convergenceConcepts - Supplemental materials for this...

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Supplemental materials for this article are available through the TAS web page at . Teacher’s Corner Understanding Convergence Concepts: A Visual-Minded and Graphical Simulation-Based Approach Pierre L AFAYE DE M ICHEAUX and Benoit L IQUET This article describes the difficult concepts of convergence in probability, convergence almost surely, convergence in law, and convergence in r th mean using a visual-minded and graphical simulation-based approach. For this purpose, each probability of events is approximated by a frequency. An R package that reproduces all of the experiments cited in this article is available in CRAN. See the online Supplement for details. KEY WORDS: Convergence almost surely; Convergence in law; Convergence in probability; Convergence in r th mean; Dy- namic graphics; Monte Carlo simulation; R language; Visual- ization. 1. INTRODUCTION Most departments of statistics teach at least one course on the difficult concepts of convergence in probability ( P ), almost sure convergence ( a.s. ), convergence in law ( L ), and convergence in r th mean ( r ) at the graduate level (see Sethuraman 1995 ). In- deed, as pointed out by Bryce et al. ( 2001 ), “statistical theory is an important part of the curriculum, and is particularly impor- tant for students headed for graduate school.” Such knowledge is prescribed by learned statistics societies (e.g., the Accredi- tation of Statisticians by the Statistical Society of Canada and Curriculum Guidelines for Undergraduate Programs in Statisti- cal Science by the American Statistical Association). The main textbooks (e.g., Chung 1974 ; Billingsley 1986 ; Ferguson 1996 ; Lehmann 2001 ; Serfling 2002 ) devote about 15 pages to defin- ing these convergence concepts and their interrelations. Very often, these concepts are provided as definitions, and students are exposed only to some basic properties and to the universal implications displayed in Figure 1 . The aim of this article is to clarify these convergence con- cepts for master’s students in mathematics and statistics, and Pierre Lafaye de Micheaux is Associate Professor, Départment Mathéma- tiques et de Statistique, Universitè de Montréal, Canada (E-mail: lafaye@ ). Benoit Liquet is Associate Professor, INSERM U897, ISPED, University of Bordeaux 2, Bordeaux, France (E-mail: Benoit.Liquet@ ). The authors thank R. Drouilhet, M. Lejeune, and the ref- erees, the editor, and the associate editor for their many helpful comments. Figure 1. Universally valid implications of the four classical modes of convergence. (See Ferguson 1996 for proofs.) also to provide software useful for learning these concepts. Each convergence notion provides an essential foundation for further work. For example, convergence in law is used to obtain asymptotic confidence intervals and hypothesis tests using the central limit theorem; convergence in probability is used to obtain the limiting distribution of the Z test re- placing an unknown variance with its estimate (through Slut-
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This note was uploaded on 08/23/2010 for the course ECON 583 taught by Professor Zivot during the Fall '09 term at University of West Alabama-Livingston.

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convergenceConcepts - Supplemental materials for this...

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