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We give a new proof of the identifiably of the MPH model. This proof is constructive: it is a recipe for constructing the triple—regression function, base‐line hazard, and distribution of the individual effect—from the observed cumulative distribution functions. We then prove that the triples do not depend continuously on the observed cumulative distribution functions. Uniformly consistent estimators do not exist. Finally we show that the MPH model is even identifiable from two‐sided censored observations. This proof is constructive, too.

When considering a mathematical theorem one ought not only to know how to prove it but also why and whether any given conditions are necessary. All too often little attention is paid to to this side of the theory and in writing this account of the theory of real functions the authors hope to rectify matters. They have put the classical theory of real functions in a modern setting and in so doing have made the mathematical reasoning rigorous and explored the theory in much greater depth than is customary. The subject matter is essentially the same as that of ordinary calculus course and the techniques used are elementary (no topology, measure theory or functional analysis). Thus anyone who is acquainted with elementary calculus and wishes to deepen their knowledge should read this.

In this paper we introduce the notion of Riesz homomorphism on Archimedean directed partially ordered groups and use it to study the vector lattice cover of such groups. There are close relations between the theories of Boolean algebras, distributive lattices, abelian lattice ordered groups and the like on one hand and vector lattices (= Riesz spaces) on the other hand. In the case of Boolean algebras such a relation is made explicit by considering the elements of a Boolean algebra as so called place functions. The set of these place functions generates a vector lattice and this vector lattice can then be used to study the Boolean algebra (see Fremlin's book [8]).In the other cases mentioned, one simply "forgets" some of the structure of a Riesz space if one considers it as a lattice ordered group or a distributive lattice. Conrad in [6] introduced the notion of a vector lattice cover for Archimedean lattice ordered groups and indeed, in categorical language, he therefore studied "the adjoint of the forgetful functor". Shortly thereafter, Bleier in [3] proved a somewhat more general result than Conrad's: For every Archimedean lattice ordered group G there exists an (essentially) unique Archimedean Riesz space E(G) that contains G as a lattice ordered subgroup and which is minimal with that property; furthermore, such a G is automatically large in E{G) (that is, for every 0 < g e E(G) there exists

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