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Update

Experimenting with descriptive statistics, I determined that if rank the scaled weighted average quantity for each set, computed as follows:

$$Mean(Weighted Quantity) * \frac {Median ( Weighted Quantity)} {Range(Weighted Quantity)}$$

the results match my intuition. Although I've solved my immediate problem, I'd still like an explanation for what I'm doing in more formal terms. I'd like to start learning more about these types of problems.


Update 2

Context, per request: The data are inputs into a parametric (aka variance-covariance) interest rate value at risk (IR VaR) computation (the associated correlation matrices are not considered here)

  • Each matrix is per security holding.
  • Each row is per key rate tenor (i.e. 3 months, 1 year, 5 years, etc.) The number of tenors varies.

For each tenor (row):

  • f is dv01 - or the $ change in market value of the holding corresponding to a 1bp shift in the yield curve at the row's tenor
  • v is the spot rate at that tenor, interpolated from the yield curve
  • q is the volatility of the the spot rate at that tenor

The scaled weighted average volatility is a measure of, or proxy for, the degree of non-linearity of the security's pricing model. It accurately ranks holdings (based on test data) in order of the percentage variance of the holding's parametric IR VaR result from its Monte Carlo IR VaR result.


I wish to rank sets of data by some form of average quantity as weighted by a factor adjusted by a value. For example, consider the two sets of data:

Set One: \begin{matrix} f[1]_1 & v[1]_1 & q[1]_1\cr f[2]_1 & v[2]_1 & q[2]_1\cr f[3]_1 & v[3]_1 & q[3]_1 \end{matrix}

and

Set Two: \begin{matrix} f[1]_2 & v[1]_2 & q[1]_2\cr f[2]_2 & v[2]_2 & q[2]_2\cr f[3]_2 & v[3]_2 & q[3]_2\cr f[4]_2 & v[4]_2 & q[4]_2\cr f[5]_2 & v[5]_2 & q[5]_2 \end{matrix}

My initial thought was to rank them by weighted arithmetic mean of quantity, i.e.

$$\frac {\sum (f * v * q)}{\sum(f * v)}$$

But the results don't fit my intuition (which may be faulty.)

How can I tell whether I need to normalize before ranking? Which normalization methods should I consider?

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migrated from math.stackexchange.com Mar 4 '13 at 21:48

This question came from our site for people studying math at any level and professionals in related fields.

    
Can you rewrite your question to provide some context as to what your variables represent? I'm failing to connect $f$, $v$, and $q$ to any conventional notation. Please write this in a manner that would be appropriate to a non-mathematician, not that statisticians don't know math, but to curtail ambiguity. –  AdamO Mar 5 '13 at 19:49
    
@AdamO - context added. Let me know if you need more details. –  marfarma Mar 6 '13 at 17:40
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