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Estimating functions, such as the score or quasiscore,can have more than one root. In many of these cases, theory tells us that there is a unique consistent root of the estimating function. However, in practice, there may be considerable doubt as to which root is appropriate as a parameter estimate. The problem is of practical importance to data analysts and theoretically challenging as well. In this paper, we review the literature on this problem. A variety of examples are provided to illustrate the diversity of situations in which multiple roots can arise. Some methods are suggested to investigate the possibility of multiple roots, search for all roots and compute the distributions of the roots. Various approaches are discussed for selecting among the roots. These methods include (1) iterating from consistent estimators, (2) examining the asymptotics when explicit formulas for roots are available, (3) testing the consistency of each root, (4) selecting by bootstrapping and (5) using information-theoretic methods for certain parametric models. As an alternative approach to the problem, we consider how an estimating function can be modified to reduce the number of roots. Finally, we survey some techniques of artificial likelihoods for semiparametric models and discuss their relationship to the multiple root problem.
We discuss a linearized model to analyze the errors in the reconstruction of the relative motion of two tectonic plates using marine magnetic anomaly data. More complicated geometries, consisting of several plates, can be analyzed by breaking the geometry into its stochastically independent parts and repeatedly applying a few simple algorithms to recombine these parts. A regression version of Welch’s solution to the Behrens-Fisher problem is needed in the recombination process. The methodology is illustrated using data from the Indian Ocean. Through a historical perspective we show how improving data density and improving statistical techniques have led to more sophisticated models for the Indo-Australian plate.
We propose an influencebased regression diagnostic for tectonic data. A generalization of the standardized influence matrix of Lu, Ko and Chang is applied to study the influence of a group of data points on a subparameter of interest. This methodology could also be used in treatment-block designs to analyze the influence of the blocks on the estimated treatment effects.
We present the Bayesian approach to estimating parameters associated with animal survival on the basis of data arising from mark recovery and recapture studies. We provide two examples, beginning with a discussion of band-return models and examining data gathered from observations of blue winged teal (Aas discors), ringed as nestlings. We then look at open population recapture models, focusing on the Cormack- Jolly-Seber model, and examine this model in the context of a data set on European dippers (Cinclus cinclus). The Bayesian procedures are shown to be straightforward and provide a convenient framework for model-averaging, which incorporates the uncertainty due to model selection into the inference process. Sufficient detail is provided so that readers who wish to employ the Bayesian approach in this field can do so with ease. An example of BUGS code is also provided.
We explore the tasks where sensitivity analysis (SA) can be useful and try to assess the relevance of SA within the modeling process. We suggest that SA could considerably assist in the use of models, by providing objective criteria of judgement for different phases of the modelbuilding process: model identification and discrimination; model calibration; model corroboration.
We review some new global quantitative SA methods and suggest that these might enlarge the scope for sensitivity analysis in computational and statistical modeling practice. Among the advantages of the new methods are their robustness, model independence and computational convenience.
Johannes H. B. Kemperman, born in 1924, received his Bachelor of Science in 1945 and Ph.D. in 1950, each in mathematics and physics, from the University of Amsterdam. From 1948 to 1951 he was a Research Associate at the Mathematical Centre in Amsterdam. During 1951–1953 he was a Visiting Professor on a Fulbright grant at Purdue University, and subsequently he joined the faculty and stayed at Purdue for 10 years, becoming a full Professor in 1959.
In 1961 he went to the University of Rochester, becoming the Fayerweather Professor of Mathematics in 1970. He stayed at Rochester for 25 years. In 1985 he joined the faculty at Rutgers University as a Professor in the Statistics Department and also a voting member of the Mathematics Department. He retired to emeritus status at Rutgers in 1995.
He served in editorial posts at the Annals of Mathematical Statistics, Annals of Probability, Annals of Statistics, Aequationes Mathematicae and Stochastic Processes and Applications. He is a Fellow of the Institute of Mathematical Statistics and the American Association for the Advancement of Sciences and is a Correspondent of the Royal Dutch Academy of Sciences (Amsterdam) .
He has produced 23 Ph.D. students, 3 books and over 100 publications in analysis, number theory, group theory, probability, statistics, functional equations, mathematical biology and other areas.