Elizabeth L. Scott

OUTLIER PRONENESS OF PHENOMENA AND OF RELATED DISTRIBUTIONS

Publisher Summary This chapter discusses the outliner proneness of the phenomena and of related distributions. It discusses the distinction between cases where the tendency to suspect and to eliminate outlier observations may be justifiable and those in which it is not. The chapter discusses three concepts: (1) the concept of a k-outlier, (2) the concept of families of distributions that are outlier-resistant, and (3) the concept of families that are outlier-prone. If a substantial previous experience in a particular domain of study appears sufficient for the statistician to act on the assumption that the observable variables follow an outlier-resistant distribution, then the efforts to seek out and, possibly, to eliminate the outliers are justified. In statistical practice, when applying a test, it is important to see that the theory underlying the test is not in conflict with the phenomenon studied. The customary requirements on the test of a hypothesis H are two: (i) if H is true, the use of the test should ensure the maintenance of the desired level of significance and (ii) if H is false and some contemplated alternative hypothesis H1 is true, the power with regard to H1 should be high.

ASYMPTOTICALLY OPTIMAL TESTS OF COMPOSITE HYPOTHESES FOR RANDOMIZED EXPERIMENTS WITH NONCONTROLLED PREDICTOR VARIABLES

Abstract The paper is concerned with randomized experiments with one treatment. Two randomization schemes are considered: randomized pairs and unrestricted randomization. If effective at all, the treatment is supposed to affect the conditional distribution of the “experimental” variable Y given another variable X, called “predictor”. The distribution of X is not affected by the treatment. Using the general theory published elsewhere, the paper deduces the locally asymptotically optimal test of the hypothesis that the treatment has no effect. Apart from the usual difficulties connected with asymptotic tests (how large must N be?), the theory is easily applicable in many “live” cases even though the conditional distribution of Y given X may contain nuisance parameters and be of unusual form.

Further Comments on the “Final Report of the Advisory Committee on Weather Control”

Abstract The Final Report contains only two assertative conclusions: A(1) that commercial cloud seeding in mountainous areas produced 10–15 per cent increase in precipitation, and A(3) that no seeding project meant to increase precipitation was found in which the indicated effect of cloud seeding was negative. With reference to A(1) evidence is introduced showing that the apparent 10–15 per cent increase in precipitation is likely to be due to a combination of factors other than cloud seeding. Conclusion A(3) was found inconsistent with some of the Advisory Committees own findings and with some other published results. Section 8 indicates that a Bayes decision rule whether to seed or not, published in Volume II of the Final Report as minimizing the risks, may well maximize these risks. Section 9 indicates a misrepresentation and a significant omission in Volume II of the Final Report.

Large Scale Organization of the Distribution of Galaxies

In reading the astronomical literature concerned with the distribution of galaxies in space, one encounters a combination of two basic ideas. One of these ideas is that the galaxies are clustered, and the other is that the large-scale organization of the distribution of galaxies is statistically uniform. Occasionally, the possibility of higher order clustering of galaxies is also mentioned.

Field Galaxies and Cluster Galaxies: Abundances of Morphological Types and Corresponding Luminosity Functions

With reference to theory published earlier, formulas are given for the estimation of (i) abundances of morphological types among field galaxies, (ii) of selection probabilities, and (iii) of ‘space luminosity functions’. Strictly, the theory applies to ‘homogeneous classes’ of galaxies. This term designates a category of galaxies, say C, so finely defined that the probability, say Φ(m | C), that a galaxy of category C will be included in the catalogue depends on its photographic apparent magnitude m and on nothing else. The practical use of the theory is illustrated on data in the HMS Catalogue. It appears that certain combinations of the Hubble morphological types satisfy the definition of a homogeneous class. Such, for example, is the case for combinations of ellipticals E0–E3 and, separately, of spirals Sc, Sep, SBc. However, the combination of these two categories is not a homogeneous class.

Estimating the Increase in Skin Cancer Caused by Increases in Ultraviolet Radiation

There is substantial evidence that ultraviolet radiation causes skin cancer. Any impact to the environment that will increase ultraviolet radiation, such as depletion of the stratospheric ozone shield, is expected to produce an increase in skin cancer. The usual method of estimating the increase in skin cancer involves estimating the relation between the occurrence of skin cancer at different localities and the intensity of ultraviolet radiation at the same localities. We study the influence of the difficulties in estimating the relationship: (i) the observations of skin cancer, and also of ultraviolet radiation, are uncertain and incomplete, (ii) the carcinogenesis of skin cancer is unclear and possibly complicated by a complex ultraviolet repair mechanism, (iii) there are other associations that may be interacting.