In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, sieve estimators are a class of
non-parametric
Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distri ...
estimators
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...
which use progressively more complex models to estimate an unknown high-dimensional function as more data becomes available, with the aim of asymptotically reducing error towards zero as the amount of data increases. This method is generally attributed to
Ulf Grenander
Ulf Grenander (23 July 1923 – 12 May 2016) was a Swedish statistician and professor of applied mathematics at Brown University.
His early research was in probability theory, stochastic processes, time series analysis, and statistical theory (pa ...
.
Method of sieves in
Positron emission tomography
Positron emission tomography (PET) is a functional imaging technique that uses radioactive substances known as radiotracers to visualize and measure changes in Metabolism, metabolic processes, and in other physiological activities including bl ...
Sieve estimators have been used extensively for estimating density functions in high-dimensional spaces such as in
Positron emission tomography
Positron emission tomography (PET) is a functional imaging technique that uses radioactive substances known as radiotracers to visualize and measure changes in Metabolism, metabolic processes, and in other physiological activities including bl ...
(PET). The first exploitation of Sieves in PET for solving the maximum-likelihood
Positron emission tomography#Image reconstruction problem was by Donald Snyder and Michael Miller, where they stabilized the time-of-flight PET problem originally solved by Shepp and Vardi.
Shepp and Vardi's introduction of Maximum-likelihood estimators in emission tomography exploited the use of the Expectation-Maximization algorithm, which as it ascended towards the maximum-likelihood estimator developed a series of artifacts associated to the fact that the underlying emission density was of too high a dimension for any fixed sample size of Poisson measured counts. Grenander's method of sieves was used to stabilize the estimator, so that for any fixed sample size a resolution could be set which was consistent for the number of counts. As the observe PET imaging time would go to infinity, the dimension of the sieve would increase as well in such a manner that the density was appropriate for each sample size.
See also
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Nonparametric regression
Nonparametric regression is a category of regression analysis in which the predictor does not take a predetermined form but is constructed according to information derived from the data. That is, no parametric form is assumed for the relationship ...
References
External links
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Estimator
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