The formula from Finley is reproduced in Johnson, Kotz, and
Balakrishnan's "Continuous Distributions: Vol. 1" in the beginning of
their Log Normal chapter. I am not clear that the recursive formula in
W. Huber's spreadsheet is a correct representation of the iterative
version there, but cannot claim to have disproven it either. (I am
suggesting checking it against references.)
It certainly seems to deliver results close to the results of the
psi() approximation, which is also reproduced in that text. (And on a
three year-old computer, the "converges slowly" comment does not seem
to have meaning.) Results have no discernible time lag, so I think
speed of convergence in 1941 (at the end of the mechanical computation
era) may have quite a different meaning in the electronic era.
--
David.
On Jul 10, 2011, at 7:42 PM, Durant, James T. (ATSDR/DTEM/PRMSB) wrote:
Ah, thanks so much.
I found the excel spreadsheet almost right after I posted to the r
group. I had concerns about using Chebyshev and wanted to reproduce
Dr. Whubers simulation to see for myself how it performs. To be
clear I normally use Lands exact or bootstrap for the UCL (or
sometimes take a Bayesian approach with an uninformed prior).
Someone wanted me to use proUCL from EPA for something and I had
never seen a Chebyshev inequality until then, so I was curious about
its performance.
Thanks to you both! You have made my day.
Jim
----- Original Message -----
From: David Winsemius [mailto:dwinsem...@comcast.net]
Sent: Sunday, July 10, 2011 03:14 PM
To: ted.hard...@wlandres.net <ted.hard...@wlandres.net>
Cc: Durant, James T. (ATSDR/DTEM/PRMSB); r-help@r-project.org <r-help@r-project.org
>
Subject: Re: [R] Chebyshev Inequality — MVUE
On Jul 10, 2011, at 2:49 PM, (Ted Harding) wrote:
On 10-Jul-11 16:27:04, Durant, James T. (ATSDR/DTEM/PRMSB) wrote:
Hello,
I was interested in trying to write an R script to calculate a
UCL for a lognormal distribution using the Chebyshev Inequality
-- MVUE Approach (based on EPA’s guidance found in
http://www.epa.gov/oswer/riskassessment/pdf/ucl.pdf).
This looks like it should be straight forward, but I am need to
calculate an MVUE for the population mean and an MVUE for the
population variance, which requires a value (g_n) from a table A7,
found in Aitchison and Brown (1969): The lognormal distribution.
I have looked across the RSiteSearch and can not seem to find a
function that will give me g_n or the MVUE for mean and variance
of lognormal distribution.
Is there an R function that will give me g_n or will calculate
an MVUE for the population mean and variance for the lognormal
distribution?
VR
Jim
James T. Durant, MSPH CIH
Emergency Response Coordinator
US Agency for Toxic Substances and Disease Registry
Atlanta, GA 30341
770-378-1695
Some quick comments. I will try to repond more fully later.
1. The Chebyshev inequality is usually very conservative.
As a simple example, consider X with a negative exponential
distribution with density exp(x), so that the population
mean is 1 and the population variance is also 1.
Then, for a factor K, Chebyshev says that
Prob(|X-1] > K*1) < 1/(K^2).
This is only informative if K>1. So (e.g.) take K=2. Then the
Chebyshev
result is that this Prob < 1/4. HOwever, because X is positive, the
event in question is X > 1 + 2 = 3 so Prob is exp(-3) = 0.0498 <
1/20.
The reference you cite suggests ("Exhibit 5") applying the method to
log-transformed data, which for lognormal data would be normally
distributed. So apply Chebyshev to N(0,1) (mean=0, var=1). Then
Prob(|X-0| > K*1) < 1/(K^2) as before.
Now take K=2 again (i.e. outside +/- 2 SDs, so Prob approx=0.05).
But Chebyshev still says "Prob < 1/4 = 0.25".
So, as a first comment, I am seriously wondering about the wisdom
of basing an approach on Chebyshev's inequality. Note also the
comments in your reference at the end of that section (bottom of
page 12) headed "Caveats about the Chebyshev method.", which is
essentially a warning on similar lines to the above.
2. The function in the reference you cite is not "g_n" but "psi_n",
and the Table cited from Aitchison and Brown is not A7 but A2.
On page 45 of Aitchison and Brown (1969), section 5.41 "The Method
of Maximum Likelihood", the function psi_n is defined (Eqn 5.38)
so as to be applicable to the sufficient statistics mean(log(X))
and var(log(X)) to yield unbiased estimators of the population
mean of X and the population variance of X (Eqns (5.40) and (5.42)).
psi_n is defined as an infinite series which, according to A&B
(page 46) "converges only slowly", and they exhibit a finite-form
asymptotic approximation to it (Eqn (5.43)) which is accurate
asyn=mptotically to O(1/(n^3)). This fairly simple expression
would be easy to define as a function in R:
psi <- function(t,n){
exp(t)*(1 - t*(t+1)/n + (t^2)*(3(t^2) + 22*t + 21)/(6*(n^2)))
}
ITYM:
psi <- function(t,n){
exp(t)*(1 - t*(t+1)/n + (t^2)*(3*(t^2) + 22*t + 21)/(6*(n^2)))
}
I was doing a bit of searching an found some VB code that whuber (and
am wondering if it's the same whuber as frequently makes cogent posts
on stats.stackexchange.com ?) had posted in an Excel macro about ten
years ago that claimed to have reproduced the A9 Table in Gilbert.
http://www.quantdec.com/envstats/software/ln_mvue.xls
His macro was named Finney and I transposed it into R:
Finney <- function(m , z){
aTol <- 0.0000000001
iterMax <- 1000
if (m <= -1) {# issue an error
error("Finney = 0#")}
x <- z * m * m / (m + 1)
if (abs(x) < aTol) { return(Finney = 1L)}
# This is the correct answer.
iMax = abs(trunc(z) + 1) + 20
if (iMax > iterMax) {error("iMax > iterMax")}
# Init
a = 1L
g = a # Lead terms
for ( i in seq(iMax) ) {
# Test for convergence
if (abs(a) <= aTol * abs(g)) {
break()} # Compute the next term
a <- a * x / (m + 2 * (i - 1)) / i
#'
#' Accumulate terms
#'
g = g + a} # Next
return(g)
}
The order of the arguments is reversed but they seem to offer similar
results:
psi(2, 30)
[1] 6.332695
Finney(30, 2)
[1] 6.254139
Finney(60, 1.5)
[1] 4.230381
psi(1.5, 60)
[1] 4.229944
I would think that a conservative statistical method _should_ be used
when assessing toxic risks as the OP might to be doing, given his
address and title.
--
David.
Hoping this helps. As I say, I hope to find time later to look
at this in more detail.
Ted.
--------------------------------------------------------------------
E-Mail: (Ted Harding) <ted.hard...@wlandres.net>
Fax-to-email: +44 (0)870 094 0861
Date: 10-Jul-11 Time: 19:49:39
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David Winsemius, MD
West Hartford, CT
David Winsemius, MD
West Hartford, CT
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