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Including advanced statistics in an elementary statistics course  Doctorow
Posted:
Jan 6, 2001 12:11 PM


From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sat. Jan. 6, 2001 7:38AM
G. L. Yang's paper "Some recent developments in nonparametric inference for right censored and randomly truncated data," pages 441457 of the volume Statistics For the 21st Century, Editors C. R. Rao and G. J. Szekely, Marcel Dekker: New York 2000, is a very tempting paper to enable elementary statistics students to learn about the latest research in the remarkable fields of censored and truncated statistics. There are things to be said on both sides of the argument about including advanced statistics or latest research in an elementary mathematics course, but I think that in the long run there will be some tradeoff between the large time required to introduce students to basic elementary statistics and the creativity which is stimulated by challenging some of the world's most difficult problems. By the way, there is a "companion volume", Perplexing Problems in Probability, Editors M. Bramson and R. Durrett, Birkhauser: Boston 1999, which does the analogous thing for probability. I recommend both for the college library (reserve).
Yang, of the University of Maryland, mentions that there are about 4800 titles on censored data (not including "random truncation" as a keyword) in the Math. Sci. database. S. Csorgo's 1996 paper in Ann. Statist. 24, 27442778, goes into more detail on this. A right censored sample of a random variable X with unknown distribution function F cuts off the sample above by using a sample of another random variable C with unknown distribution function L such that C and X are independent and thereby generating iid random variables Zj = min(Xj, Cj). The indicator or characteristic function dj = I(Xj < = Cj) is also used, and the right censored sample is the sample of n iid pairs (Zj, dj), j = 1 to n. Yang isolates two basic theorems on censored data involving mean zero Gaussian processes asymptotically. A third theorem involves bounded and uniformly bounded processes, while a fourth theorem involves stopped and/or Gaussian processes and order statistics. The fifth and sixth theorems involve Brownian bridges and/or KaplanMeier (KM) estimate. I have discussed Brownian bridges (BBs) on anzapl and elsewhere. They are very closely related to the uniform distribution via the uniform empirical process and are part of an "explosive" field of modern research that includes fractional and fractal and snake BBs and Brownian motion. They constitute an excellent example for the student of how the uniform probability distribution is of great importance in modern pure and applied probability/statistics. The KM estimate is one minus the product of factors of the form (1  (dk,n divided by (n  k + 1)) )taken to the powers I(Zk,n < = z) where Zk,n are the ordered Zj (order statistics) and dk,n the corresponding censoring indicators. The proof of the sixth theorem depends strongly on the discrete uniform distribution (equiprobable distribution) and relates it to reverse super martingales. Most of the remaining three theorems relate to the uniform empirical distribution function/process. Random truncation concerns bivariate distribution of (X,Y) on the lower half plane, and is commonly useed for astronomy data where X is the random luminosity of a galaxy and Y is a function of the red shift.and we have to estimate the distribution function of X and the proportion of missing data in region X > Y (which cannot be deterministically computed).
My "obsession" with the uniform distribution function (and also with the gamma distribution function) comes from my field of logicbased probability (LBP), which introduced in 1980 with my wife Marleen J. Doctorow, Ph.D. My most recent published paper on this is in the volume edited by B. N. Kursunuglu (Ph.D. Cambridge University under Paul Dirac), S. L. Mintz, and A. Perlmutter 2000, Quantum Gravity, Generalized Theory of Gravitation, and Superstring TheoryBased Unification, Kluwer Academic/Plenum. Abstracts of 49 of my papers are on the internet at http://www.logic.univie.ac.at, Institute for Logic of the University of Vienna (select ABSTRACTS, then select BY AUTHOR, then select my name).
Osher Doctorow West Los Angeles College, Ventura College, etc.
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