A Robust Variance

A robust variance for non-sample case

The refoundation of the Data Science must do accounting with the robust variance metric here suggested. Robustness, a sample estimator property with respect to tone down hypothesis, here about the deviation squaring metric, is intended respect the variance abnormal effects already reported. The variance adopts a metric that has a superiority complex. At the same time, the absolute deviation suffers from the opposite problem: an inferiority complex: it certainly appears timid and haughty, while, in a certain sense, it fails to take a stand. With this idea in mind we suggests, for an X variable observed on N-unit collective, a class of robust-variances: , with, , where is an increasing function in the negative real segment and a decreasing one in the positive one thus producing opposite effects on the deviations with respect to the variance. This would amount to a re-match victory for the deviations which occupy central positions in order to snub the lateral deviations, as defined afterwards. In we prefer what can be called a robust variance: where α is a modulator parameter of the squaring defects using the ratio between the reciprocal of the absolute deviation, and the absolute deviation, .

First, let's see how two different types of deviations behave: standardized, zX , and normalized, tX, with PnP transformation (from non-preferred to preferred), in percentage, relative to the five parameters provided by Brussels and already considered previously.

zX				min≤max	%||<1
zPil	1	0.695	V	-1.4≤3.8	74	0.390	V	0.389
zDeficit	1	0.782	III	-1.9≤2.0	67	0.564	III	0.228
zDebt	1	0.816	II	-1.5≤2.3	67	0.632	II	0.210
zInflaction	1	0.752	IV	-1.2≤3.0	70	0.504	IV	0.778
zEmployment	1	0.830	I	-2.0≤1.9	63	0.660	I	0.091

We can call those deviations whose absolute value is less than its unity, here typically 66%, central deviations; those deviations which have an absolute value greater than one, here 34%, however, we will call lateral deviations. The central deviations are strongly scaled down by the squaring produced by the variance and the lateral deviations are overstated. This is the impact of variance as a metric. Therefore, the robust variance may be preferable, suggested as an alternative, or, if preferred, as a culmination. We note that α almost halved the squared defects (0.43 precisely). The European parameter with greater variability, like Inflection and Debt, needed priority interventions, rather than Employment and Debt, as absolute deviation indicates. In fact,robust deviation, do not agrees with the absolute deviation, δ.The overestimation effect can be summarized as follows: the standard deviation, σ, is on average 1.29 times the absolute deviation, δ, and 2.28 times the robust deviation,, while δ is, on average, 1.77 times bigger . When considering deviations, or rather PnP on a percentage scale, things are different on worthwhile variables such as classifications (a scale which ranges from non-preferable to preferable):

				min≤max	%|x|<|1|
tPil	19.50	13.57	V	-27≤74	0	7.64	V	0.558
tDeficitD*	30.46	25.06	I	-62≤38	0	19.66	I	0.290
tDebt	26.66	21.76	III	-60≤40	3.7	16.86	III	0.219
tInflaction	24.65	18.53	IV	-72≤28	0	12.41	IV	0.782
tEmployment	26.65	22.13	II	-52≤49	3.7	17.60	II	0.090

* tDeficitD = tDeficit distance from zero

In this second case it can be verified that the quadratic effect contracts 1.5% of central deviations, on average, while it typically dilates 98.5% of lateral deviations. The robust deviation, , do not agrees with the absolute deviation,, as in the previous case. The overestimation effect agrees with the previous one. The robust Pnp case requires priority interventions on Deficit and Inflation (not in Inflation and Pil like in standardized case).