A note on stochastic domination for discrete exponential families

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

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KR20 and KR21 for Some Nondichotomous Data (It’s Not Just Cronbach’s Alpha)

Educational and Psychological Measurement ◽

10.1177/0013164421992535 ◽

2021 ◽

pp. 001316442199253

Author(s):

Robert C. Foster

Keyword(s):

Exponential Family ◽

Exponential Families ◽

Natural Exponential Family ◽

Cronbach’S Alpha ◽

Natural Exponential Families ◽

Cronbach's Alpha ◽

Generalized Framework ◽

The Common ◽

Poisson Data ◽

Poisson Count

This article presents some equivalent forms of the common Kuder–Richardson Formula 21 and 20 estimators for nondichotomous data belonging to certain other exponential families, such as Poisson count data, exponential data, or geometric counts of trials until failure. Using the generalized framework of Foster (2020), an equation for the reliability for a subset of the natural exponential family have quadratic variance function is derived for known population parameters, and both formulas are shown to be different plug-in estimators of this quantity. The equivalent Kuder–Richardson Formulas 20 and 21 are given for six different natural exponential families, and these match earlier derivations in the case of binomial and Poisson data. Simulations show performance exceeding that of Cronbach’s alpha in terms of root mean square error when the formula matching the correct exponential family is used, and a discussion of Jensen’s inequality suggests explanations for peculiarities of the bias and standard error of the simulations across the different exponential families.

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‘Exponential mixtures and quadratic exponential families’

Biometrika ◽

10.1093/biomet/83.1.248 ◽

1996 ◽

Vol 83 (1) ◽

pp. 248-248

Author(s):

PETER MCCULLAGH

Keyword(s):

Exponential Families

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EXPRESS: Overcoming the Cold Start Problem of CRM using a Probabilistic Machine Learning Approach

Journal of Marketing Research ◽

10.1177/00222437211032938 ◽

2021 ◽

pp. 002224372110329

Author(s):

Nicolas Padilla ◽

Eva Ascarza

Keyword(s):

Machine Learning ◽

Cold Start ◽

Customer Relationship ◽

Exponential Families ◽

Modeling Framework ◽

Wide Range ◽

The Moment ◽

Probabilistic Machine Learning ◽

Marketing Actions ◽

Cold Start Problem

The success of Customer Relationship Management (CRM) programs ultimately depends on the firm's ability to identify and leverage differences across customers — a very diffcult task when firms attempt to manage new customers, for whom only the first purchase has been observed. For those customers, the lack of repeated observations poses a structural challenge to inferring unobserved differences across them. This is what we call the “cold start” problem of CRM, whereby companies have difficulties leveraging existing data when they attempt to make inferences about customers at the beginning of their relationship. We propose a solution to the cold start problem by developing a probabilistic machine learning modeling framework that leverages the information collected at the moment of acquisition. The main aspect of the model is that it exibly captures latent dimensions that govern the behaviors observed at acquisition as well as future propensities to buy and to respond to marketing actions using deep exponential families. The model can be integrated with a variety of demand specifications and is exible enough to capture a wide range of heterogeneity structures. We validate our approach in a retail context and empirically demonstrate the model's ability at identifying high-value customers as well as those most sensitive to marketing actions, right after their first purchase.

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