Recall from Volume I that in the field of statistics we represent important data-related problems and questions in terms of questions about distributions and their parameters. Thus our goal is to use data X ∈ 𝒳 to estimate or draw conclusions about aspects of the probability distribution P of X. The probability distribution P is assumed to belong to a class 𝒫 of distributions called the model. Examining what models are useful for answering data-related questions is an important part of statistics. In Volume I we considered three cases with a focus on the first:

P is a member of a parametric class of distributions {P_θ : θ ∈ Θ}, Θ ⊂ R^d, and our interest is in θ or some vector q(θ).

P is arbitrary except for regularity conditions, such as finite second moments or continuity of the distribution function, and our interest is in functionals ν(P) that may be real valued, vectors, or functions.

Our class of distributions is neither smoothly parametrizable by a Euclidean parameter nor essentially unrestricted.