Underlying Probability Model Research Articles

A unification of tail estimators

This work considers estimators of the tails of the cdf, quantile function and edf, which are generally applicable because they use only those observations in the sample which exceed a high threshold value. Three seemingly different approaches have been proposed by Hill (1975), Hall (1982), and Pickands (1975). Herein the tail behavior of the underlying probability model is expressed using the density-quantile function. This general tail behavior model is shown to be a common origin for the parametric models proposed by Hill, Hall, and Pickands' GPD model. Further, this tail behavior model motivates a representation for the quantile function of the exceedences which is used to obtain a parametric model for tail estimates using the exceedences which unifies the seemingly different tail estimators mentioned above.

Saturation models: A brief survey and critique

AbstractThe authors review and compare the papers in this issue on saturation and logistic growth with special emphasis on the structure of the underlying probability models. Influence diagrams are used to illustrate the dependence and independence assumptions and the way in which uncertainty is reflected in key model assumptions and parameters characterizing the models.

Further results on the asymptotic agreement problem

The problem of reaching a concensus between two decision-makers provided with different information is considered. The problem in which the decision-makers may have different underlying probability models is studied. Results are developed to characterize the likelihood of an agreement being reached eventually in terms of the nature of the inter-decision-maker communications. The problem in which the decision-makers are aware of the possibility that they may have different models is treated. It is found that in this case a deadlock can be reached where neither decision maker can learn additional information from the concensus process and they cannot reach a concensus decision. This result indicates that incorporating human uncertainty in probability assessment into the asymptotic agreement problem can lead to outcomes not anticipated in the general theory previously developed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Statistical Analysis of Hominoid Molecular Evolution

The core data of molecular biology consists of DNA sequences. We will show how DNA sequences may be used to infer the evolution of the primates, human, chimpanzee, ape, orangutan and gibbon. The underlying probability models are taken to be Markov processes on trees. Some dependencies along the sequence due to the genetic code are also considered.

Sensitivity of the test error probabilities with respect to the level of contamination in general model of contaminacy

Characteristics of sensitivity of test error probabilities with respect to changes of underlying probability model are proposed. Explicit expressions for these characteristics are derived in the framework of a model in which a family of composite hypothese is determined by a family of capacities. Earlier results are shown to be special cases of these general ones. Also, a numerical illustration is presented.

Cluster validity profiles

The quantitative evaluation of clusters has lagged far behind the development of clustering algorithms. This paper introduces a new procedure, based on probability profiles, for judging the validity of clusters established from rank-order proximity data. Probability profiles furnish a comprehensive picture of the compactness and isolation of a cluster, scaled in probability units. Given a rank-order proximity matrix and a cluster to be examined, profiles compare the cluster's upper bounds on the best compactness and isolation indices one would expect in a randomly chosen graph. After reviewing the pertinent literature this paper explains the background from graph theory and cluster analysis needed to treat cluster validity. The probabilities and bounds needed to form cluster profiles are derived and strategies for using profiles are suggested. Special attention is given to the underlying probability models. Profiles are demonstrated on four artificially generated data sets, two of which have good hierarchical structure, and on data from a speaker recognition project. They reject spurious clusters and accept apparently valid clusters. Since profiles quantify the interaction between a cluster and its environment, they provide a much richer source of information on cluster structure than single-number indices proposed in the literature.

Estimating Sample Size in Computing Simulation Experiments

A method is described for estimating and collecting the sample size needed to estimate the mean of a process (with a specified level of statistical precision) in a simulation experiment. Steps are also discussed for incorporating the determination and collection of the sample size into a computer library routine that can be called by the ongoing simulation program. We present the underlying probability model that enables us to denote the variance of the sample mean as a function of the autoregressive representation of the process under study and describe the estimation and testing of the parameters of the autoregressive representation in a way that can easily be “built into” a computer program. Several reliability criteria are discussed for use in determining sample size. Since these criteria assume that the variance of the sample mean is known, an adjustment is necessary to account for the substitution of an estimate for this variance. It is suggested that Student's distribution be used as the sampling distribution, with “equivalent degrees of freedom” determined by analogy with a sequence of independent observations. A bias adjustment is described that can be applied to the beginning of the collected data to reduce the influence of initial conditions on events in the experiment. Four examples are presented using these techniques, and comparisons are made with known theoretical solutions. One unfortunate shortcoming of the proposed procedure is that its performance is directly linked to the initially chosen sample size. Our results show that as this sample size increases, the procedure gives results which agree more closely with predicted results.

On the Hypotheses of 'No Interaction' in Contingency Tables

The statistical analysis of data in multi-dimensional contingency tables is discussed in terms of appropriate underlying probability models. Emphasis is placed on the distinction between 'factors' (such as treatments or blocks) which have fixed marginal totals and 'responses' (such as category of performance) which have random marginal totals. Hence, four principal cases arise: (i) the 'multi-response, no factor' tables, (ii) the 'multi-response, uni-factor' tables, (iii) the 'multi-response, multifactor' tables, (iv) the 'uni-response, multi-factor' tables. For situations (i) and (ii), the concept of 'no interaction' is related to questions regarding the pattern of association among responses. However, for situation (iv), it is related to how factors combine (e.g., additively) to determine the response distribution. Finally, for situation (iii), both types of questions arise. For each of the different types of tables, the problem of formulating appropriate hypotheses of 'no interaction' is considered. The corresponding test statistics are based upon a general and computationally simple criterion of Wald [1943]. The suggested methods are illustrated with several numerical examples.