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Journal ArticleDOI

Mathematical Contributions to the Theory of Evolution. XIX. Second Supplement to a Memoir on Skew Variation

01 Jan 1901-Philosophical Transactions of the Royal Society A (The Royal Society)-Vol. 216, pp 429-457
TL;DR: In this article, the authors presented a series of curves such that one or other of them would agree with any observational or theoretical frequency curve of positive ordinates to the following extent: (i) the areas should be equal; (ii) the mean abscissa or centroid vertical should be the same for the two curves; (iii) the standard deviation (or, what amounts to the same thing, the second moment coefficient) about this centroid horizontal vertical, and (iv) to (v) the third and fourth moment coefficients about this horizontal vertical should also be the
Abstract: In a memoir presented to the Royal Society in 1894, I dealt with skew variation in homogeneous material. The object of that memoir was to obtain a series of curves such that one or other of them would agree with any observational or theoretical frequency curve of positive ordinates to the following extent :—(i) The areas should be equal; (ii) the mean abscissa or centroid vertical should be the same for the two curves; (iii) the standard deviation (or, what amounts to the same thing, the second moment coefficient) about this centroid vertical should be the same, and (iv) to (v) the third and fourth moment coefficients should also be the same. If μs be the s th moment coefficient about the mean vertical, N the area, x ¯ be the mean abscissa, σ = √ μ 2 the standard deviation, β 1 = μ 32/ μ 23, β 4 = μ 4/ μ 22, then the equality for the two curves of N, x ¯, σ, β 1 and β 2 leads almost invariably in the case of frequency to excellency of fit. Indeed, badness of fit generally arises from either heterogeniety, or the difficulty in certain cases of accurately determining from the data provided the true values of the moment coefficients, e. g ., especially in J- and U-shaped frequency distributions, or distributions without high contact at the terminals ; here the usual method of correcting the raw moments for sub-ranges of record fails. Having found a curve which corresponded to the skew binomial in the same manner as the normal curve of errors to the symmetrical binomial with finite index, it occurred to me that a development of the process applied to the hypergeometrical series would achieve the result I was in search of, i. e ., a curve whose constants would be determined by the observational values of N, x ¯, σ, β 1 and β 2.

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Journal ArticleDOI
TL;DR: It is shown analytically and without recourse to the scaling hypothesis that the probability density function for the fluctuations of the magnetic order parameter in the low-temperature phase of the XY model of finite size is non-Gaussian and of universal form, independent of both system size and critical exponent eta.
Abstract: We study the probability density function for the fluctuations of the magnetic order parameter in the low-temperature phase of the $\mathrm{XY}$ model of finite size In two dimensions, this system is critical over the whole of the low-temperature phase It is shown analytically and without recourse to the scaling hypothesis that, in this case, the distribution is non-Gaussian and of universal form, independent of both system size and critical exponent $\ensuremath{\eta}$ An exact expression for the generating function of the distribution is obtained, which is transformed and compared with numerical data from high-resolution molecular dynamics and Monte Carlo simulations The asymptotes of the distribution are calculated and found to be of exponential and double exponential form The calculated distribution is fitted to three standard functions: a generalization of Gumbel's first asymptote distribution from the theory of extremal statistics, a generalized log-normal distribution, and a ${\ensuremath{\chi}}^{2}$ distribution The calculation is extended to general dimension and an exponential tail is found in all dimensions less than 4, despite the fact that critical fluctuations are limited to $D=2$ These results are discussed in the light of similar behavior observed in models of interface growth and for dissipative systems driven into a nonequilibrium steady state

135 citations

Journal ArticleDOI
TL;DR: In this article, the authors trace the historical developments emphasizing the optimality features of tests based on scores and their usefulness in practical problems in statistics and econometrics, and give some new results, present easier computation of score-based tests and alternative derivations of some known results.

134 citations

Journal ArticleDOI
TL;DR: In this paper, the authors focus on approaches for representing error heteroscedasticity with respect to simulated streamflow, i.e., the pattern of larger errors in higher streamflow predictions.
Abstract: Reliable and precise probabilistic prediction of daily catchment-scale streamflow requires statistical characterization of residual errors of hydrological models. This study focuses on approaches for representing error heteroscedasticity with respect to simulated streamflow, i.e., the pattern of larger errors in higher streamflow predictions. We evaluate 8 common residual error schemes, including standard and weighted least squares, the Box-Cox transformation (with fixed and calibrated power parameter λ) and the log-sinh transformation. Case studies include 17 perennial and 6 ephemeral catchments in Australia and USA, and two lumped hydrological models. Performance is quantified using predictive reliability, precision and volumetric bias metrics. We find the choice of heteroscedastic error modelling approach significantly impacts on predictive performance, though no single scheme simultaneously optimizes all performance metrics. The set of Pareto optimal schemes, reflecting performance trade-offs, comprises Box-Cox schemes with λ of 0.2 and 0.5, and the log scheme (λ=0, perennial catchments only). These schemes significantly outperform even the average-performing remaining schemes (e.g., across ephemeral catchments, median precision tightens from 105% to 40% of observed streamflow, and median biases decrease from 25% to 4%). Theoretical interpretations of empirical results highlight the importance of capturing the skew/kurtosis of raw residuals and reproducing zero flows. Paradoxically, calibration of λ is often counterproductive: in perennial catchments, it tends to overfit low flows at the expense of abysmal precision in high flows. The log-sinh transformation is dominated by the simpler Pareto optimal schemes listed above. Recommendations for researchers and practitioners seeking robust residual error schemes for practical work are provided. This article is protected by copyright. All rights reserved.

113 citations


Cites background from "Mathematical Contributions to the T..."

  • ...Much of this excess kurtosis is a consequence of the skew, as indicated by the theoretical relationship ekurt skew222 [Pearson, 1916]....

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Journal ArticleDOI
TL;DR: In this article, the authors examined the effect of partial 40Ar/39Ar resetting on the interpretation of Apollo-era impact histories and concluded that the assignment of apparent 40Ar /39Ar plateau ages bears an undesirably high degree of subjectivity.
Abstract: The Late Heavy Bombardment (LHB), a hypothesized impact spike at ∼3.9 Ga, is one of the major scientific concepts to emerge from Apollo-era lunar exploration. A significant portion of the evidence for the existence of the LHB comes from histograms of 40Ar/39Ar “plateau” ages (i.e., regions selected on the basis of apparent isochroneity). However, due to lunar magmatism and overprinting from subsequent impact events, virtually all Apollo-era samples show evidence for 40Ar/39Ar age spectrum disturbances, leaving open the possibility that partial 40Ar* resetting could bias interpretation of bombardment histories due to plateaus yielding misleadingly young ages. We examine this possibility through a physical model of 40Ar* diffusion in Apollo samples and test the uniqueness of the impact histories obtained by inverting plateau age histograms. Our results show that plateau histograms tend to yield age peaks, even in those cases where the input impact curve did not contain such a spike, in part due to the episodic nature of lunar crust or parent body formation. Restated, monotonically declining impact histories yield apparent age peaks that could be misinterpreted as LHB-type events. We further conclude that the assignment of apparent 40Ar/39Ar plateau ages bears an undesirably high degree of subjectivity. When compounded by inappropriate interpretations of histograms constructed from plateau ages, interpretation of apparent, but illusory, impact spikes is likely.

88 citations

Book ChapterDOI
TL;DR: In this paper, a review of probability distributions that have been and can be applied to problems arising in finance and examines some of these applications is presented. But, while the normal or lognormal distributions may provide an adequate representation for many financial series, other series are not so conveniently modeled.
Abstract: Publisher Summary This chapter reviews probability distributions that have been and can be applied to problems arising in finance and examines some of these applications. Viewed from a purely statistical perspective, financial data provides a rich source of variables with diverse distributional characteristics ranging from normally distributed variates to variables characterized by various degrees of skewness and kurtosis. While the normal or lognormal distributions may provide an adequate representation for many financial series, other series are not so conveniently modeled. This chapter reviews some important alternatives to the normal, log-normal, and stable paretian distributions. Financial data are of great interest to individual investors, corporate planners, politicians, and government policy makers. Financial data are constantly changing and are highly visible in daily reports on stock prices, interest rates, currency exchange rates, and gold prices. Many of these data are characterized by a high degree of uncertainty, and changes have the potential to generate huge gains or losses.

79 citations