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Including advanced statistics in an elementary statistics course - Doctorow
Posted:
Jan 6, 2001 12:11 PM
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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 441-457 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 trade-off 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,
2744-2778, 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 Kaplan-Meier (KM) estimate. I have discussed
Brownian bridges (BBs) on anzap-l 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 logic-based 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 Theory-Based 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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