2 NOTATIONS

Let us consider a probability set $\Omega$ and the associated r.v. (random variable) $\xi\in\Omega$. When relevant, the $pd\! f$ of $\xi$ is noted $\varphi \left(\xi\right)$. For a given $n\geq2$, a sample $\omega$ of size $n$ drawn "at random" from $\Omega$ will be an element of the set $\Phi \doteq\Omega ^{n}$. By construction, sampling with replacement is assumed, ensuring that variables $x_{i}\in\omega$ are i.i.d.

Notation 2.1   The following equations summarize our notations :

$$\mu =E\left(\xi\right),\quad\mu_{2}=\sigma ^{2}=var\left(\xi\righ... ...\xi-\mu \right)^{2}\right),\quad\mu_{4}=E\left(\left(\xi-\mu \right)^{4}\right)$$ (1)

$$m=E_{\omega }\left(x\right),\quad m_{2}=s^{2}=\frac{n}{n-1}\,var_... ...\right),\quad m_{4}=\frac{n}{n-1}\,E_{\omega }\left(\left(x-m\right)^{4}\right)$$ (2)

The expectations are noted by letter $E$. Without subscript, $E$ denotes the $\Omega$-expectation of a function of the random variable $\xi\in\Omega$. With subscript $\omega$, $E_{\omega }$ denotes, for a given fixed sample $\omega$, the ordinary mean value of a function of $x\in\omega$, so that $E_{\omega }\left(f\left(x\right)\right)=\sum_{x\in\omega }f\left(x\right)/n$. With subscript $\Phi$, $E_{\Phi }$ denotes the $\Phi$-expectation of a function of the sample $\omega$, where the usual product measure is used over the set $\Phi$.

The moments are noted by letters $\mu$ and $m$. Without subscript, $\mu$ denotes the expectation of variable $\xi\in\Omega$. With a subscript $i>1$, $\mu _{i}$ denotes the corresponding centered moment. The symbol $\mu _{1}$ will never be used. Letter $m$ will be used in a similar manner to describe the mean and the corrected centered moments of variable $x\in\omega$ for a given sample $\omega \in\Phi$. Symbols $\sigma ,\,s$ will sometimes be used, when useful to avoid square roots.

When a formula doesn't contain $\mu$, its proof is quite ever easier when assuming $\mu =0$. This will be done without further mention.

Well Known Result 2.2   There are two usual measures for the skewness of a distribution. The Pearson's skewness is defined as $3\left(mean-median\right)/\sigma$ and ranges in $\left[-3..+3\right]$ while the Fisher's skewness, used throughout this paper and defined by :

$$\gamma_{1}\doteq E\left(\left(\xi-\mu \right)^{3}\right)/\sigma ^{3},$$

is not bounded. Common values are $\gamma_{1}\left(normal\right)=0$, $\gamma_{1}\left(exponential\right)=2$ and $\gamma_{1}\left(\chi_{\nu}^{2}\right)=\sqrt{8/\nu}$ where $\nu$ is the d.o.f. number.

Well Known Result 2.3   Let $A_{0},\, A_{1},\,\cdots,\, A_{\nu}$ be a partition of $\Omega$ such that $\forall j\,:\, p_{j}\doteq Pr\left(\xi\in A_{j}\right)>0$. The $\chi_{Pearson}^{2}$ statistic of sample $\omega \in\Phi =\Omega ^{n}$ is defined by :

$$\chi_{Pearson}^{2}\left(\omega \right)=\sum_{j=0}^{\nu}\frac{\left(n\, p_{j}-n_{j}\right)^{2}}{n\, p_{j}}$$

where $n_{j}$ is the number of $x_{i}$ that have fallen into $A_{j}$. Then, without any other assumptions, we have :

$$E_{\Phi }\left(\chi_{Pearson}^{2}\left(\omega \right)\right)=\nu,... ...+\frac{1}{n}\left(3-\left(\nu+2\right)^{2}+\sum_{0}^{\nu}\frac{1}{p_{j}}\right)$$

giving a meaning to the standardized value $\chi_{std}^{2}=\left(\chi_{Pearson}^{2}-\nu\right)/\sqrt{2\nu}$ even when the $\chi_{Pearson}^{2}$ statistic is not $\chi_{\nu}^{2}$ distributed.