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Some days ago I asked this question: Unbiased estimators for the moments of 2 non-independent random variables

Now, using the same framework employed for the previous question, I'm facing with the problem of estimating the statistical variance of one of these unbiased estimator.

Honestly, in this case I'm not sure about how to proceed. Anyway, I started with the following thoughts:

  1. I observe random variables $X$ and $Y$ over time. I collect $N$ couples of realizations $(X_i, Y_i)$, where $i$ denotes the time.
  2. Using the notation used by wolfies in its answer, I estimate $E[(X-\mu_X)(Y-\mu_Y)]$ using this formula: $$ h_{1,1} = \frac{N s_{1,1} - s_{1,0}s_{0,1}}{N(N-1)}$$ where $$s_{1,0} = \sum_{i=1}^{N}X_i,~ s_{0,1} = \sum_{i=1}^{N}Y_i,~ s_{1,1} = \sum_{i=1}^{N}X_iY_i $$
  3. This is the crucial point. I pose: $$Z_i = \frac{NX_iY_i - X_iY_i}{N-1 }$$ In practice, I substitute $s_{1,0}$ with $X_i$, $s_{0,1}$ with $Y_i$ and $s_{1,1}$ with $X_iY_i$. In this way, I obtain a new sample dataset, formed by $Z_i$, and I say that: $$h_{1,1} = \frac{1}{N} \sum_{i=1}^N Z_i$$ and the variance is: $$\frac{1}{N-1} \sum_{i=1}^N (Z_i - h_{1,1})^2$$

Is this correct?

the_candyman
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1 Answers1

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The Question: You have the h-statistic $h_{1,1}$ (i.e. unbiased estimator of the population central moment $\mu _{1,1}$):

... where $s_{a, b}=\sum _{i=1}^n X_i^a Y_i^b$. You now seek $Var(h_{1,1})$.

Solution: Since the variance operator is just the 2nd CentralMoment of the population, the answer is given by:

where each $\mu _{r,t}$ term denotes:

$$\mu _{r,t} = E\left[(X-E[X])^r (Y-E[Y])^t\right]$$

All done.

OP wrote: Is there some book/reference where I can find something about the mathematics behind h-statistic?

To get into the moments of moments literature ... that is, to solve your own problems by hand ...., you will probably first need to become familiar with augmented symmetric functions. These are difficult to derive/convert by hand, so without automated computational tools, one is then left to using 20th C tables. For a brief intro, see, for instance, Section 7.4 A and C of Chapter 7 of our book, "Mathematical Statistics with Mathematica". A free download of the chapter is available here:

http://www.mathStatica.com/book/Rose_and_Smith_2002edition_Chapter7.pdf

... which also provides many references. As to theory, in books, the best reference would be to Stuart and Ord: Kendall's Advanced Theory of Statistics (volume 1 - Distribution Theory) ... Chapters 12 and 13 ... although this is set up mostly in terms of k-statistics rather than h-statistics. Even here, after 6 editions, there are still some mistakes in the listed tables of formulae, and deriving one's own example by hand can be tough work.

Notes: (i) CentralMomentToCentral is a function from the mathStatica package. (ii) I am an author of the latter.

Glorfindel
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wolfies
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