P1.T2.308. Coskewness and cokurtosis

[Fran]

AIMs: Interpret the skewness and kurtosis of a statistical distribution, and interpret the concepts of coskewness and cokurtosis. Define and interpret the best linear unbiased estimator (BLUE).

Questions:

308.1. The thee joint outcomes of two variables (X,Y) are characterized by the following probability distribution:

Which is nearest to the co-skew (XXY) and co-kurtosis (XYYY) of this bivariate distribution (please note this question is clearly more difficult than an exam-level question; it is meant to give concrete practice to the concept)?

a. Co-skew(XXY) = -0.245, Co-kurtosis(XYYY) = -7.931
b. Co-skew(XXY) = -0.245, Co-kurtosis(XYYY) = 1.6250
c. Co-skew(XXY) = +0.411, Co-kurtosis(XYYY) = +3.027
d. Co-skew(XXY) = +1.588, Co-kurtosis(XYYY) = +5.682

308.2. In regard to skew, kurtosis, co-skew and co-kurtosis, each of the following is true EXCEPT which is technically false (this is a difficult question meant for training purposes)?

a. The coskew between A and B, S(AAB) = E[(A-mu[A])^2*(B-mu)], where mu[A] is the mean of (A)
b. In the case of a bivariate distribution between two (2) random variables, we can compute two (2) nontrivial co-skew and three (3) nontrivial cokurtosis statistics
c. If a univariate population skew is adjusted to its sample-based equivalent, for a given (n), the sample skew might be greater or less than the population skew
d. For ten (10) random variables, there are 45 non-trivial second (2nd) cross central moment, where non-trivial refers to covariance(X,X)

308.3. Analyst Rob has identified an estimator, denoted T(.), which qualifies as the best linear unbiased estimator (BLUE). If T(.) is BLUE, which of the following must also necessarily be TRUE?

a. T(.) must have the minimum variance among all possible estimators
b. T(.) must be the most efficient (the "best") among all possible estimators
c. It is possible that T(.) is the maximum likelihood (MLE) estimator of variance; i.e., SUM([X - average (X)]^2)/(n-1)
d. Among the class of unbiased estimators that are linear, T(.) has the smallest variance

Answers:

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[brian.field]

For 308.2, I am in some disagreement. I thought that skewness was defined as the third central moment and that it is/was separate and distinct from the standardized or normalized skewness (which divides skewness by the standard deviation cubed.)

I swear my studies have stated this previously, although now it seems most web references define skewness as the standardized version!

So, there is some support for the notion that A in 308.2 is technically correct! Perhaps it should say "In regard to standardized skewness, ..."

(I also realize that the assigned reading defines skewness as "standardized skewness". Personally, I think skewness defined simply as the third central moment and normalized or standardized skewness as the third central moment divided by standard deviation cubed to be most precise...or even Pearson's Standardized Skewness!)