Mixture distributions as exponential families

In many real-world applications, datasets contain a mixture of different types of information. For example, a dataset may combine continuous and discrete components or contain a large proportion of zeros alongside a distribution of positive values. Such mixed structures are common in fields such as engineering, ecology, biostatistics, finance, and reliability studies. In such situations, classical single-component distributions may not describe the data well, and mixture distributions provide a useful alternative. One phenomenon that has received considerable attention is zero-inflation, where the frequency of zeros in the data exceeds what is expected under a chosen model distribution. In case of a continuous distribution, the probability of zeros is zero, but zeros may be observed in a real-world application. To capture this feature, zero-inflated models were developed. These are typically formulated as two-component mixtures: one component is a degenerate distribution at zero, and the other is a non-degenerate probability distribution. If the latter belongs to an exponential family, then, with suitable parameterizations, the resulting zero-inflated model can form an exponential family, too. This aspect has received only limited attention in the existing literature. Exponential families and, more restrictively, regular exponential families exhibit particular favorable properties in statistical analysis. When a regular exponential family is present, a canonical representation leads directly to minimal sufficient and complete statistics and smooth mean value and cumulant functions, which facilitate the derivation of uniformly minimum-variance unbiased estimators, the characterization of the existence and uniqueness of maximum likelihood estimators, and the study of their asymptotic behavior. Moreover, a regular exponential family allows for the construction of optimal tests and optimal confidence intervals in a natural and elegant way. This doctoral thesis studies a broad class of two-component mixture models which form regular exponential families. The starting point is a formulation with functionally unrelated parameters; under appropriate assumptions, this formulation is shown to yield a regular exponential family. By imposing functional relationships among the parameters within a single component or across components, or by fixing some parameters, regular exponential families of lower orders are derived. The developed construction approaches apply not only to zero-inflated and zero-altered models, but also to a much wider range of two-component mixtures, including models with multiple inflation or alteration points. Statistical inference within the two-component mixtures as regular exponential families is then studied. Maximum likelihood estimation is analyzed, with particular emphasis on conditions ensuring existence and uniqueness. In parallel, uniformly minimum-variance unbiased estimators are derived, including explicit expressions for the mixture parameter and the distribution mean under appropriate assumptions. Several testing problems are examined in detail. Among them is the assessment of whether a two-component mixture is required in place of a single-component model. Four scenarios are considered in this context. The first concerns models in which certain mixture distributions can be viewed as the untruncated form of one of their components, with the testing problem being whether the simpler untruncated model should be used instead. The second handles discrete models with multiple-point alterations, where a likelihood ratio test is developed; within the same framework, a uniformly most powerful unbiased test is established for testing alteration at a single point. The third scenario focuses on discrete models with one-point alteration, for which uniformly most powerful unbiased tests are derived to assess alteration, inflation, or deflation at that point. The fourth scenario considers mixtures of a discrete and a continuous distribution, and a uniformly most powerful test is constructed for this case. An important advantage of regular exponential families is that the existence of most powerful tests can often be stated directly. Both uniformly most powerful and uniformly most powerful unbiased tests for selected model parameters are derived, taking into account scenarios with and without functional restrictions on parameters. Utilizing the duality between confidence intervals and hypothesis tests, corresponding most accurate confidence intervals are obtained, and the characterization of interval bounds is discussed. The theoretical framework is illustrated through three applications: a zero-altered Poisson model, a zero-to-m-altered Poisson model, and a geometric-exponential mixture model. Extensive simulation studies demonstrate the practical performance of the proposed inference methods. Finally, the scope of the analysis is broadened to include a class of two-component mixtures that are exponential families but not necessarily regular. Particular attention is given to the full-rank but non-regular scenario, in which favorable inferential properties, similar to those in the regular case, are present and demonstrated using three concrete examples.