History of statistics is alive to me as I fondly recall myData Modeling, Quantile/Quartile Functions, Confidence Intervals, Introductory Statistics Reform

Emanuel ParzenTexas A&M University

interaction with Eric Lehmann since receiving my Ph.D. in Berkeley

in 1953. The 2003 Economics Nobel Prize (awarded for fundamental

research in statistical time series analysis) reminds me of my

joking complaint to diverse applied researchers: "why do you call

it theory if I know it and applied research when you practice

it?" I have continued to learn a lot about Quantiles and

Nonparametric Data Modeling since my 1979 JASA paper. New methods

have been developed (that some applied researchers consider a

gold mine). Quantile data modeling is not practiced by most

statisticians who are limited to sample median Q2, interquartile

range IQR, and Q-Q probability plots. To estimate and test a

parameterµone starts with the natural estimatorµ^; we define a

statistic T(µ,µ^), an increasing function ofµand with distribution

(whenµis true parameter) equal to distribution of a random

variable T (usually Normal(0,1), Student, or inverse average chi-square).

To test H_0:µ=µ_0 one computes or bounds P-value(µ_0)=F_T(observed T(µ_0,µ^));

it is a distribution function ofµ_0 (whose probability density one

could derive). Define its inverseµ^(u) by Q_T(u)=T(µ^(u),µ^);µ^(u),

called the parameter with P-value u, is a quantile function which has

a pseudo-Bayesian interpretation as the conditional quantile ofµ

given the data. The conventional confidence level 1-a confidence interval

can be shown to beµ^(a/2)<µ<µ^(1-(a/2)). A table ofµ^(u) for many

conventional statistical problems (for u, 1-u=.05,.025,.01,.005) can teach

introductory statistics in less words and closer to the frontier of

modern statistical thinking (including quantile based Bayesian credible

intervals and bootstrap). For random variable X with quantile Q(u) define

(and plot on same graph with exponential and normal) informative q

uantile/quartile function Q/Q(u)=(Q(u)-midquartile)/2 IQR.

Talk could also discuss confidence Q-Q plots, conditional quantile,

comparison distribution, mid-distribution, and definition of sample

quantiles, linear rank statistics, and sample variance.