Canonical Exponential-Family Mixtures for Latent Class Choice Models: A Unified Closed-Form EM Framework, with the Gaussian–Geometric Case

Latent class choice models (LCCMs) become increasingly complex when class membership depends on covariates of mixed measurement type, since the standard practice of fitting each covariate block with a maximally flexible density (e.g., a negative binomial mixture for count attributes, as in the Gaussian–negative-binomial LCCM of Sajjad et al. typically leaves at least one class-membership parameter—such as a dispersion parameter—without a closed-form estimator. We first establish a precise result: whenever every block of a class-membership mixture is drawn from a canonical (non-curved) exponential family, (i) the class-membership M-step is additively separable across blocks, (ii) each block’s stationary point is characterized by moment matching, (iii) that stationary point is the block’s unique global maximizer, and (iv) an explicit closed-form update is available whenever the block’s mean-to-natural-parameter map has a closed-form inverse—which we verify directly for the Gaussian, Geometric, Poisson, Bernoulli, and Binomial-with-known-trials families used below. Only the class-specific choice-model utilities, which never admit a closed form in any LCCM, require numerical optimization. We then instantiate the framework with the Geometric distribution, chosen for its added structural properties (it is the unique memoryless distribution on the positive integers, and its variance is a deterministic function of its mean, σ2=μ2−μ), and establish the model-specific results needed to use it in practice: the moment-matching M-step, the Fisher information and asymptotic variance of the Geometric parameter, log-concavity of the class-conditional density, identifiability for the two-block Gaussian–Geometric model, and a generalized-EM monotone-ascent guarantee. A Monte Carlo study (R=60 replications per sample size, three times the pilot design) confirms accurate parameter recovery and finite-sample coverage of the closed-form standard error close to nominal (within Monte Carlo error), and an additional overdispersed data-generating scenario quantifies, rather than assumes, the cost of the Geometric mean–variance restriction relative to a negative-binomial alternative. Application to the Greene–Hensher travel-mode-choice dataset (210 travelers, with income and party size as class-membership covariates) , isolates the dataset’s top-coded income bracket as a candidate latent class, and shows this class-3 solution to be a boundary (near-singular-covariance) optimum of the Gaussian mixture likelihood: we quantify this directly, report its sensitivity to a variance floor and to starting values, and correspondingly qualify the model’s out-of-sample discrimination (repeated 70/30 splits give an AUC of 0.85±0.04 versus 0.86±0.03 for the strongest covariate-free benchmark, a small gap, reported descriptively and without a formal significance claim) against a directly fitted Gaussian–negative-binomial competitor, for which the conventional χ2 likelihood-ratio approximation provides no strong evidence in favour of the additional dispersion parameters.