A characterization of the uniform distribution on the circle in the analysis of directional data

1978 ◽  
Vol 15 (04) ◽  
pp. 852-857
Author(s):  
M. S. Bingham
Keyword(s):  
Uniform Distribution ◽  
Directional Data ◽  
Regularity Conditions ◽  
Sample Sizes ◽  
Fixed Size ◽  
Sample Mean ◽  
Random Samples ◽  
Resultant Length

It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.

A characterization of the uniform distribution on the circle in the analysis of directional data

1978 ◽  
Vol 15 (4) ◽  
pp. 852-857
Author(s):  
M. S. Bingham
Keyword(s):  
Uniform Distribution ◽  
Directional Data ◽  
Regularity Conditions ◽  
Sample Sizes ◽  
Fixed Size ◽  
Sample Mean ◽  
Random Samples ◽  
Resultant Length

It was conjectured some time ago by K. V. Mardia that if, for random samples of some fixed size N ≧ 2 from a given non-degenerate circular population, the sample mean direction and the sample resultant length are independently distributed, then the population must be uniformly distributed round the circle. In this paper it is shown that, apart from one minor exception, Mardia's conjecture is true in the case N = 2. No regularity conditions are necessary for the proof. The same problem has been studied, subject to regularity conditions, by Kent, Mardia and Rao (1976) for all sample sizes N ≧ 2 except N = 4.

Implications of Small Samples for Generalization: Adjustments and Rules of Thumb

2016 ◽  
Vol 41 (5) ◽  
pp. 472-505 ◽  
Author(s):  
Elizabeth Tipton ◽  
Kelly Hallberg ◽  
Larry V. Hedges ◽  
Wendy Chan
Keyword(s):  
Observational Studies ◽  
Small Sample ◽  
Average Treatment Effect ◽  
Small Samples ◽  
Sample Sizes ◽  
Random Samples ◽  
Rules Of Thumb ◽  
Large Populations ◽  
Small Sample Sizes ◽  
Combine Information

Background: Policy makers and researchers are frequently interested in understanding how effective a particular intervention may be for a specific population. One approach is to assess the degree of similarity between the sample in an experiment and the population. Another approach is to combine information from the experiment and the population to estimate the population average treatment effect (PATE). Method: Several methods for assessing the similarity between a sample and population currently exist as well as methods estimating the PATE. In this article, we investigate properties of six of these methods and statistics in the small sample sizes common in education research (i.e., 10–70 sites), evaluating the utility of rules of thumb developed from observational studies in the generalization case. Result: In small random samples, large differences between the sample and population can arise simply by chance and many of the statistics commonly used in generalization are a function of both sample size and the number of covariates being compared. The rules of thumb developed in observational studies (which are commonly applied in generalization) are much too conservative given the small sample sizes found in generalization. Conclusion: This article implies that sharp inferences to large populations from small experiments are difficult even with probability sampling. Features of random samples should be kept in mind when evaluating the extent to which results from experiments conducted on nonrandom samples might generalize.

scholarly journals High-accuracy individual identification using a “thin slice” of the functional connectome

2019 ◽  
Vol 3 (2) ◽  
pp. 363-383 ◽  
Author(s):  
Lisa Byrge ◽  
Daniel P. Kennedy
Keyword(s):  
High Accuracy ◽  
Individual Identification ◽  
Brain Functioning ◽  
Replication Sample ◽  
Thin Slice ◽  
Functional Connections ◽  
Random Samples ◽  
Human Connectome Project ◽  
Independent Replication

Connectome fingerprinting—a method that uses many thousands of functional connections in aggregate to identify individuals—holds promise for individualized neuroimaging. A better characterization of the features underlying successful fingerprinting performance—how many and which functional connections are necessary and/or sufficient for high accuracy—will further inform our understanding of uniqueness in brain functioning. Thus, here we examine the limits of high-accuracy individual identification from functional connectomes. Using ∼3,300 scans from the Human Connectome Project in a split-half design and an independent replication sample, we find that a remarkably small “thin slice” of the connectome—as few as 40 out of 64,620 functional connections—was sufficient to uniquely identify individuals. Yet, we find that no specific connections or even specific networks were necessary for identification, as even small random samples of the connectome were sufficient. These results have important conceptual and practical implications for the manifestation and detection of uniqueness in the brain.

The Kruskal-Katona Theorem and a Characterization of System Signatures

2015 ◽  
Vol 52 (02) ◽  
pp. 508-518 ◽  
Author(s):  
Alessandro D'Andrea ◽  
Luca De Sanctis
Keyword(s):  
Finite Number ◽  
Structure Function ◽  
Uniform Distribution ◽  
Simplicial Complexes ◽  
Number Of Components ◽  
Full Characterization ◽  
Signature Vector

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

scholarly journals About the benefits of poststratification in forest inventories

2008 ◽  
Vol 53 (No. 4) ◽  
pp. 139-148 ◽  
Author(s):  
J. Saborowski ◽  
J. Cancino
Keyword(s):  
Large Scale ◽  
Systematic Sampling ◽  
Inventory Data ◽  
Sample Sizes ◽  
Forest Inventories ◽  
Virtual Population ◽  
Random Samples ◽  
Gis Data ◽  
Forest District ◽  
Unconditional Variance

A large virtual population is created based on the GIS data base of a forest district and inventory data. It serves as a population where large scale inventories with systematic and simple random poststratified estimators can be simulated and the gains in precision studied. Despite their selfweighting property, systematic samples combined with poststratification can still be clearly more efficient than unstratified systematic samples, the gain in precision being close to that resulting from poststratified over simple random samples. The poststratified variance estimator for the conditional variance given the within strata sample sizes served as a satisfying estimator in the case of systematic sampling. The differences between conditional and unconditional variance were negligible for all sample sizes analyzed.

Almost sure limit sets of random samples in ℝd

1988 ◽  
Vol 20 (3) ◽  
pp. 573-599 ◽  
Author(s):  
Richard A. Davis ◽  
Edward Mulrow ◽  
Sidney I. Resnick
Keyword(s):  
Regular Variation ◽  
Almost Sure Convergence ◽  
Random Sets ◽  
Regularity Conditions ◽  
Limit Set ◽  
Multivariate Regular Variation ◽  
Exponential Form ◽  
Limit Sets ◽  
Random Samples ◽  
Distribution Tail

If {Xj, } is a sequence of i.i.d. random vectors in , when do there exist scaling constants bn > 0 such that the sequence of random sets converges almost surely in the space of compact subsets of to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.

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