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[ai-geostats] Least biased variance estimate

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  • JW
    Hello List, On October 7, 2005, I pointed out that the correct formula for a set of test results determined in samples of variable weights was implemented in
    Message 1 of 1 , Oct 23, 2005
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      Hello List,
       
      On October 7, 2005, I pointed out that the correct formula for a set of test results determined in samples of variable weights was implemented in several Excel templates posted under Documents on ai-geostats.org. However, on October 20, 2005, a flawed formula was accepted as the one most likely formula to give the least biased or best unbiased variance estimate.
       
      Please peruse "Variances variable weights" (also posted under Documents) in which the correct formula is once again applied. The basic difference between these formulae is that the term: (1/sum(wi^2))-1, in which sum(wi)=1 in the correct formula, unlike the term: 1/sum(wi^2) in the accepted formula, is the number of degrees of freedom for a set of "n" samples of variable weights. Evidently,  the number of degrees of freedom in the correct formula is no longer a positive integer but a positive irrational. I introduced this formula in "Sampling in Mineral Processing", a paper that was published in 2002 and is posted on my website.
       
      Look at the variance estimates in each of the three columns, and note that the least biased variance estimate in the first column is lower than the correct variance estimate in the second column, which, in turn, is lower than the variance estimate in the third column. Change the variable weights of 1, 2, 3, 4 and 5 to the same weight and note that all columns give the same variance estimates and the same central values. If one were to worry about the veracity of Excel's VAR-function for the variance of a randomly distributed or randomized set of test data with identical weights, then the heuristic proof with constant weights indicates that this is one Excel stat function that does meet rigorous QC requirements.
       
      The variance of the central value of any set of measured values (its arithmetic mean or some weighted average) is obtained by multiplying var(x), the variance of the set, with the term: sum(wi^2). Therefore, var(xbar)=var(x)*sum(wi^2) applies to area-, count-, density-, DISTANCE-, length-, mass- and volume-weighted averages. This formula converges on the Central Limit Theorem when all of the weighing factors converge on 1/n.
       
      Here are a few true facts that derive from mathematical statistics:
       
           1. Each and every distance-weighted average has its own variance,
           2. Distance-weighted averages became honorific kriged estimates,
           3. Each and every kriged estimate has its own variance,
           4. Variances and covariances of SETS of kriged estimates are invalid.
       
      All I want to know is which of these four facts is false and why.
       
      Kind regards,
      Jan W Merks   
       

       
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