A new procedure is discussed which represents the asymmetric relationships between N objects. These relationships must be defined at the interval levels of measurement in Steven’s terminology. This method not only reveals the clustering of the objects but also enables us to give information about both the magnitude and the orientation of the “skewness” between objects in “a” configuration. The procedure optimizes the fit of the model directly to the data by an alternating least squares procedure. It is found to be robust, as the Hessian Matrix of the loss function is positive definite at least in two dimensional case, except for a special case. The method is illustrated with an artificial data and three empirical data.

Summary

This paper is devoted to a comparison of the asymptotic tests of separate families of hypotheses proposed byCox [1961, 1962] and byAtkinson [1970]. The adequacy of the asymptotic results for finite samples is investigated and some conclusions reached. An examination of the terms which differentiate the two procedures is made. Empirical simulated results are discussed for cases involving the exponential, the lognormal and the Weibull distributions.

Summary

Horvitz andThompson [1952] considered varying probability sampling method in general and furnished an unbiased estimator of the population total.Rao, Hartley andCochran [1962] proposed a simple procedure of unequal probability sampling with replacement. It leads to an estimator of the population total having smaller variance than is obtained by sampling with replacement. An attempt has been made in the present paper to compare efficiencies ofHorvitz-Thompson's estimator with that due toRao, hartley andCochran. It is demonstrated that the generalized πps sampling strategy consisting of the design with π_i, the probability of inclusion of thei-th population unit in the sample proportional to the modified size together withHorvitz-Thompson's estimator is superior toRao, Hartley andCochran's sampling strategy under a general super-population model.

Zusammenfassung

Als Einstieg in den zweiten Teil der Untersuchung „Internationale Verbrauchsangleichungen“, der Einzeldarstellungen ausgewählter Verbrauchs- und Gebrauchsgüterbereiche gewidmet ist, wollen wir zwei Berichte (1) über die Ergebnisse einer großangelegten Querschnittsanalyse zitieren, dieim Jahre 1963 von verschiedenen Marktforschungsinstituten im Auftrag des Verlages „DAS BESTE aus Readers Digest“, Düsseldorf, durchgeführt wurde.