Inference for the diffusion branching process

1975 ◽  
Vol 12 (3) ◽  
pp. 588-594 ◽  
Author(s):  
B. M. Brown ◽  
J. I. Hewitt
Keyword(s):  
Brownian Motion ◽  
Branching Process ◽  
Exponential Family ◽  
Minimum Variance ◽  
Sufficient Statistics ◽  
Sufficient Statistic ◽  
Unbiased Estimators ◽  
Standard Problem

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we are left with a complete sufficient statistic whose distribution belongs to an exponential family, and can therefore derive minimum variance unbiased estimators, etc.

Inference for the diffusion branching process

1975 ◽  
Vol 12 (03) ◽  
pp. 588-594 ◽  
Author(s):  
B. M. Brown ◽  
J. I. Hewitt
Keyword(s):  
Brownian Motion ◽  
Branching Process ◽  
Exponential Family ◽  
Minimum Variance ◽  
Sufficient Statistics ◽  
Sufficient Statistic ◽  
Unbiased Estimators ◽  
Standard Problem

For the diffusion branching process, we consider a method of inference that is essentially sequential in nature. The method allows us to simplify the natural sufficient statistics involved, and we are able to get their distributions quite easily by translating our problem into a standard problem in Brownian motion. Under certain circumstances, we are left with a complete sufficient statistic whose distribution belongs to an exponential family, and can therefore derive minimum variance unbiased estimators, etc.

scholarly journals Complete and sufficient statistics and perfect families in orthogonal and error orthogonal normal models

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Aníbal Areia ◽  
Francisco Carvalho ◽  
João T. Mexia
Keyword(s):  
Jordan Algebra ◽  
Algebraic Structure ◽  
Minimum Variance ◽  
Jordan Algebras ◽  
Symmetric Matrices ◽  
Sufficient Statistics ◽  
Unbiased Estimators

AbstractWe will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters. Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters.

Curved Exponential Models in Econometrics

1997 ◽  
Vol 13 (6) ◽  
pp. 771-790 ◽  
Author(s):  
Kees Jan van Garderen
Keyword(s):  
Structural Equation ◽  
Equation Model ◽  
Econometric Models ◽  
Demand Systems ◽  
Likelihood Ratios ◽  
Sufficient Statistics ◽  
Sufficient Statistic ◽  
Exponential Models ◽  
Minimal Sufficient Statistic ◽  
The Difference

Curved exponential models have the property that the dimension of the minimal sufficient statistic is larger than the number of parameters in the model. Many econometric models share this feature. The first part of the paper shows that, in fact, econometric models with this property are necessarily curved exponential. A method for constructing an explicit set of minimal sufficient statistics, based on partial scores and likelihood ratios, is given. The difference in dimension between parameterand statistic and the curvature of these models have important consequences for inference. It is not the purpose of this paper to contribute significantly to the theory of curved exponential models, other than to show that the theory applies to many econometric models and to highlight some multivariate aspects. Using the methods developed in the first part, we show that demand systems, the single structural equation model, the seemingly unrelated regressions, and autoregressive models are all curved exponential models.

scholarly journals Exponential-family Random Graph Models for Rank-order Relational Data

2017 ◽  
Vol 47 (1) ◽  
pp. 68-112 ◽  
Author(s):  
Pavel N. Krivitsky ◽  
Carter T. Butts
Keyword(s):  
General Framework ◽  
Exponential Family ◽  
Rank Order ◽  
Relative Volume ◽  
Relational Data ◽  
Interpersonal Interaction ◽  
Graph Models ◽  
Cross Sectional ◽  
Sufficient Statistics ◽  
Random Graph Models

Rank-order relational data, in which each actor ranks other actors according to some criterion, often arise from sociometric measurements of judgment or preference. The authors propose a general framework for representing such data, define a class of exponential-family models for rank-order relational structure, and derive sufficient statistics for interdependent ordinal judgments that do not require the assumption of comparability across raters. These statistics allow estimation of effects for a variety of plausible mechanisms governing rank structure, both in a cross-sectional context and evolving over time. The authors apply this framework to model the evolution of liking judgments in an acquaintance process and to model recall of relative volume of interpersonal interaction among members of a technology education program.

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