Does the Moment-Generating Function Characterize a Distribution?

It is shown that two visibly distinct distributions can have almost identical moment-generating functions.     

A theory of statistical models for Monte Carlo integration

Summary. The task of estimating an integral by Monte Carlo methods is formulated as a statistical model using simulated observations as data. The difficulty in this exercise is that we ordinarily have at our disposal all of the information required to compute integrals exactly by calculus or numerical integration, but we choose to ignore some of the information for simplicity or computational feasibility. Our proposal is to use a semiparametric statistical model that makes explicit what information is ignored and what information is retained. The parameter space in this model is a set of measures on the sample space, which is ordinarily an infinite dimensional object. None-the-less, from simulated data the base-line measure can be estimated by maximum likelihood, and the required integrals computed by a simple formula previously derived by Vardi and by Lindsay in a closely related model for biased sampling. The same formula was also suggested by Geyer and by Meng and Wong using entirely different arguments. By contrast with Geyer's retrospective likelihood, a correct estimate of simulation error is available directly from the Fisher information. The principal advantage of the semiparametric model is that variance reduction techniques are associated with submodels in which the maximum likelihood estimator in the submodel may have substantially smaller variance than the traditional estimator. The method is applicable to Markov chain and more general Monte Carlo sampling schemes with multiple samplers.     

Invariance and factorial models

Two factors having the same set of levels are said to be homologous. This paper aims to extend the domain of factorial models to designs that include homologous factors. In doing so, it is necessary first to identify the characteristic property of those vector spaces that constitute the standard factorial models. We argue here that essentially every interesting statistical model specified by a vector space is necessarily a representation of some algebraic category. Logical consistency of the sort associated with the standard marginality conditions is guaranteed by category representations, but not by group representations. Marginality is thus interpreted as invariance under selection of factor levels ( I -representations), and invariance under replication of levels ( S -representations). For designs in which each factor occurs once, the representations of the product category coincide with the standard factorial models. For designs that include homologous factors, the set of S -representations is a subset of the I -representations. It is shown that symmetry and quasi-symmetry are representations in both senses, but that not all representations include the constant functions (intercept). The beginnings of an extended algebra for constructing general I -representations is described and illustrated by a diallel cross design.     

Equivalence class formation in non-hearing impaired children and hearing impaired children

The relationship between verbal behavior and stimulus equivalence was examined using three sets of children differing in chronological age and verbal ability: (1) non-hearing impaired three and four year olds who had verbal skills generally consistent with their chronological ages; (2) partially hearing (severe to profoundly deaf) children who were rated with verbal ages of above 2 years; and (3) partially hearing children (also severely to profoundly deaf) who were rated with verbal ages of below 2 years. All children were taught a series of four conditional discriminations using unfamiliar stimuli. The children were then tested to determine whether classes of equivalent stimuli had formed. Although all the children were able to learn the conditional discriminations equally well and all the verbally-able children (normal and partially hearing) formed equivalence classes, only one of the verbally-impaired children reliably demonstrated stimulus equivalence formation. These results are consistent with the suggestion that stimulus equivalence and human verbal behavior are closely related.

     

Survival models and health sequences

Survival studies often generate not only a survival time for each patient but also a sequence of health measurements at annual or semi-annual check-ups while the patient remains alive. Such a sequence of random length accompanied by a survival time is called a survival process. Robust health is ordinarily associated with longer survival, so the two parts of a survival process cannot be assumed independent. This paper is concerned with a general technique—reverse alignment—for constructing statistical models for survival processes, here termed revival models. A revival model is a regression model in the sense that it incorporates covariate and treatment effects into both the distribution of survival times and the joint distribution of health outcomes. The revival model also determines a conditional survival distribution given the observed history, which describes how the subsequent survival distribution is determined by the observed progression of health outcomes.     

Response to discussants of “survival models and health sequences”

Survival studies often generate not only a survival time for each patient but also a sequence of health measurements at annual or semi-annual check-ups while the patient remains alive. Such a sequence of random length accompanied by a survival time is called a survival process. Robust health is ordinarily associated with longer survival, so the two parts of a survival process cannot be assumed independent. This paper is concerned with a general technique—reverse alignment—for constructing statistical models for survival processes, here termed revival models. A revival model is a regression model in the sense that it incorporates covariate and treatment effects into both the distribution of survival times and the joint distribution of health outcomes. The revival model also determines a conditional survival distribution given the observed history, which describes how the subsequent survival distribution is determined by the observed progression of health outcomes.