Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Completeness (statistics)
Summary
In statistics, completeness is a property of a statistic in relation to a parameterised model for a set of observed data.
A complete statistic T is one for which any proposed distribution on the domain of T is predicted by one or more prior distributions on the model parameter space. In other words, the model space is 'rich enough' that every possible distribution of T can be explained by some prior distribution on the model parameter space. In contrast, a sufficient statistic T is one for which any two prior distributions will yield different distributions on T. (This last statement assumes that the model space is identifiable, i.e. that there are no 'duplicate' parameter values. This is a minor point.)
Put another way: assume that we have an identifiable model space parameterised by , and a statistic (which is effectively just a function of one or more i.i.d. random variables drawn from the model). Then consider the map which takes each distribution on model parameter to its induced distribution on statistic . The statistic is said to be complete when is surjective, and sufficient when is injective.
Consider a random variable X whose probability distribution belongs to a parametric model Pθ parametrized by θ.
Say T is a statistic; that is, the composition of a measurable function with a random sample X1,...,Xn.
The statistic T is said to be complete for the distribution of X if, for every measurable function g,:
The statistic T is said to be boundedly complete for the distribution of X if this implication holds for every measurable function g that is also bounded.
The Bernoulli model admits a complete statistic. Let X be a random sample of size n such that each Xi has the same Bernoulli distribution with parameter p. Let T be the number of 1s observed in the sample, i.e. . T is a statistic of X which has a binomial distribution with parameters (n,p). If the parameter space for p is (0,1), then T is a complete statistic. To see this, note that
Observe also that neither p nor 1 − p can be 0.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (4)
MATH-413: Statistics for data science
Statistics lies at the foundation of data science, providing a unifying theoretical and methodological backbone for the diverse tasks enountered in this emerging field. This course rigorously develops
COM-302: Principles of digital communications
This course is on the foundations of digital communication. The focus is on the transmission problem (rather than being on source coding).