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In asymptotic analysis in general, one sequence () that converges to a limit is said to asymptotically converge to with a faster order of convergence than another sequence () that converges to in a shared metric space with distance metric | |, such as the real numbers or complex numbers with the ordinary absolute difference metrics, if
Sieve estimators have been used extensively for estimating density functions in high-dimensional spaces such as in Positron emission tomography (PET). The first exploitation of Sieves in PET for solving the maximum-likelihood image reconstruction problem was by Donald Snyder and Michael Miller, [1] where they stabilized the time-of-flight PET problem originally solved by Shepp and Vardi. [2]
The rate of convergence of the MSE to 0 is the necessarily the same as the MISE rate noted previously O(n −4/(d+4)), hence the convergence rate of the density estimator to f is O p (n −2/(d+4)) where O p denotes order in probability. This establishes pointwise convergence.
A sieve analysis (or gradation test) is a practice or procedure used in geology, civil engineering, [1] and chemical engineering [2] to assess the particle size distribution (also called gradation) of a granular material by allowing the material to pass through a series of sieves of progressively smaller mesh size and weighing the amount of material that is stopped by each sieve as a fraction ...
The rate of convergence must be chosen carefully, though, usually h ∝ n −1/5. In many cases, highly accurate results for finite samples can be obtained via numerical methods (i.e. computers); even in such cases, though, asymptotic analysis can be useful. This point was made by Small (2010, §1.4), as follows.
The Goldston–Pintz–Yıldırım sieve (also called GPY sieve or GPY method) is a sieve method and variant of the Selberg sieve with generalized, multidimensional sieve weights. The sieve led to a series of important breakthroughs in analytic number theory. It is named after the mathematicians Dan Goldston, János Pintz and Cem Yıldırım. [1]
In number theory, the fundamental lemma of sieve theory is any of several results that systematize the process of applying sieve methods to particular problems. Halberstam & Richert [ 1 ] : 92–93 write:
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value = (). In essence, given the value of A ( h ) {\displaystyle A(h)} for several values of h {\displaystyle h} , we can estimate A ∗ {\displaystyle A^{\ast }} by extrapolating the ...