scholarly journals Asymptotic approximation of central binomial coefficients with rigorous error bounds

2021 ◽  
Vol 5 (1) ◽  
pp. 380-386
Author(s):  
Richard P. Brent ◽  
Keyword(s):  
Theta Function ◽  
Error Bounds ◽  
Asymptotic Approximation ◽  
Asymptotic Series ◽  
Binomial Coefficient ◽  
Binomial Coefficients ◽  
Rigorous Error Bounds ◽  
Historical Remarks ◽  
Rigorous Error ◽  
Central Binomial Coefficient

We show that a well-known asymptotic series for the logarithm of the central binomial coefficient is strictly enveloping in the sense of Pólya and Szegö, so the error incurred in truncating the series is of the same sign as the next term, and is bounded in magnitude by that term. We consider closely related asymptotic series for Binet's function, for \(\ln\Gamma(z+\frac12)\), and for the Riemann-Siegel theta function, and make some historical remarks.

scholarly journals Rigorous Error Bounds for the Optimal Value in Semidefinite Programming

2008 ◽  
Vol 46 (1) ◽  
pp. 180-200 ◽  
Author(s):  
Christian Jansson ◽  
Denis Chaykin ◽  
Christian Keil
Keyword(s):  
Semidefinite Programming ◽  
Error Bounds ◽  
Rigorous Error Bounds ◽  
Optimal Value ◽  
Rigorous Error

scholarly journals INTEGRAL REPRESENTATIONS AND INEQUALITIES OF EXTENDED CENTRAL BINOMIAL COEFFICIENTS

2021 ◽  
Author(s):  
Chunfu Wei
Keyword(s):  
Binomial Coefficient ◽  
Integral Representations ◽  
Binomial Coefficients ◽  
Exponential Functions ◽  
Central Binomial Coefficient

In the paper, the author presents three integral representations of extended central binomial coefficient, proves decreasing and increasing properties of two power-exponential functions involving extended (central) binomial coefficients, derives several double inequalities for bounding extended (central) binomial coefficient, and compares with known results.

scholarly journals Tight and Rigorous Error Bounds for Basic Building Blocks of Double-Word Arithmetic

2017 ◽  
Vol 44 (2) ◽  
pp. 1-27 ◽  
Author(s):  
Mioara Joldes ◽  
Jean-Michel Muller ◽  
Valentina Popescu
Keyword(s):  
Error Bounds ◽  
Building Blocks ◽  
Rigorous Error Bounds ◽  
Rigorous Error

scholarly journals Robust estimation of risks from small samples

2017 ◽  
Vol 375 (2100) ◽  
pp. 20160299 ◽  
Author(s):  
Simon H. Tindemans ◽  
Goran Strbac
Keyword(s):  
Error Bounds ◽  
Probability Distributions ◽  
Simulated Data ◽  
Small Samples ◽  
Estimation Errors ◽  
Electricity Grid ◽  
Rigorous Error Bounds ◽  
Probability Mass ◽  
Rigorous Error ◽  
The Impact

Data-driven risk analysis involves the inference of probability distributions from measured or simulated data. In the case of a highly reliable system, such as the electricity grid, the amount of relevant data is often exceedingly limited, but the impact of estimation errors may be very large. This paper presents a robust non-parametric Bayesian method to infer possible underlying distributions. The method obtains rigorous error bounds even for small samples taken from ill-behaved distributions. The approach taken has a natural interpretation in terms of the intervals between ordered observations, where allocation of probability mass across intervals is well specified, but the location of that mass within each interval is unconstrained. This formulation gives rise to a straightforward computational resampling method: Bayesian interval sampling. In a comparison with common alternative approaches, it is shown to satisfy strict error bounds even for ill-behaved distributions. This article is part of the themed issue ‘Energy management: flexibility, risk and optimization’.

Sign in / Sign up

Export Citation Format

Share Document