# Chebyshev Inequality — MVUE

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## Chebyshev Inequality — MVUE

 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         [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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## Re: Chebyshev Inequality  MVUE

 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))) } Hoping this helps. As I say, I hope to find time later to look at this in more detail. Ted. -------------------------------------------------------------------- E-Mail: (Ted Harding) <[hidden email]> Fax-to-email: +44 (0)870 094 0861 Date: 10-Jul-11                                       Time: 19:49:39 ------------------------------ XFMail ------------------------------ ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-helpPLEASE do read the posting guide http://www.R-project.org/posting-guide.htmland provide commented, minimal, self-contained, reproducible code.
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