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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
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.
This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, all ...
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 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 convergence of the sum of reciprocals of twin primes follows from bounds on the density of the sequence of twin primes. Let π 2 ( x ) {\displaystyle \pi _{2}(x)} denote the number of primes p ≤ x for which p + 2 is also prime (i.e. π 2 ( x ) {\displaystyle \pi _{2}(x)} is the number of twin primes with the smaller at most x ).
Two classical techniques for series acceleration are Euler's transformation of series [1] and Kummer's transformation of series. [2] A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Katahiro Takebe in 1722; the ...
Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate. [ 1 ] The basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s.