4 Information value of parameters

4.1 Uncertainties

In fact, an exact knowledge of $\mu$ and of either $\sigma$ or $\delta$ can hardly be assumed. If we reexamine the data and assume their independence then, without any other assumption, the distribution of $\mathrm{est}\left(\mu \right)$ is roughly normal, with mean $\mu$ and variance $\sigma^{2}/n\approx5135$, while the distribution of $\mathrm{est}\left(\sigma^{2}\right)$ is also roughly normal, with mean $\sigma^{2}$ and variance $\left(M_{4}-\sigma^{4}\left(n-3\right)/\left(n-1\right)\right)/n\approx4.289 10^{8}$, where $M_{4}$ is the fourth centered moment. In other words, our knowledge is :

$$\mu \in\left[958\pm72 k_{\mu }\right]\quad;\quad\sigma^{2}\in\left[82166\pm20710 k_{\sigma}\right]$$

where the $k$ are the coverage factors.

From the Student's distribution, we know that $k\leq2$ is an optimistic choice, and all what we actually know is that $\mu$ should stay somewhere in an interval at least as wide as $\left[814, 1101\right]$, while $\sigma$ should stay somewhere in an interval at least as wide as $\left[201, 352\right]$.

Tab. 4: Information values concerning the mean $\mu$ (assuming $\sigma$ fixed and $c/r=0.3$).

$$\begin{array}{c} k_{\mu }\end{array}$$	$$\begin{array}{c} \mathrm{Factual}\\ \mu \end{array}$$	$$\begin{array}{c} \mathrm{Newsboy's}\\ \mathrm{belief}\end{array}$$	$$\begin{array}{c} \mathrm{best}\:y\:\mathrm{for}\:\mathrm{this}\\ \mathrm{belief}\end{array}$$	$$\begin{array}{c} \mathrm{worst}\\ \mathrm{case}\end{array}$$	$G\left(y, \Phi \right)$	$$\begin{array}{c} \mathrm{Value}\end{array}$$
$-2$	$814$	$958$	$1083$	$\theta =0.16$	$4277$
	$814$	$814$	$939$	$\theta =0.3$	$4390$	$113$
$+1$	$1030$	$958$	$1083$	$\theta =0.41$	$5857$
	$1030$	$1030$	1155	$\theta =0.3$	$5895$	$38$
$+2$	$1101$	$958$	$1083$	$\theta =0.53$	$6237$
	$1101$	$1101$	$1227$	$\theta =0.3$	$6397$	$159$

In Tab. 4, the information value of the coverage factor $k_{\mu }$ is discussed when assuming that $\sigma$ is exactly known. The values obtained in this Table and in Tab. 1 are casually of the same magnitude, but they are not of the same nature since those of Tab. 4 do depend from the size of the history while the others don't.

It must be noticed that having historical data from the $n=16$ last independent periods is out of question in many domains, the textile industry among them. Moreover, disposing of a $n=64$ history will only reduce the uncertainties by a factor $2$ and the influence of these uncertainties will remain dominant. In such a case, the search of a robust solution must be enlarged to the family of all the distributions whose parameters fall in the intervals of uncertainty.

4.2 Influence of the Dispersion

As said before, the information values concerning the shape of the distribution computed in Tab.  were quite negligible due to the quite small value of $\sigma/\mu \approx0.3$. Let us now consider what will happen if the dispersion is multiplied by a proportionality factor $\omega$. All the deviations $y^{*}-\mu$ (except from those relative to lognormal shapes) will be multiplied by $\omega$, and therefore the differences between all these "best decisions" (relative to various assumptions) will also be multiplied by $\omega$.

An information value being the variation of a function in the vicinity of an extremum, the information values will rather be multiplied by $\omega^{2}$. Therefore, the phenomenon described above will therefore be amplified if the dispersion increase, but the relative orders of magnitude will not change.