Arcsine Distribution Research Articles

Comparison of MCM and GUM Method for Evaluating Measurement Uncertainty of Wind Speed by Pitot Tube

“The Guide to the Expression of Uncertainty in Measurement” (GUM) method has many disadvantages when evaluating the uncertainty of wind speed measured by pitot tube, and the “Monte Carlo method” (MCM) is put forward to evaluate the uncertainty of wind speed. The measured quantity value of wind speed by the S-shaped pitot tube is the research object, and the uncertainty is evaluated by GUM and MCM, respectively, and the simulation test is carried out. GUM has errors due to its adoption of approximate linear model. MCM adopts real simulation strategy and has higher credibility than the GUM method, so the GUM method can be validated by the MCM method. When the partial input quantity is changed into triangular distribution, uniform distribution and arcsine distribution, the GUM method is found to be inapplicable. In the evaluation of MCM method, the complex calculation of sensitivity coefficient is avoided, and some uncertainty with small influencing quantity is not needed to be discarded. The evaluation result is more complete. Through comparing the uncertainty evaluation results of MCM before and after the water vapor correction term is discarded, the influence of the water vapor correction term is quantified. Therefore, compared with the GUM method, the MCM method has more advantages when wind speed is measured by pitot tube.

Multiray generalization of the arcsine laws for occupation times of infinite ergodic transformations

We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti’s generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.

A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM.

It is known that if (p n ) n ∈ℕ is a sequence of orthogonal polynomials in L 2([-1,1],w(x)dx), then the roots are distributed according to an arcsine distribution π -1(1 - x 2)-1 dx for a wide variety of weights w(x). We connect this to a result of the Hilbert transform due to Tricomi: if f(x)(1 - x 2)1/4 ∈ L 2(-1,1) and its Hilbert transform Hf vanishes on (-1,1), then the function f is a multiple of the arcsine distribution We also prove a localized Parseval-type identity that seems to be new: if f(x)(1-x 2)1/4 ∈ L2(-1, 1) and has mean value 0 on (-1, 1), then .

Modeling of the Flows of Admixtures in a Random Layered Strip with Probable Arrangement of Inclusions Near the Boundaries of the Body

We study the random flow of admixtures in a two-phase stochastically inhomogeneous strip with the most probable arrangement of inclusions in the vicinity of the surfaces of the body. A mathematical model is formulated for the function of diffusion flow with nonzero constant initial concentration. A random diffusion flow is represented in the form of a Neumann series. The procedure of averaging of the random mass flow over the ensemble of phase configurations with arcsine distribution function is performed. The influence of the characteristics of the medium on the distribution of mass flow is analyzed. It is shown that if the diffusion coefficient of admixtures in the inclusion is higher than for the matrix, then the increase in the characteristic thickness of the layers causes a decrease in the value of the diffusion flow, whereas the mass flow in the entire body increases with the volume fraction of the inclusions.

Ordered random variables

Given two random variables, what effect does a linear ordering have? We answer this question for more than thirty commonly known families of distributions, including the arcsine, Cauchy, exponential, Frechet, Gumbel, half normal, logistic, lognormal, Lomax, normal, Pareto, uniform and Weibull distributions. A real data application is given.

A new bivariate beta distribution of Kind-1 of Type-A

This paper proposes a new form of bivariate beta distribution of Kind-1 of Type-A as well as the density function is visualized in terms of Generalised hypergeometric function and the marginals are univariate beta distribution of Kind-1. Moreover, conditional distributions and constants of the conditionals are also computed. Generating functions such as moment, Cumulant, characteristic functions and the Shannon's differential entropy are also determined. The Bivariate correlations between the random variables for different combination of shape parameters are tabulated and three dimensional probability surfaces are used to visualize the shape of the derived densities graphically. Finally, the special cases of the distribution such as Bivariate Beta distribution of Kind-2 of Type-A, Bivariate F-distribution of Kind-1, Bivariate uniform distribution, Different forms of Bivariate Kumaraswamy distribution, Bivariate exponential and exponentiated distribution, Bivariate Wigner distribution and Bivariate Arcsine distribution are also discussed.

TRANSMUTED ARCSINE DISTRIBUTION PROPERTIES AND APPLICATION

The distribution of ArcSine will be developed to another new distribution using the Quadratic Rank Transmutation (QRT) method proposed by Shaw and Buckley (2007). The new distribution will be called the Transmuted ArcSine distribution, some of its mathematical characteristics such as variance, expectation, residual function, risk function, moments, moment generating function and characteristic function will be presented. The model parameters will be estimated by the maximum likelihood method. Finally, two real data sets are analyzed to illustrates the usefulness of the TAS distribution.

On probabilistic aspects of Chebyshev polynomials

The main goal of this note is to provide new, mostly multidimensional densities, compactly supported and list many of its properties that enable effective calculations. The idea of obtaining such densities is firstly to build some one-dimensional densities depending on many parameters and then treat the constructed in this way distributions as conditional ones. Then of course by imposing certain distribution on the parameters and multiplying the two distributions we arrive at new multivariate distribution. To enable effective calculations, we utilize nice, simple and widely known properties of Chebyshev polynomials. Thus, in particular, the one-dimensional distribution mentioned above will have a form of arcsine distribution multiplied by some rational function. The fact that we use Chebyshev polynomials allows us to calculate all moments of this one-dimensional distribution as well as to find a family of polynomials that are orthogonal with respect to this distribution.

Universality in Random Moment Problems

Let $\mathcal{M} _n(E)$ denote the set of vectors of the first $n$ moments of probability measures on $E\subset \mathbb{R} $ with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in the recent literature. For instance, it has been shown that a uniformly distributed moment sequence in $\mathcal M_n([0,1])$ converges in the large $n$ limit to the moment sequence of the arcsine distribution. In this article we provide a unifying viewpoint by identifying classes of more general distributions on $\mathcal{M} _n(E)$ for $E=[a,b],\,E=\mathbb{R} _+$ and $E=\mathbb{R} $, respectively, and discuss universality problems within these classes. In particular, we demonstrate that the moment sequence of the arcsine distribution is not universal for $E$ being a compact interval. Rather, there is a universal family of moment sequences of which the arcsine moment sequence is one particular member. On the other hand, on the moment spaces $\mathcal{M} _n(\mathbb{R} _+)$ and $\mathcal{M} _n(\mathbb{R} )$ the random moment sequences governed by our distributions exhibit for $n\to \infty $ a universal behaviour: The first $k$ moments of such a random vector converge almost surely to the first $k$ moments of the Marchenko-Pastur distribution (half line) and Wigner’s semi-circle distribution (real line). Moreover, the fluctuations around the limit sequences are Gaussian. We also obtain moderate and large deviations principles and discuss relations of our findings with free probability.

Convergence of ray sequences of Frobenius-Padé approximants

Let be a Cauchy transform of a possibly complex-valued Borel measure and a system of orthonormal polynomials with respect to a measure , where . An th Frobenius-Padé approximant to is a rational function , , , such that the first Fourier coefficients of the remainder function vanish when the form is developed into a series with respect to the polynomials . We investigate the convergence of the Frobenius-Padé approximants to along ray sequences , , when and are supported on intervals of the real line and their Radon-Nikodym derivatives with respect to the arcsine distribution of the corresponding interval are holomorphic functions. Bibliography: 30 titles.

Extended arcsine distribution to proportional data: Properties and applications

We propose a new two-parameter continuous model called the extended arcsine distribution restricted to the unit interval. It is a very competitive model to the beta and Kumaraswamy distributions for modeling percentages, rates, fractions and proportions. We provide a mathematical treatment of the new distribution including explicit expressions for the ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating and quantile functions, Shannon entropy and order statistics. Maximum likelihood is used to estimate the model parameters and the expected information matrix is determined. We demonstrate by means of two applications to proportional data that it can give consistently a better fit than other important statistical models.

Expected number of real zeros for random orthogonal polynomials

AbstractWe study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only (2/π + o(1))logn expected real zeros in terms of the degree n. If the basis is given by the orthonormal polynomials associated with a compactly supported Borel measure on the real line, or associated with a Freud weight defined on the whole real line, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the same asymptotic relation holds for all random orthogonal polynomials on the real line associated with a large class of weights, and give local results on the expected number of real zeros. We also show that the counting measures of properly scaled zeros of these random polynomials converge weakly to either the Ullman distribution or the arcsine distribution.

Recent results on time-dependent Hamiltonian oscillators

Time-dependent Hamilton systems are important in modeling the nondissipative interaction of the system with its environment. We review some recent results and present some new ones. In time-dependent, parametrically driven, one-dimensional linear oscillator, the complete analysis can be performed (in the sense explained below), also using the linear WKB method. In parametrically driven nonlinear oscillators extensive numerical studies have been performed, and the nonlinear WKB-like method can be applied for homogeneous power law potentials (which e.g. includes the quartic oscillator). The energy in time-dependent Hamilton systems is not conserved, and we are interested in its evolution in time, in particular the evolution of the microcanonical ensemble of initial conditions. In the ideal adiabatic limit (infinitely slow parametric driving) the energy changes according to the conservation of the adiabatic invariant, but has a Dirac delta distribution. However, in the general case the initial Dirac delta distribution of the energy spreads and we follow its evolution, especially in the two limiting cases, the slow variation close to the adiabatic regime, and the fastest possible change – a parametric kick, i.e. discontinuous jump (of a parameter), where some exact analytic results are obtained (the so-called PR property, and ABR property). For the linear oscillator the distribution of the energy is always, rigorously, the arcsine distribution, whose variance can in general be calculated by the linear WKB method, while in nonlinear systems there is no such universality. We calculate the Gibbs entropy for the ensembles of noninteracting nonlinear oscillator, which gives the right equipartition and thermostatic laws even for one degree of freedom.

Some characterizations of Arcsine distribution

Abstract Some distributional properties of the symmetric arcsine distribution on (−1, 1) is presented. Based on the distributional properties, several characterizations of the arcsine distribution are given.

Spectral density of generalized Wishart matrices and free multiplicative convolution.

We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas for the level densities are derived for s=3 and s=1/3. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions, is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.

An interesting property of the arcsine distribution and its applications

Representative points obtained by number-theoretic method (RPs-NTM) are used to construct a discrete approximation to the arcsine distribution. An interesting property of the discrete approximation is presented. In addition, another type of representative points of minimizing mean squared error (RPs-MSE) is also considered as an approximation. The performance of the two discrete approximations is evaluated.

Monotone and boolean unitary Brownian motions

The additive monotone (respectively boolean) unitary Brownian motion is a non-commutative stochastic process with monotone (respectively boolean) independent and stationary increments which are distributed according to the arcsine law (respectively Bernoulli law). We introduce the monotone and boolean unitary Brownian motions and derive a closed formula for their associated moments. This provides a description of their spectral measures. We prove that, in the monotone case, the multiplicative analog of the arcsine distribution is absolutely continuous with respect to the Haar measure on the unit circle, whereas in the boolean case the multiplicative analog of the Bernoulli distribution is discrete. Finally, we use quantum stochastic calculus to provide a realization of these processes as the stochastic exponential of the correspending additive Brownian motions.

Nuttall’s theorem with analytic weights on algebraic S-contours

Given a function f holomorphic at infinity, the nth diagonal Padé approximant to f, denoted by [n/n]f, is a rational function of type (n,n) that has the highest order of contact with f at infinity. Nuttall’s theorem provides an asymptotic formula for the error of approximation f−[n/n]f in the case where f is the Cauchy integral of a smooth density with respect to the arcsine distribution on [−1,1]. In this note, Nuttall’s theorem is extended to Cauchy integrals of analytic densities on the so-called algebraic S-contours (in the sense of Nuttall and Stahl).

Distributional Behavior of Time Averages of Non- $$L^1$$ L 1 Observables in One-dimensional Intermittent Maps with Infinite Invariant Measures

In infinite ergodic theory, two distributional limit theorems are well-known. One is characterized by the Mittag-Leffler distribution for time averages of \(L^1(m)\) functions, i.e., integrable functions with respect to an infinite invariant measure. The other is characterized by the generalized arc-sine distribution for time averages of non-\(L^1(m)\) functions. Here, we provide another distributional behavior of time averages of non-\(L^1(m)\) functions in one-dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure. Observation functions considered here are non-\(L^1(m)\) functions which vanish at the indifferent fixed point. We call this class of observation functions weak non-\(L^1(m)\) function. Our main result represents a first step toward a third distributional limit theorem, i.e., a distributional limit theorem for this class of observables, in infinite ergodic theory. To prove our proposition, we propose a stochastic process induced by a renewal process to mimic a Birkoff sum of a weak non-\(L^1(m)\) function in the one-dimensional intermittent maps.

Estimating time delays for constructing dynamical networks

Abstract. Dynamical networks – networks inferred from multivariate time series – have been widely applied to climate data and beyond, resulting in new insights into the underlying dynamics. However, these inferred networks can suffer from biases that need to be accounted for to properly interpret the results. Here, we report on a previously unrecognized bias in the estimate of time delays between nodes in dynamical networks inferred from cross-correlations, a method often used. This bias results in the maximum correlation occurring disproportionately often at large time lags. This is of particular concern in dynamical networks where the large number of possible links necessitates finding the correct time lag in an automated way. We show that this bias can arise due to the similarity of the estimator to a random walk, and are able to map them to each other explicitly for some cases. For the random walk there is an analytical solution for the bias that is closely related to the famous Lévy arcsine distribution, which provides an upper bound in many other cases. Finally, we show that estimating the cross-correlation in frequency space effectively eliminates this bias. Reanalysing large lag links (from a climate network) with this method results in a distribution peaked near zero instead, as well as additional peaks at the originally assigned lag. Links that are reassigned smaller time lags tend to have a smaller distance between them, which indicates that the new time delays are physically reasonable.