Search results
Results From The WOW.Com Content Network
Non-asymptotic rates of convergence do not have the common, standard definitions that asymptotic rates of convergence have. Among formal techniques, Lyapunov theory is one of the most powerful and widely applied frameworks for characterizing and analyzing non-asymptotic convergence behavior.
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.
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function f ( n ) as n becomes very large.
The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...
In particular, the bootstrap is useful when there is no analytical form or an asymptotic theory (e.g., an applicable central limit theorem) to help estimate the distribution of the statistics of interest. This is because bootstrap methods can apply to most random quantities, e.g., the ratio of variance and mean.
In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical estimators .
The asymptotic distribution can be further characterized in several different ways. First, the central limit theorem states that pointwise, ^ has asymptotically normal distribution with the standard rate of convergence: [2]
The definition of the asymptotic equipartition property can also be extended for certain classes of continuous-time stochastic processes for which a typical set exists for long enough observation time. The convergence is proven almost sure in all cases.