Convex order monotonicity of conditional mean predictors in latent mixture models
(2026) , 32 pages
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- Abstract
- This paper studies the monotonicity, in the convex order, of conditional mean predictors based on aggregate statistics in latent mixture models. For partial sums Sj= X1 +···+ Xj, we give general conditions under which E[Xi |Sj] is dominated by E[Xi |Sj−1] in the convex order, i < j. Such an ordering means that enlarging the conditioning sum reduces the dispersion, and hence the informativeness, of the corresponding conditional mean predictor. The first set of results is based on a projection identity for conditional expectations and includes the classical case where an independent noise term is added to the signal. We thenshow that the same convex-order reduction follows from a Markov dependence structure. This provides a tractable route for latent mixture models: when the variables are conditionally independent given a common factor and belong to natural exponential families with a common canonical parameter, the sufficiency of partial sums yields the required Markov property. A complementary analysis characterizes situations in which the conditional mean predictor is affine or proportional to the conditioning sum. In particular, for convolution semigroups associated with infinitely divisible distributions, we identify the latent scaling structures leading to proportional predictors. The results apply to a range of standard models, including Poisson, binomial, negative binomial, Gamma, normal, inverse Gaussian, Tweedie, compound Poisson and tempered stable distributions.
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Dendoncker, V., Denuit, M., & Robert, C. (2026). Convex order monotonicity of conditional mean predictors in latent mixture models (LIDAM Discussion Paper ISBA 2026/20). https://hdl.handle.net/2078.5/276679