Yuzisee

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Unbiased weighted sample covariance

Hey there,

I saw that you reverted my changes about unbiased weighted sample covariance for frequency type weights on this page.

Could you please explain the rationale in more details? Because for the moment I am not convinced by the explanations you did (you said it was the same as for the variance, but for the variance two subsections were made for the two different kinds of weights, whereas only the reliability weights are accounted for in the covariance section).

For example, let's say that we have four weights (1, 2, 3, 3).

By using ${\frac {\sum _{i=1}^{N}w_{i}}{\left(\sum _{i=1}^{N}w_{i}\right)^{2}-\sum _{i=1}^{N}w_{i}^{2}}}$ we obtain 0.155

But when using ${\frac {1}{\sum _{i=1}^{N}w_{i}-1}}$ we obtain 0.125.

In this case, how can the equation you kept be possibly equivalent and working for both frequency type weights and reliability type weights?

Also, see the discussion here, this is the origin of my modifications on wikipedia.

Note that I tried the frequency-type equation programmatically against non-weighted sample covariance results, and they are exactly the same, whereas the reliability-type equation does not give the same result (maybe I did something wrong?).

Thank you very much for taking the time to reply.

Best, --Lrq3000 (talk) 14:13, 28 June 2016 (UTC)

@Lrq3000: Hey good catch. Yes, my derivation was assuming purely non-random weights. Looks like the section was further clarified by User:Btyner on April 2015

--Yuzisee (talk) 00:20, 30 November 2016 (UTC)

@Yuzisee: Thank you for the clarification Yuzisee. However I don't understand what do you mean by "non-random"? Did you mean non-normal?

--Lrq3000 (talk) 01:45, 8 January 2017 (UTC)

@Lrq3000: When I say non-random I mean a constant that does not have a probability distribution (i.e. one could think of a "non-random variable" as equivalent to a random variable whose probability distribution takes the shape of a Dirac delta function). So, in this case, a non-random variable has the same outcome everywhere in the event space (or, has no event space at all). On the other hand, "non-normal" would mean any distribution that is not Gaussian shaped. Since a non-random variable has a trivial probability distribution, it could be said to be neither normal nor non-normal. Hope that helps?

--Yuzisee (talk) 04:14, 19 July 2017 (UTC)