A random sample is drawn from a population of animals of various species. (The theory may also be applied to studies of literary vocabulary, for example.) If a particular species is represented r times in the sample of size N, then r / N is not a good estimate of the population frequency, p, when r is small. Methods are given for estimating p, assuming virtually nothing about the underlying population. The estimates are expressed in terms of smoothed values of the numbers n, (r= 1, 2, 3, ...), where n, is the number of distinct species that are each represented r times in the sample. (n, may be described as 'the frequency of the frequency r'.) Turing is acknowledged for the most interesting formula in this part of the work. An estimate of the proportion of the population represented by the species occurring in the sample is an immediate corollary. Estimates are made of measures of heterogeneity of the population, including Yule's 'characteristic'and Shannon's 'entropy '. Methods are then discussed that do depend on assumptions about the underlying population. I t is here that most work has been done by other writers. I t is pointed out that a hypothesis can give a good fit to the numbers r~, but can give quite the wrong value for Yule's characteristic. An example of this is Fisher's fit to some data of Williams's on Macrolepidoptera.

Summary This paper deals first with the relationship between the theory of probability and the theory of rational behaviour. A method is then suggested for encouraging people to make accurate probability estimates, a connection with the theory of imformation being mentioned. Finally Wald's theory of statistical decision functions is summarised and generalised and its relation to the theory of rational behaviour is discussed.

From replicate trials of experimental gingivitis in four periodontally healthy subjects, 166 bacterial species and subspecies were detected among 3,034 randomly selected isolates from 96 samples. Of these bacteria, Actinomyces naeslundii (serotype III and phenotypically similar strains that were unreactive with available antisera), Actinomyces odontolyticus (serotype I and phenotypically similar strains that were unreactive with available antisera), Fusobacterium nucleatum, Lactobacillus species D-2, Streptococcus anginosus, Veillonella parvula, and Treponema species A appeared to be the most likely etiological agents of gingivitis. Statistical interpretations indicated that the greatest source of microbiological variation of the total flora observed was person-to-person differences in the floras. The next greatest source of variation was the inflammatory status of the sample sites. Person-to-person differences were smallest at experimental day 4. The floras became more diverse with time and as gingivitis developed and progressed. Analyses indicated that sequential colonization by certain species was repeatable and therefore probably predictable. Variation was relatively small between replicate trials, between two sites on the same teeth sampled on the same day, and between the same sites sampled at the same relative time in a replicate trial.

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JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Given a number of observations xl, ..., xN, a nonparametric method is suggested for estimating the entire probability density curve. The method is to subtract a roughness penalty from the log likelihood, where the roughness penalty is a certain functional of the assumed density function f. Those used are linear combinations off y'2dx andf y"2dx, where y = If. The method appears to be consistent under wide conditions, although consistent methods can be rough. Numerical examples are given and show that for certain values of the coefficients in this linear expression the density function turns out to be very smooth even when N is small. Multivariate extensions are proposed, including one to distributions having some continuous and some discrete components, but numerical examples of these have not been tried. Some of the techniques are borrowed from quantum mechanics and tensor calculus.

Data are presented on the distribution of 101 bacterial species and subspecies among 1,442 isolates from 25 fecal specimens from three men on: (i) their normal diet and normal living conditions, (ii) normal living conditions but eating the controlled metabolic diet designed for use in the Skylab simulation and missions, and (iii) the Skylab diet in simulated Skylab (isolation) conditions. These bacteria represent the most numerous kinds in the fecal flora. Analyses of the kinds of bacteria from each astronaut during the 5-month period showed more variation in the composition of the flora among the individual astronauts than among the eight or nine samples from each person. This observation indicates that the variations in fecal flora reported previously, but based on the study of only one specimen from each person, more certainly reflect real differences (and not daily variation) in the types of bacteria maintained by individual people. The proportions of the predominant fecal species in the astronauts were similar to those reported earlier from a Japanese-Hawaiian population and were generally insensitive to changes from the normal North American diet to the Skylab diet; only two of the most common species were affected by changes in diet. However, one of the predominant species (Bacteroides fragilis subsp. thetaiotaomicron) appeared to be affected during confinement of the men in the Skylab test chamber. Evidence is presented suggesting that an anger stress situation may have been responsible for the increase of this species simultaneously in all of the subjects studied. Phenotypic characteristics of some of the less common isolates are given. The statistical analyses used in interpretation of the results are discussed.

The notion of fractional dimensions is one which is now well known. The object of the present paper is the investigation of the dimensional numbers of sets of points which, when expressed as continued fractions, obey some simple restriction as to their partial quotients. The sets considered are naturally of linear measure zero. Those properties of the partial quotients which hold for almost all continued fractions make up the subject called by Khintchine ‘the measure theory of continued fractions’.