F. Steiner

P-Norm based Statistical Procedures are More Efficient then the L 1 -based Ones for all Error-Types of the Complete Supermodel f c ( x )

The paper shows that P-norm based procedures are significantly more robust than the L1-based ones. For these reason it was introduced the notion and the defining formulae for the indices of relative robustness. (These definitions are usable also generally, i.e., not only for P and L1.)

Investigations Concerning Semi-Interquantile Ranges

The paper shows that the general use of the semi-intersextile range (Q) as error characteristics is more advantageous as that of the semi-interquartile range q. (Denoting the general interquantile range by q(p), Q = q(1/6) and q =q(1/4).) In Section 3 it is shown that three differing characteristics: Q, the A asymptotic scatter of the standard most frequent value determinations based on samples and finally the U uncertainty (the minimum value of the P-norm) approximately coincide for fa(x)-probability distributions of the Cauchy-Gaussian error-type interval.

The fulfilment of the Law of Large Numbers for Arithmetic Means in Case of Infinite Asymptotic Scatter

It is quantitatively presented for a large error-type domain (from the Gaussian type to the Cauchy-distribution) how the accuracy of the arithmetic means increases (as an estimate of the location parameter) if the n sample size gets larger and larger. An even theoretically interesting result is that the law of large numbers can be fulfilled if the asymptotic scatter is infinite.

Insufficiency of Asymptotic Results Demonstrated on Statistical Efficiencies of the L 1 Norm Calculated for Some Types of the Supermodel f p(x)

We can mislead ourselves if we take only asymptotic efficiencies into consideration. Detailed Monte Carlo-investigations for finite samples are in this respect urgently necessary.

Error-Types Characterized by Arbitrary Short or Heavy Flanks

The paper defines the fc(x)-supermodel; the error-types of it realize all relative heavinesses of flanks between zero and infinity. Using fc(x) it is possible to give a more sophisticated definition for the type distance between any actual error-distribution and the Gaussian type.

A More Sophisticated Definition of the Sample Median

As the conventional definition of sample medians is not apt for special investigations belonging to the behaviours of the L1 norm in case of small samples, the authors propose a more sophisticated way to determine the sample median. If the latter values are denoted by MED and Qa is the asymptotic semi-intersextile range (calculated as 0.9674 times the asymptotic scatter), the authors have found the curious result in case of Cauchy-type parent distribution that

TWO TESTS OF "NORMALITY"

Q_{⤪ MED}=Q_{a}/sqrt{n}

ERROR CHARACTERISTICS OF MAD-S (OF SAMPLE MEDIANS) IN CASE OF SMALL SAMPLES FOR SOME PARENT DISTRIBUTION TYPES CHOSEN FROM THE SUPERMODELS fp(x) AND fa(x)

holds not only asymptotically but also for very small samples, too.

Relative Efficiency of the Conventional Statistics Compared to the L 1-Norm based Statistical Procedures, Tested on Distribution Types of the Generalized Gaussian Supermodel

The χ2-test cannot be recommended for the normality tests of different distributions occurring in the practice. Even if our samples are quite different from the Gaussian distribution, the χ2-test accepts them as normally distributed ones with high probabilities at the most frequently used significance levels.When applying the χ2-test the seemingly predominant presence of Gaussian parent distribution may contribute to the survival of the traditional (not robust and not resistant) statistical algorithms.For measured data sets it can be suggested the use of the Csernyák test as a first step of type-determination (using the actual Fa distribution function instead of Φ in Eq. (10) by calculating the critical values) if according to our zero hypothesis the distribution originates from the fa supermodel and an outlier-free case can be justifiedly supposed.

Analytical Determination of QMed ( I.E., of the Error of Sample Medians) for Arbitrary \(\underset{\raise0.3em\hbox{

The error of the sample medians is given in the paper for 14 parent distribution types (for 9 cases the results are also visualized as curves). On the basis of these results one can easily decide for every given task if the asymptotic approximation can be accepted or not. An example is also treated showing the importance of such investigation belonging to finite (or even small) samples.