and cannot reproduce). We illustrate their utility with an analysis of data taken from an experimental study of Echinacea angustifolia sampled from remnant prarie populations in western Minnesota. Statistically, they are graphical models with some resemblance to generalized linear models and survival analysis. They have directed acyclic graphs with nodes having no more than one parent. The conditional distribution of each node given the parent is a oneparameter exponential family with the parent variable the sample size. The model may be heterogeneous, each node having a different exponential family. We show that the joint distribution is a flat exponential family and derive its canonical parameters, Fisher information, and other properties. These models are implemented in an R package ‘aster ’ available from CRAN.
Exponential Families
, 1990
"... General methods for obtaining maximum likelihood estimates in exponential families are demonstrated using a constrained autologistic model for estimating relatedness from DNA fingerprint data. The novel features are the use of constrained optimization and two new algorithms for maximum likelihood es ..."
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Cited by 10 (4 self)
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General methods for obtaining maximum likelihood estimates in exponential families are demonstrated using a constrained autologistic model for estimating relatedness from DNA fingerprint data. The novel features are the use of constrained optimization and two new algorithms for maximum likelihood estimation. The first, the "phase I " algorithm determines the support of the MLE in the closure of the exponential family (a distribution in the family conditioned on a face of the convex support of the natural statistic) when the MLE does not exist in the traditional sense (a point in the natural parameter space). The second, the maximum Monte Carlo likelihood algorithm uses the Metropolis algorithm or the Gibbs sampler to obtain estimates when exact calculation of the likelihood is not possible. Separate papers on each algorithm accompany
Tail Exactness of Multivariate Saddlepoint Approximations
"... We consider a logconcave density f in R m satisfying certain weak conditions, particularly on the Hessian matrix of \Gamma log f . For such a density, we prove tail exactness of the multivariate saddlepoint approximation. The proof is based on a local limit theorem for the exponential family gene ..."
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Cited by 8 (4 self)
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We consider a logconcave density f in R m satisfying certain weak conditions, particularly on the Hessian matrix of \Gamma log f . For such a density, we prove tail exactness of the multivariate saddlepoint approximation. The proof is based on a local limit theorem for the exponential family generated by f . However, the result refers not to asymptotic behaviour under repeated sampling, but to a limiting property at the boundary of the domain of f . Our approach does not apply any complex analysis, but relies totally on convex analysis and exponential models arguments. Running headline: Multivariate saddlepoint approximations. AMS 1991 Subject Classifications: primary: 62E17, 62E20, 62F11, 62H10 secondary: 60E10, 60F05 Keywords: asymptotic normality, convex analysis, exponential models, local limit theorem, moment generating function, Legendre transform, saddlepoint approximation. Department of Mathematical Sciences, Aarhus University, DK8000 Aarhus C, Denmark, email: atsoebn@mi...
Likelihood Inference in Exponential Families and Directions of Recession
"... url: www.stat.umn.edu/geyer/gdor Abstract: When in a full exponential family the maximum likelihood estimate (MLE) does not exist, the MLE may exist in the BarndorffNielsen completion of the family. We propose a practical algorithm for finding the MLE in the completion based on repeated linear prog ..."
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Cited by 7 (3 self)
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url: www.stat.umn.edu/geyer/gdor Abstract: When in a full exponential family the maximum likelihood estimate (MLE) does not exist, the MLE may exist in the BarndorffNielsen completion of the family. We propose a practical algorithm for finding the MLE in the completion based on repeated linear programming using the R contributed package rcdd and illustrate it with two generalized linear model examples. When the MLE for the null hypothesis lies in the completion, likelihood ratio tests of model comparison are almost unchanged from the usual case. Only the degrees of freedom need to be adjusted. When the MLE lies in the completion, confidence intervals are changed much more from the usual case. The MLE of the natural parameter can be thought of as having gone to infinity in a certain direction, which we call a generic direction of recession. We propose a new onesided confidence interval which says how close to infinity the natural parameter may be. This maps to onesided confidence intervals for mean values showing how close to the
A Note on Multivariate Logistic Models for Contingency Tables
 Austral. J. Statist
, 1997
"... Loglinear models are a widely accepted tool for modeling discrete data given in a contingency table. Although their parameters reflect the interaction structure in the joint distribution of all variables, they do not give information about structures appearing in the margins of the table. This is i ..."
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Loglinear models are a widely accepted tool for modeling discrete data given in a contingency table. Although their parameters reflect the interaction structure in the joint distribution of all variables, they do not give information about structures appearing in the margins of the table. This is in contrast to multivariate logistic parameters recently introduced by Glonek & McCullagh (1995). They have as parameters the highest order log odds ratios derived from the joint table and from each marginal table. The link between the cell probabilities and the multivariate logistic parameters is given in Glonek & McCullagh in an algebraic fashion. In this paper we focus on this link, showing that it is derived by general parameter transformations in exponential families. In particular, the connection between the natural, the expectation and the mixed parameterization in exponential families (BarndorffNielsen, 1978) is used. This also yields the derivatives of the likelihood equation and shows properties of the Fisher matrix. Further emphasis is paid to the analysis of independence hypotheses in margins of a contingency table.
A recursive modelreduction method for approximate inference in Gaussian Markov random fields
 IEEE TRANS. IMAG. PROC
, 2008
"... This paper presents recursive cavity modeling—a principled, tractable approach to approximate, nearoptimal inference for large Gauss–Markov random fields. The main idea is to subdivide the random field into smaller subfields, constructing cavity models which approximate these subfields. Each cavit ..."
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Cited by 4 (3 self)
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This paper presents recursive cavity modeling—a principled, tractable approach to approximate, nearoptimal inference for large Gauss–Markov random fields. The main idea is to subdivide the random field into smaller subfields, constructing cavity models which approximate these subfields. Each cavity model is a concise, yet faithful, model for the surface of one subfield sufficient for nearoptimal inference in adjacent subfields. This basic idea leads to a treestructured algorithm which recursively builds a hierarchy of cavity models during an “upward pass ” and then builds a complementary set of blanket models during a reverse “downward pass. ” The marginal statistics of individual variables can then be approximated using their blanket models. Model thinning plays an important role, allowing us to develop thinned cavity and blanket models thereby providing tractable approximate inference. We develop a maximumentropy approach that exploits certain tractable representations of Fisher information on thin chordal graphs. Given the resulting set of thinned cavity models, we also develop a fast preconditioner, which provides a simple iterative method to compute optimal estimates. Thus, our overall approach combines recursive inference, variational learning and iterative estimation. We demonstrate the accuracy and scalability of this approach in several challenging, largescale remote sensing problems.
Exponential Family Inference for Diffusion Models
"... We consider ergodic diffusion processes for which the class of invariant measures is an exponential family, and study inference based on the class of invariant probability measures when the diffusion has been observed at discrete time points. When the drift depends linearly on the parameters, the in ..."
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Cited by 3 (2 self)
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We consider ergodic diffusion processes for which the class of invariant measures is an exponential family, and study inference based on the class of invariant probability measures when the diffusion has been observed at discrete time points. When the drift depends linearly on the parameters, the invariant measures form an exponential family. It is investigated how the usual exponential family inference, which can be done by means of standard statistical computer packages, works when the observations are from a diffusion process. In particular, the limit distributions of estimators and test statistics are derived. As an example, we consider classes of diffusions with generalized inverse Gaussian marginals. A particular instance is the wellknown CoxIngersollRoss model from mathematical finance. Key Words: Asymptotic Normality; Consistency; CoxIngersollRoss model; Discrete time observation; Inference for Diffusion Processes; Estimating Functions; Exponential families; Generalized in...