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bio website quantdec.com
location Northeastern US
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visits member for 4 years, 1 month
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Consultant (environmental and spatial stats a specialty), expert witness, and teacher. I can be reached through (outdated but still valid) links posted on my web site.

Twitter: @WilliamAHuber // ASA-P website: http://amstatphilly.org/


Why waste time learning, when ignorance is instantaneous?

--T(iger) Hobbes.

For any complex problem there is a simple solution. And it's always wrong.

--[Mis?]attributed to H.L. Mencken by Dava Sobel, Longitude.


Sep
9
comment Mean of squared “sum of squared errors”
See Advanced Theory of Statistics, Volume I, chapters 3 ("Moments and Cumulants") and 12 ("Cumulants of Sampling Distributions--(2)").
Sep
9
comment Mean of squared “sum of squared errors”
Jonas, the very first comment to your question indicates how such derivations can be done. Your example, when fully expanded, is a quartic form in the data and therefore the expectation (because it's a linear operator) becomes a homogeneous polynomial (in a suitable sense) of the first four moments of $X_1$. Mathematica merely is doing that routine algebra under the hood. An algebraic theory has been developed; it is explained in great detail in Kendall & Stuart (5th Ed.).
Sep
8
comment How to find a mapping function from n dimensional space to m dimensional space
Could you perhaps add some information to this question to show readers why it might be of interest on this site?
Aug
25
awarded  Yearling
Aug
21
revised Is there a name (and use) for an average based on the unique values of a set of data?
Improved theTeX markup
Aug
4
comment Does affine equivariance implies shape unbiasedness?
(Due to lack of answers on CV, this question has been migrated to Math at the OP's request.)
Jul
30
revised How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?
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Jul
30
revised How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?
added 176 characters in body
Jul
30
answered How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$?
Jul
5
comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
You have hit on the crux of the matter: the question, to be unambiguous, must somehow include enough information to enable calculation of all probabilities. In some settings those probabilities are given explicitly, but usually some particular process whereby the randomization is achieved is described. When the process consists of placing each blue ball within one of the spaces between red balls, all with equal probability (of $1/15$), and doing so in a way that the placement of one blue ball affects no other blue balls, you get the multinomial probabilities.
Jul
5
comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
Perhaps an illustration would help. In the case $n=k=2$ there are two distinguishable configurations modulo symmetries of the circle: RBRB (red-blue-red-blue in counterclockwise order) and RBBR. In the multinomial scenario both are assumed to have probability $1/2$, while in the permutation scenario they are assumed to have probabilities $1/3$ and $2/3$, respectively. Some people have argued that absent any additional information, one should assume the maximum-entropy distribution. That's the multinomial one in this case--not the permutation one!
Jul
5
comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
Yes I can still come up with two (and even more) solutions. In this post I have explained two ways of looking at the situation that are perfectly consistent with the problem. If you disagree, the burden is on you to demonstrate how one (or even both!) violate some explicitly-stated criterion in the problem statement. I would even go so far as to accept that there are some conventional implicit criteria in questions like this one: the obvious one is that the distribution ought to be invariant under symmetries of the configuration. That is the case in both scenarios.
Jul
5
comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
No, that's not right. "Random" is not a quantitative characterization: you have to specify the process involved. The problem statement does not give enough information to "assess" a unique distribution. Both my answers are correct given their assumptions about how the balls are distributed randomly. One of them happens to agree with yours, but that only means it likely adopted the same assumptions you did.
Jul
5
revised Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
added 4099 characters in body
Jul
3
answered Combinatorics: Mean and Variance of an indicator function of items arranged in a circle.
Jul
1
comment How to understand Demartines theorem
This is a minor variant of the Central Limit Theorem, which asserts the standardized version of $||X||_2^2$ is asymptotically Normal. It requires that $X$ have finite variance.
Jun
26
comment Transforming vector elements to element indices
Because matrix operations represent linear transformations and the relationship between Y and X is not linear, no such Q exists.
Jun
18
comment Does affine equivariance implies shape unbiasedness?
Some things to consider: when $\sigma(X)=0$ (the $p\times p$ zero matrix) the procedure is trivially equivariant but obviously not unbiased. More generally, when $\sigma$ is equivariant unbiased and $p\gt 1$, the procedure $\tau(X)=\lambda\sigma(X)$ is also equivariant but when $\lambda^p\ne \lambda$ it must be biased. This suggests that dividing $\sigma(X)$ by its determinant might not be the right definition of "unbiased" to be using.
Apr
7
comment Holder's inequality
Your second statement is not well defined until you assume both $X$ and $Y$ are strictly positive.
Mar
7
reviewed Approve suggested edit on another question on surds and how to use math symbols in this site