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- Evolutionary game theory (1)
- Fair division problems; indivisible objects; lottery; fair lottery; independent lottery; deterministic distribution; randomized allocation; optimal share (1)
- Normal numbers; Significant digits; Benford's law; Digit-regular random variable; Significant-digit-regular random variable; Law of least significant digits; Floating-point numbers; Nonleading digits; Trailing digits (1)
- Population dynamics (1)
- Selectivity variability principle (1)
Articles 1 - 30 of 55
Full-Text Articles in Mathematics
Modeling The Evolution Of Differences In Variability Between Sexes, Theodore P. Hill
Modeling The Evolution Of Differences In Variability Between Sexes, Theodore P. Hill
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An elementary mathematical theory based on a “selectivity-variability” principle is proposed to address a question raised by Charles Darwin, namely, how one sex of a sexually dimorphic species might tend to evolve with greater variability than the other sex. Two mathematical models of the principle are presented: a discrete-time one-step probabilistic model of the short-term behavior of the subpopulations of a given sex, with an example using normally distributed perceived fitness values; and a continuous-time deterministic coupled ODE model for the long-term asymptotic behavior of the expected sizes of the subpopulations, with an example using exponentially distributed fitness levels.
Finite-State Markov Chains Obey Benford’S Law, Babar Kaynar, Arno Berger, Theodore P. Hill, Ad Ridder
Finite-State Markov Chains Obey Benford’S Law, Babar Kaynar, Arno Berger, Theodore P. Hill, Ad Ridder
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A sequence of real numbers (xn) is Benford if the significands, i.e. the fraction parts in the floating-point representation of (xn) are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with probability transition matrix P and limiting matrix P* is Benford if every component of both sequences of matrices (Pn - P*) and (Pn+1-Pn) is Benford or eventually zero. Using recent tools that established Benford behavior both for Newton's method and for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 …
An Optimal Method To Combine Results From Different Experiments, Theodore P. Hill, Jack Miller
An Optimal Method To Combine Results From Different Experiments, Theodore P. Hill, Jack Miller
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This article describes an optimal method (conflation) to consolidate data from different experiments, and illustrates the advantages of conflation by graphical examples involving gaussian input distributions, and by a concrete numerical example involving the values of lattice spacing of silicon crystals used in determination of the current values of Planck's constant and the Avogadro constant.
Ham Sandwich With Mayo: A Stronger Conclusion To The Classical Ham Sandwich Theorem, John H. Elton, Theodore P. Hill
Ham Sandwich With Mayo: A Stronger Conclusion To The Classical Ham Sandwich Theorem, John H. Elton, Theodore P. Hill
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The conclusion of the classical ham sandwich theorem of Banach and Steinhaus may be strengthened: there always exists a common bisecting hyperplane that touches each of the sets, that is, intersects the closure of each set. Hence, if the knife is smeared with mayonnaise, a cut can always be made so that it will not only simultaneously bisect each of the ingredients, but it will also spread mayonnaise on each. A discrete analog of this theorem says that n finite nonempty sets in n-dimensional Euclidean space can always be simultaneously bisected by a single hyperplane that contains at least one …
Conflations Of Probability Distributions, Theodore P. Hill
Conflations Of Probability Distributions, Theodore P. Hill
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The conflation of a finite number of probability distributions P1,..., Pn is a consolidation of those distributions into a single probability distribution Q=Q(P1,..., Pn), where intuitively Q is the conditional distribution of independent random variables X1,..., Xn with distributions P1,..., Pn, respectively, given that X1= ... =Xn. Thus, in large classes of distributions the conflation is the distribution determined by the normalized product of the probability density or probability mass functions. Q is shown to be the …
Scale-Distortion Inequalities For Mantissas Of Finite Data Sets, Arno Berger, Theodore P. Hill, Kent E. Morrison
Scale-Distortion Inequalities For Mantissas Of Finite Data Sets, Arno Berger, Theodore P. Hill, Kent E. Morrison
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In scientific computations using floating point arithmetic, rescaling a data set multiplicatively (e.g., corresponding to a conversion from dollars to euros) changes the distribution of the mantissas, or fraction parts, of the data. A scale-distortion factor for probability distributions is defined, based on the Kantorovich distance between distributions. Sharp lower bounds are found for the scale-distortion of n-point data sets, and the unique data set of size n with the least scale-distortion is identified for each positive integer n. A sequence of real numbers is shown to follow Benford’s Law (base b) if and only if the scale-distortion (base b) …
A Better Definition Of The Kilogram, Ronald F. Fox, Theodore P. Hill
A Better Definition Of The Kilogram, Ronald F. Fox, Theodore P. Hill
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Fixing the value of Avogadro's constant, the number of atoms in 12 grams of carbon-12, at exactly 844468863 would imply that one gram is the mass of exactly 18x140744813 carbon-12 atoms. This new definition of the gram, and thereby also the kilogram, is precise, elegant and unchanging in time, unlike the current 118-year-old artifact kilogram in Paris and the proposed experimental definitions of the kilogram using man-made silicon spheres or the watt balance apparatus.
Newton's Method Obeys Benford's Law, Arno Berger, Theodore P. Hill
Newton's Method Obeys Benford's Law, Arno Berger, Theodore P. Hill
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No abstract provided.
Regularity Of Digits And Significant Digits Of Random Variables, Theodore P. Hill, Klaus Schürger
Regularity Of Digits And Significant Digits Of Random Variables, Theodore P. Hill, Klaus Schürger
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A random variable X is digit-regular (respectively, significant-digit-regular) if the probability that every block of k given consecutive digits (significant digits) appears in the b-adic expansion of X approaches b-k as the block moves to the right, for all integers b>1 and k≥1. Necessary and sufficient conditions are established, in terms of convergence of Fourier coefficients, and in terms of convergence in distribution modulo 1, for a random variable to be digit-regular (significant-digit-regular), and basic relationships between digit-regularity and various classical classes of probability measures and normal numbers are given. These results provide a theoretical basis for analyses …
One-Dimensional Dynamical Systems And Benford's Law, Arno Berger, Leonid A. Bunimovich, Theodore P. Hill
One-Dimensional Dynamical Systems And Benford's Law, Arno Berger, Leonid A. Bunimovich, Theodore P. Hill
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Near a stable fixed point at 0 or ∞, many real-valued dynamical systems follow Benford's law: under iteration of a map T the proportion of values in {x, T(x), T2(x), ... , Tn(x)} with mantissa (base b) less than t tends to logbt for all t in [1,b) as n→ ∞, for all integer bases b>1. In particular, the orbits under most power, exponential, and rational functions (or any successive combination thereof), follow Benford's law for almost all sufficiently large initial values. For linearly-dominated systems, convergence to Benford's distribution …
Constructing Random Probability Distributions, Theodore P. Hill, David E.R. Sitton
Constructing Random Probability Distributions, Theodore P. Hill, David E.R. Sitton
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This article surveys several classes of iterative methods for constructing random probability distributions (or random convex functions, or random homeomorphisms), and includes illustrative applications in statistics, optimal-control theory, and game theory. Computer simulations of these methods are fast and easy to implement
Necessary And Sufficient Condition That The Limit Of Stieltjes Transforms Is A Stieltjes Transform, Jeffrey S. Geronimo, Theodore P. Hill
Necessary And Sufficient Condition That The Limit Of Stieltjes Transforms Is A Stieltjes Transform, Jeffrey S. Geronimo, Theodore P. Hill
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The pointwise limit S of a sequence of Stieltjes transforms (Sn) of real Borel probability measures (Pn) is itself the Stieltjes transform of a Borel p.m. P if and only if iy S(iy) →−1as y →∞, in which case Pn converges to P in distribution. Applications are given to several problems in mathematical physics.
Levy-Like Continuity Theorems For Convergence In Distribution, Theodore P. Hill, Ulrich Krengel
Levy-Like Continuity Theorems For Convergence In Distribution, Theodore P. Hill, Ulrich Krengel
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Levy’s classical continuity theorem states that if the pointwise limit of a sequence of characteristic functions exists, then the limit function itself is a characteristic function if and only if the limit function satisfies a single universal limit condition (in his case, the limit at zero is one), in which case the underlying measures converge weakly to the probability measure represented by the limit function. It is the purpose of this article to give a number of direct analogs of L´evy’s theorem for other probability-representing functions including moment sequences, maximal moment sequences, mean-residual-life functions, Hardy-Littlewood maximal functions, and failure-rate functions. …
Random Probability Measures With Given Mean And Variance Running Title: Random Probability Measures, Lisa Bloomer, Theodore P. Hill
Random Probability Measures With Given Mean And Variance Running Title: Random Probability Measures, Lisa Bloomer, Theodore P. Hill
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This article describes several natural methods of constructing random probability measures with prescribed mean and variance, and focuses mainly on a technique which constructs a sequence of simple (purely discrete, finite number of atoms) distributions with the prescribed mean and with variances which increase to the desired variance. Basic properties of the construction are established, including conditions guaranteeing full support of the generated measures, and conditions guaranteeing that the final measure is discrete. Finally, applications of the construction method to optimization problems such as Plackett’s Problem are mentioned, and to experimental determination of average-optimal solutions of certain control problems.
Extreme-Value Moment Goodness-Of-Fit Tests, Theodore P. Hill, Victor Perez-Abreu
Extreme-Value Moment Goodness-Of-Fit Tests, Theodore P. Hill, Victor Perez-Abreu
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A general goodness-of-fit test for scale-parameter families of distributions is introduced, which is based on quotients of expected sample minima. The test is independent of the mean of the distribution, and, in applications to testing for exponentiality of data, compares favorably to other goodness-of-fit tests for exponentiality based on the empirical distribution function, regression methods and correlation statistics. The new minimal-moment method uses ratios of easily-calculated, unbiased, strongly consistent U-statistics, and the general technique can be used to test many standard composite null hypotheses such as exponentiality, normality or uniformity (as well as simple null hypotheses).
Alternative Empirical Distributions Based On Weigted Linear Combinations Of Order Statistics, Theodore P. Hill, James Mann
Alternative Empirical Distributions Based On Weigted Linear Combinations Of Order Statistics, Theodore P. Hill, James Mann
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A class of empirical distributions is introduced which are based on various weighted linear combinations of order statistics, and which have convergence properties the classical empirical distribution does not, or which stochastically or convexly dominate the classical empirical distribution
On The Basic Representation Theorem For Convex Domination Of Measures, J. Elton, Theodore P. Hill
On The Basic Representation Theorem For Convex Domination Of Measures, J. Elton, Theodore P. Hill
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A direct, constructive proof is given for the basic representation theorem for convex domination of measures. The proof is given in the finitistic case (purely atomic measures with a finite number of atoms), and a simple argument is then given to extend this result to the general case, including both probability measures and finite Borel measures on infinite-dimensional spaces. The infinite-dimensional case follows quickly from the finite-dimensional case with the use of the approximation property.
Constructions Of Random Distributions Via Sequential Barycenters, Theodore P. Hill, Michael Monticino
Constructions Of Random Distributions Via Sequential Barycenters, Theodore P. Hill, Michael Monticino
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This article introduces and develops a constructive method for generating random probability measures with a prescribed mean or distribution of the means. The method involves sequentially generating an array of barycenters which uniquely defines a probability measure. Basic properties of the generated measures are presented, including conditions under which almost all the generated measures are continuous or almost all are purely discrete or almost all have finite support. Applications are given to models for average-optimal control problems and to experimental approximation of universal constants.
A Note On Distributions Of True Versus Fabricated Data, Theodore P. Hill
A Note On Distributions Of True Versus Fabricated Data, Theodore P. Hill
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New empirical evidence and statistical derivations of Benford’s Law have led to successful goodness-of fit tests to detect fraud in accounting data. Several recent case studies support the hypothesis that fabricated data does not conform to expected true digital frequencies.
Strongly-Consistent, Distribution-Free Confidence Intervals For Quantiles, David Gilat, Theodore P. Hill
Strongly-Consistent, Distribution-Free Confidence Intervals For Quantiles, David Gilat, Theodore P. Hill
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Strongly-consistent, distribution-free confidence intervals are derived to estimate the fixed quantiles of an arbitrary unknown distribution, based on order statistics of an iid sequence from that distribution. This new method, unlike classical estimates, works for totally arbitrary (including discontinuous) distributions, and is based on recent one-sided strong laws of large numbers.
Strong Laws For L- And U-Statistics, J. Aaronson, R. Burton, H. Dehling, D. Gilat, Theodore P. Hill, B. Weiss
Strong Laws For L- And U-Statistics, J. Aaronson, R. Burton, H. Dehling, D. Gilat, Theodore P. Hill, B. Weiss
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Strong laws of large numbers are given for L-statistics (linear combinations of order statistics) and for U-statistics (averages of kernels of random samples) for ergodic stationary processes, extending classical theorems of Hoeffding and of Helmers for iid sequences. Examples are given to show that strong and even weak convergence may fail if the given sufficient conditions are not satisfied, and an application is given to estimation of correlation dimension of invariant measures.
A Statistical Derivation Of The Significant-Digit Law, Theodore P. Hill
A Statistical Derivation Of The Significant-Digit Law, Theodore P. Hill
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The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a summary of recent invariant-measure characterizations. Then a new statistical derivation of the law in the form of a CLT-like theorem for significant digits is presented. If distributions are selected at random (in any "unbiased" way) and random samples are then taken from each of these distributions, the significant digits of the combines sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant0digit phenomenon in many different empirical contexts and helps justify its recent application …
The Significant-Digit Phenomenon, Theodore P. Hill
The Significant-Digit Phenomenon, Theodore P. Hill
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No abstract provided.
Base-Invariance Implies Benford's Law, Theodore P. Hill
Base-Invariance Implies Benford's Law, Theodore P. Hill
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A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only base-invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an appropriate mantissa σ-algebra on the positive reals, and results for invariant measures on the circle.
Minimax-Optimal Strategies For The Best-Choice Problem When A Bound Is Known For The Expected Number Of Objects, Theodore P. Hill, D. P. Kennedy
Minimax-Optimal Strategies For The Best-Choice Problem When A Bound Is Known For The Expected Number Of Objects, Theodore P. Hill, D. P. Kennedy
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For the best-choice (or secretary) problem with an unknown number N of objects, minimax-optimal strategies for the observer and minimax distributions for N are derived under the assumption that N is a random variable with expected value at most M, where M is known. The solution is derived as a special case of the situation where N is constrained by Ef(N) ≤ M, where f is increasing with f(i)-f(i-1) convex.
Quantile-Locating Functions And The Distance Between The Mean And Quantiles, D. Gilat, Theodore P. Hill
Quantile-Locating Functions And The Distance Between The Mean And Quantiles, D. Gilat, Theodore P. Hill
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Given a random variable X with finite mean, for each 0 < p < 1, a new sharp bound is found on the distance between a p-quantile of X and its mean in terms of the central absolute first moment of X. The new bounds strengthen the fact that the mean of X is within one standard deviation of any of its medians, as well as a recent quantile-generalization of this fact by O'Cinneide.
Partitioning Inequalities In Probability And Statistics, Theodore P. Hill
Partitioning Inequalities In Probability And Statistics, Theodore P. Hill
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This article surveys fair-division or cake-cutting inequalities in probability statistics, including bisection inequalities, basic fairness inequalities, convexity tools, superfairness inequalities, and partitioning inequalities hypotheses testing and optimal stopping theory. The emphasis is measure theoretic, as opposed to game theoretic or economic, and a number of open problems in the area are mentioned.
On The Game Of Googol, Theodore P. Hill, Ulrich Krengel
On The Game Of Googol, Theodore P. Hill, Ulrich Krengel
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In the classical secretary problem the decision maker can only observe the relative ranks of the items presented. Recently, Ferguson — building on ideas of Stewart — showed that, in a game theoretic sense, there is no advantage if the actual values of the random variables underlying the relative ranks can be observed (game of googol). We extend this to the case where the number of items is unknown with a known upper bound. Corollary 3 extends one of the main results in [HK] to all randomized stopping times. We also include a modified, somewhat more formal argument for Ferguson's …
One-Sided Refinements Of The Strong Law Of Large Numbers And The Glivenko-Cantelli Theorem, David Gilat, Theodore P. Hill
One-Sided Refinements Of The Strong Law Of Large Numbers And The Glivenko-Cantelli Theorem, David Gilat, Theodore P. Hill
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A one-sided refinement of the strong law of large numbers is found for which the partial weighted sums not only converge almost surely to the expected value, but also the convergence is such that eventually the partial sums all exceed the expected value. The new weights are distribution-free, depending only on the relative ranks of the observations. A similar refinement of the Glivenko-Cantelli theorem is obtained, in which a new empirical distribution function not only has the usual uniformly almost-sure convergence property of the classical empirical distribution function, but also has the property that all its quantiles converge almost surely. …
A Survey Of Prophet Inequalities In Optimal Stopping Theory, Theodore P. Hill, Robert P. Kertz
A Survey Of Prophet Inequalities In Optimal Stopping Theory, Theodore P. Hill, Robert P. Kertz
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This paper surveys the origin and development of what has come to be known as "prophet inequalities" in optimal stopping theory. Included is a review of all published work to date on these problems, including extensions and variations, descriptions and examples of the main proof techniques, and a list of a number of basic open problems.